S. Duplij, Polyadic integer numbers and finite (m,n)-fields (Journal version, P-Adic Num Ultrametr Anal Appl (2017) v. 9, n.4, p. 267-291, DOI:10.1134/S2070046617040033)
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
Fixed point theorem of discontinuity and weak compatibility in non complete n...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric spaces and weak compatibility to show the maps have a common fixed point.
11.fixed point theorem of discontinuity and weak compatibility in non complet...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric functions and the given conditions to show the sequence converges, establishing a common fixed point.
Generalization of Tensor Factorization and ApplicationsKohei Hayashi
This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
Fixed point theorem of discontinuity and weak compatibility in non complete n...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric spaces and weak compatibility to show the maps have a common fixed point.
11.fixed point theorem of discontinuity and weak compatibility in non complet...Alexander Decker
The document presents a theorem about fixed points for six self-maps in a non-complete non-Archimedean Menger PM-space. The theorem proves that the six maps have a unique common fixed point under certain conditions, including:
1) The maps satisfy inequality conditions involving probabilistic metric functions.
2) One of the subspaces induced by two of the maps is complete.
3) The pairs of maps are R-weakly compatible.
The proof constructs a Cauchy sequence and uses properties of probabilistic metric functions and the given conditions to show the sequence converges, establishing a common fixed point.
Generalization of Tensor Factorization and ApplicationsKohei Hayashi
This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
PaperNo14-Habibi-IJMA-n-Tuples and ChaoticityMezban Habibi
This document presents theorems and definitions related to n-tuples of operators on a Frechet space and conditions for chaoticity. It begins with definitions of key concepts such as the orbit of a vector under an n-tuple of operators and what it means for an n-tuple to be hypercyclic or for a vector to be periodic. The main results section presents two theorems, the first characterizing when an n-tuple satisfies the hypercyclicity criterion and the second proving conditions under which an n-tuple of weighted backward shifts is chaotic. The second theorem shows the equivalence of an n-tuple being chaotic, hypercyclic with a non-trivial periodic point, having a non-trivial periodic point, and a
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
The document discusses the quantified fragment of set theory called ∀π0. ∀π0 allows for restricted quantification over sets and ordered pairs. A decision procedure for the satisfiability of ∀π0 formulas works by non-deterministically guessing a skeletal representation and checking if its realization is a model of the formula. The document considers encoding the conditions on skeletal representations as first-order formulas to view ∀π0 as a first-order logic and leverage tools developed for first-order logic fragments.
A brief introduction to Hartree-Fock and TDDFTJiahao Chen
The document provides an overview of time-dependent density functional theory (TDDFT) for computing molecular excited states. It begins with an introduction to the Born-Oppenheimer approximation and variational principle. It then discusses the Hartree-Fock and Kohn-Sham equations as self-consistent field methods for calculating ground states, and linear response theory for calculating excited states within TDDFT. The contents section outlines the topics to be covered, including basis functions, Hartree-Fock theory, density functional theory, and time-dependent DFT.
We will discuss history and recent developments in the study of the phase structure of noncommutative scalar fields. Apart from the usual disorder and uniform order phases, the theory exhibits a third phase, which survives the commutative limit.
This is connected to the UV/IR mixing of the noncommutative theory. We will rewrite the fuzzy theory as a modified quartic
matrix model, with extra multitrace terms in the action and perform saddle point analysis of the theory. Our goal will be to locate the triple point of the theory and to reconstruct the numerically obtained phase diagram. This goal will be successfully reached at the end of the talk.
The document is a presentation about using model theory to prove Hilbert's Weak Nullstellensatz. It begins with introductions to model theory, including definitions of structures, embeddings, elementary extensions, theories, and model-completeness. It then states that the theory of algebraically closed fields has model-completeness. The presentation concludes with a proof of the Weak Nullstellensatz using these model-theoretic concepts, showing there is a tuple in an algebraically closed field that satisfies a given ideal of polynomials.
1. The document discusses maximum likelihood estimation and Bayesian parameter estimation for machine learning problems involving parametric densities like the Gaussian.
2. Maximum likelihood estimation finds the parameter values that maximize the probability of obtaining the observed training data. For Gaussian distributions with unknown mean and variance, MLE returns the sample mean and variance.
3. Bayesian parameter estimation treats the parameters as random variables and uses prior distributions and observed data to obtain posterior distributions over the parameters. This allows incorporation of prior knowledge with the training data.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
This document discusses key concepts in probability theory, including:
1) Markov's inequality and Chebyshev's inequality, which relate the probability that a random variable exceeds a value to its expected value and variance.
2) The weak law of large numbers and central limit theorem, which describe how the means of independent random variables converge to the expected value and follow a normal distribution as the number of variables increases.
3) Stochastic processes, which are collections of random variables indexed by time or another parameter and can model evolving systems. Examples of stochastic processes and their properties are provided.
This document summarizes the key points of a thesis oral presentation on wavelet and frame theory. It discusses systems and frames, and how frames overcome limitations of orthonormal systems by providing stable reconstructions. It then outlines contributions made in the thesis to analyzing Gabor and wavelet systems using dual Gramian analysis. This allows constructing frames with desired properties like compact support and symmetry. It presents two papers published from the work and provides background on notions like Bessel systems, frames, Riesz sequences, and how the dual Gramian analysis connects frame properties to the adjoint system through a duality principle.
The document discusses the extension principle for generalizing crisp mathematical concepts to fuzzy sets. It defines the extension principle for mappings from cartesian products to universes. An example is provided to illustrate defining a fuzzy set in the output universe based on fuzzy sets in the input universes and the mapping between them. Fuzzy numbers are defined to have specific properties including being a normal fuzzy set, closed intervals for membership levels, and bounded support. Positive and negative fuzzy numbers are distinguished based on their membership functions. Binary operations are classified as increasing or decreasing, and it is noted the extension principle can be used to define the fuzzy result of applying increasing or decreasing operations to fuzzy inputs. Notation for fuzzy number algebraic operations is introduced. Several theore
This document discusses Hilbert-Schmidt n-tuples of operators on a Banach space. It presents two main results: 1) the Hypercyclicity Criterion, which provides conditions for an n-tuple of operators to be hypercyclic, and 2) conditions under which an n-tuple of unilateral weighted backward shifts is chaotic or has a non-trivial periodic point. It also references several other works studying properties of n-tuples and hypercyclic operators.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
This document provides an introduction to random matrix theory and its applications in machine learning. It discusses several classical random matrix ensembles like the Gaussian Orthogonal Ensemble (GOE) and Wishart ensemble. These ensembles are used to model phenomena in fields like number theory, physics, and machine learning. Specifically, the GOE is used to model Hamiltonians of heavy nuclei, while the Wishart ensemble relates to the Hessian of least squares problems. The tutorial will cover applications of random matrix theory to analyzing loss landscapes, numerical algorithms, and the generalization properties of machine learning models.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
Random Matrix Theory and Machine Learning - Part 4Fabian Pedregosa
Deep learning models with millions or billions of parameters should overfit according to classical theory, but they do not. The emerging theory of double descent seeks to explain why larger neural networks can generalize well. Random matrix theory provides a tractable framework to model double descent through random feature models, where the number of random features controls model capacity. In the high-dimensional limit, the test error of random feature regression exhibits a double descent shape that can be computed analytically.
Master Thesis on the Mathematial Analysis of Neural NetworksAlina Leidinger
Master Thesis submitted on June 15, 2019 at TUM's chair of Applied Numerical Analysis (M15) at the Mathematics Department.The project was supervised by Prof. Dr. Massimo Fornasier. The thesis took a detailed look at the existing mathematical analysis of neural networks focusing on 3 key aspects: Modern and classical results in approximation theory, robustness and Scattering Networks introduced by Mallat, as well as unique identification of neural network weights. See also the one page summary available on Slideshare.
Analysis and algebra on differentiable manifoldsSpringer
This chapter discusses integration on manifolds. It defines orientation of manifolds and orientation-preserving maps between manifolds. It presents Stokes' theorem and Green's theorem, which relate integrals over boundaries to integrals of differential forms. It introduces de Rham cohomology groups, which classify closed forms modulo exact forms. Examples are given of calculating the orientability and cohomology of manifolds like the cylinder, Möbius strip, and real projective plane.
PaperNo14-Habibi-IJMA-n-Tuples and ChaoticityMezban Habibi
This document presents theorems and definitions related to n-tuples of operators on a Frechet space and conditions for chaoticity. It begins with definitions of key concepts such as the orbit of a vector under an n-tuple of operators and what it means for an n-tuple to be hypercyclic or for a vector to be periodic. The main results section presents two theorems, the first characterizing when an n-tuple satisfies the hypercyclicity criterion and the second proving conditions under which an n-tuple of weighted backward shifts is chaotic. The second theorem shows the equivalence of an n-tuple being chaotic, hypercyclic with a non-trivial periodic point, having a non-trivial periodic point, and a
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
The document discusses the quantified fragment of set theory called ∀π0. ∀π0 allows for restricted quantification over sets and ordered pairs. A decision procedure for the satisfiability of ∀π0 formulas works by non-deterministically guessing a skeletal representation and checking if its realization is a model of the formula. The document considers encoding the conditions on skeletal representations as first-order formulas to view ∀π0 as a first-order logic and leverage tools developed for first-order logic fragments.
A brief introduction to Hartree-Fock and TDDFTJiahao Chen
The document provides an overview of time-dependent density functional theory (TDDFT) for computing molecular excited states. It begins with an introduction to the Born-Oppenheimer approximation and variational principle. It then discusses the Hartree-Fock and Kohn-Sham equations as self-consistent field methods for calculating ground states, and linear response theory for calculating excited states within TDDFT. The contents section outlines the topics to be covered, including basis functions, Hartree-Fock theory, density functional theory, and time-dependent DFT.
We will discuss history and recent developments in the study of the phase structure of noncommutative scalar fields. Apart from the usual disorder and uniform order phases, the theory exhibits a third phase, which survives the commutative limit.
This is connected to the UV/IR mixing of the noncommutative theory. We will rewrite the fuzzy theory as a modified quartic
matrix model, with extra multitrace terms in the action and perform saddle point analysis of the theory. Our goal will be to locate the triple point of the theory and to reconstruct the numerically obtained phase diagram. This goal will be successfully reached at the end of the talk.
The document is a presentation about using model theory to prove Hilbert's Weak Nullstellensatz. It begins with introductions to model theory, including definitions of structures, embeddings, elementary extensions, theories, and model-completeness. It then states that the theory of algebraically closed fields has model-completeness. The presentation concludes with a proof of the Weak Nullstellensatz using these model-theoretic concepts, showing there is a tuple in an algebraically closed field that satisfies a given ideal of polynomials.
1. The document discusses maximum likelihood estimation and Bayesian parameter estimation for machine learning problems involving parametric densities like the Gaussian.
2. Maximum likelihood estimation finds the parameter values that maximize the probability of obtaining the observed training data. For Gaussian distributions with unknown mean and variance, MLE returns the sample mean and variance.
3. Bayesian parameter estimation treats the parameters as random variables and uses prior distributions and observed data to obtain posterior distributions over the parameters. This allows incorporation of prior knowledge with the training data.
NONLINEAR DIFFERENCE EQUATIONS WITH SMALL PARAMETERS OF MULTIPLE SCALESTahia ZERIZER
In this article we study a general model of nonlinear difference equations including small parameters of multiple scales. For two kinds of perturbations, we describe algorithmic methods giving asymptotic solutions to boundary value problems.
The problem of existence and uniqueness of the solution is also addressed.
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
This document discusses key concepts in probability theory, including:
1) Markov's inequality and Chebyshev's inequality, which relate the probability that a random variable exceeds a value to its expected value and variance.
2) The weak law of large numbers and central limit theorem, which describe how the means of independent random variables converge to the expected value and follow a normal distribution as the number of variables increases.
3) Stochastic processes, which are collections of random variables indexed by time or another parameter and can model evolving systems. Examples of stochastic processes and their properties are provided.
This document summarizes the key points of a thesis oral presentation on wavelet and frame theory. It discusses systems and frames, and how frames overcome limitations of orthonormal systems by providing stable reconstructions. It then outlines contributions made in the thesis to analyzing Gabor and wavelet systems using dual Gramian analysis. This allows constructing frames with desired properties like compact support and symmetry. It presents two papers published from the work and provides background on notions like Bessel systems, frames, Riesz sequences, and how the dual Gramian analysis connects frame properties to the adjoint system through a duality principle.
The document discusses the extension principle for generalizing crisp mathematical concepts to fuzzy sets. It defines the extension principle for mappings from cartesian products to universes. An example is provided to illustrate defining a fuzzy set in the output universe based on fuzzy sets in the input universes and the mapping between them. Fuzzy numbers are defined to have specific properties including being a normal fuzzy set, closed intervals for membership levels, and bounded support. Positive and negative fuzzy numbers are distinguished based on their membership functions. Binary operations are classified as increasing or decreasing, and it is noted the extension principle can be used to define the fuzzy result of applying increasing or decreasing operations to fuzzy inputs. Notation for fuzzy number algebraic operations is introduced. Several theore
This document discusses Hilbert-Schmidt n-tuples of operators on a Banach space. It presents two main results: 1) the Hypercyclicity Criterion, which provides conditions for an n-tuple of operators to be hypercyclic, and 2) conditions under which an n-tuple of unilateral weighted backward shifts is chaotic or has a non-trivial periodic point. It also references several other works studying properties of n-tuples and hypercyclic operators.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
This document provides an introduction to random matrix theory and its applications in machine learning. It discusses several classical random matrix ensembles like the Gaussian Orthogonal Ensemble (GOE) and Wishart ensemble. These ensembles are used to model phenomena in fields like number theory, physics, and machine learning. Specifically, the GOE is used to model Hamiltonians of heavy nuclei, while the Wishart ensemble relates to the Hessian of least squares problems. The tutorial will cover applications of random matrix theory to analyzing loss landscapes, numerical algorithms, and the generalization properties of machine learning models.
This document discusses quantum modes and the correspondence between classical and quantum mechanics. It provides three key principles of quantum mechanics: (1) quantum states are represented by ket vectors, (2) quantum observables are hermitian operators, and (3) the Schrodinger equation governs the causal evolution of quantum systems. It also outlines how classical quantities like position and momentum correspond to quantum operators and how they form Lie algebras through commutation relations. Representations of quantum mechanics are discussed through examples like the energy basis of the harmonic oscillator.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
Random Matrix Theory and Machine Learning - Part 4Fabian Pedregosa
Deep learning models with millions or billions of parameters should overfit according to classical theory, but they do not. The emerging theory of double descent seeks to explain why larger neural networks can generalize well. Random matrix theory provides a tractable framework to model double descent through random feature models, where the number of random features controls model capacity. In the high-dimensional limit, the test error of random feature regression exhibits a double descent shape that can be computed analytically.
Random Matrix Theory and Machine Learning - Part 4
Similar to S. Duplij, Polyadic integer numbers and finite (m,n)-fields (Journal version, P-Adic Num Ultrametr Anal Appl (2017) v. 9, n.4, p. 267-291, DOI:10.1134/S2070046617040033)
Master Thesis on the Mathematial Analysis of Neural NetworksAlina Leidinger
Master Thesis submitted on June 15, 2019 at TUM's chair of Applied Numerical Analysis (M15) at the Mathematics Department.The project was supervised by Prof. Dr. Massimo Fornasier. The thesis took a detailed look at the existing mathematical analysis of neural networks focusing on 3 key aspects: Modern and classical results in approximation theory, robustness and Scattering Networks introduced by Mallat, as well as unique identification of neural network weights. See also the one page summary available on Slideshare.
This document summarizes the key concepts from the first three chapters of a book on Noetherian rings and modules. It begins with definitions of fundamental concepts like groups, rings, modules, ideals, and homomorphisms. It then defines what a Noetherian ring and Noetherian module are, proving the equivalence of three characterizations. It states properties of Noetherian modules, such as submodules and factor modules of a Noetherian module also being Noetherian. It concludes by stating results about direct sums and exact sequences involving Noetherian modules.
This document presents a dissertation on module theory submitted in partial fulfillment of a master's degree. It contains an introduction, three chapters, and a conclusion. Chapter 1 provides preliminaries on groups, rings, vector spaces, and related concepts needed to understand modules. Chapter 2 introduces modules and submodules, discusses module homomorphisms, quotient modules, generation of modules, and direct sums. Chapter 3 examines Artinian and Noetherian modules, which have special properties regarding ascending and descending chains of submodules.
The document discusses using Plücker coordinates to determine if a set of homogeneous polynomials generate the vector space of higher degree polynomials when projected onto a quotient ring. It begins by defining Plücker coordinates and showing that for two quadratic generators, the cubic vector space has zero image if the Plücker quantity P1P3 - P22 is non-zero. It then aims to generalize this to three cubic generators and the quartic vector space.
This document discusses linear, abelian, and continuous groups and how relaxing these properties leads to more complex groups. It begins with the simplest group, the real numbers R, and progresses to integer lattices Z and Z^n, then non-abelian Lie groups like SL(n,R). Lattices in these groups like SL(n,Z) are discussed, along with properties like the congruence subgroup property. Open questions are raised regarding the irreducibility of random matrices and deciding membership in subgroups of SL(n,Z).
This document provides an overview of Math 5045: Advanced Algebra I (Module Theory). It begins with definitions of rings, including commutative and non-commutative rings, rings with unity, units, zero-divisors, integral domains, and fields. It then discusses modules, including left and right modules, submodules, module homomorphisms, kernels, images, and the module HomR(M,N). It also covers quotient modules and the composition of module homomorphisms.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
The document provides an overview of modular arithmetic and its applications to finding square roots in modular arithmetic. It defines congruences and properties of modular arithmetic. It discusses cyclic groups and their relationship to integers and modular addition/multiplication. It introduces concepts like the order of an element, Lagrange's theorem, and Sylow theorems. It also defines quadratic residues, Legendre symbols, and provides an example of finding a square root in a finite field.
Amirim Project - Threshold Functions in Random Simplicial Complexes - Avichai...Avichai Cohen
This document presents an overview of prior work on threshold functions in random simplicial complexes. It discusses questions about the threshold for the existence of cycles and collapsibility in higher dimensional simplicial complexes. Specifically, it summarizes previous results that established coarse threshold functions of 1/n for the existence of cycles in dimensions greater than 1. It also reviews upper bounds on the critical probability for this threshold. The document outlines prior studies on the collapsability threshold and presents lower bounds on the critical probability. The author's own work is aimed at improving the bounds and determining sharp threshold functions for these properties in random simplicial complexes.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Nonstandard Study of Taylor Ser.Dev.-Abstract+ Intro. M.Sc. ThesisIbrahim Hamad
This document provides an abstract for a thesis on using concepts from nonstandard analysis to study Taylor series approximations. Specifically, it aims to:
1) Define an approximation factor relating the Taylor series remainder to the next nonzero term.
2) Prove properties of the approximation factor for different index ranges: standard n, unlimited indices inside/outside the convergence disk.
3) Consider the analyticity of the approximation factor in special and general cases.
The thesis contains background on nonstandard analysis, theorems on approximating series, and studies the approximation factor for Taylor series in terms of standard and nonstandard analysis.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
This document provides an analytic-combinatoric proof of Pólya's recurrence theorem for simple random walks on lattices Zd. It introduces generating functions to represent combinatorial classes of random walks. The generating function for simple random walks yields a formula for the probability of being at a given position at a given time. Laplace's method is then used to estimate these probabilities asymptotically, showing the random walk is recurrent if d = 1, 2 and transient if d ≥ 3, proving Pólya's theorem.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures. We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
This document discusses topological string theory and Gromov-Witten invariants. It begins by introducing supersymmetric sigma models on Kähler manifolds with N=2 supersymmetry. These lead to a topological twist known as the A-model, which is independent of the target space metric. Gromov-Witten invariants count rational curves in an algebraic variety X and are unchanged by complex structure deformations of X, making them a manifestation of the A-model's independence of complex structure. The Gromov-Witten invariants are also directly related to Donaldson-Thomas invariants.
Similar to S. Duplij, Polyadic integer numbers and finite (m,n)-fields (Journal version, P-Adic Num Ultrametr Anal Appl (2017) v. 9, n.4, p. 267-291, DOI:10.1134/S2070046617040033) (20)
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU(2) using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary Σ-matrices introduced here play a role similar to ordinary matrix units, and their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The presentation of n-ary SU(2) in terms of full Σ-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q>4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4q(n-1)+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σ^het-matrices of order (4q(n-1))^4. Some examples of the lowest arities are presented.
https://arxiv.org/abs/2403.19361. *) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures, https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the unitless ternary division algebra of imaginary "half-octonions" we have introduced is ternary alternative.
https://arxiv.org/abs/2312.01366
https://www.amazon.com/s?k=duplij
This document introduces hyperpolyadic structures, which are n-ary analogs of binary division algebras like the reals, complexes, quaternions, and octonions. It proposes two constructions:
1) A matrix polyadization procedure that increases the dimension of a binary algebra to obtain a corresponding n-ary algebra by using cyclic shift block matrices.
2) An "imaginary tower" construction on subsets of binary division algebras that gives nonderived ternary division algebras of half the original dimension, called "half-quaternions" and "half-octonions."
178 pages, 6 Chapters. DOI: 10.1088/978-0-7503-5281-9. This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers. In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way. Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.Part of IOP Series in Coherent Sources, Quantum Fundamentals, and Applications. Key features • Provides new mathematical methods for quantum computing. • Presents material in a widely accessible way. • Contains methods for unconventional computing where there is computational complexity. • Provides methods to increase speed and efficiency. For the light paperback version use MyPrint service here: https://iopscience.iop.org/book/mono/978-0-7503-5281-9, also PDF, ePub and Kindle. For the libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Innovative-Quantum-Computing/?K=9780750352796. Amazon ordering: https://www.amazon.de/gp/product/0750352795
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.
https://www.mdpi.com/books/book/6455
We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.
https://arxiv.org/abs/2212.08606
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Книга «Поэфизика души» представляет собой полное, на момент издания 2022 г., собрание прозаических произведений автора. Как рассказы, так и миниатюры на полстраницы, пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга поэтическими образами, воплощенными в прозе. Также включены юмористические путевые заметки о поездке в Китай.
Книга "Поэфизика души", Степан Дуплий – полное собрание прозы 2022, 230 стр. вышла в Ridero: https://ridero.ru/books/poefizika_dushi и Kindle Edition file на Амазоне: https://amazon.com/dp/B0B9Y4X4VJ . "Бумажную" книгу можно заказать на Озоне https://ozon.ru/product/poefizika-dushi-682515885/?sh=XPu-9Sb42Q и на ЛитРес: https://litres.ru/stepan-dupliy/poefizika-dushi-emocionalnaya-proza-kitayskiy-shtrih-punktir . Google books: https://books.google.com/books?id=9w2DEAAAQBAJ .
Книгу можно заказать из-за рубежа на AliExpress: https://aliexpress.com/item/1005004660613179.html .
Книга «Гравитация страсти» представляет собой полное собрание стихотворений автора на момент издания (август, 2022). Стихотворения пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга необычными поэтическими образами.
Книга "Гравитация страсти", Степан Дуплий - полное собрание стихотворений 2022, 338 стр. вышла в Ridero: https://ridero.ru/books/gravitaciya_strasti
. Книга в мягкой обложке доступна для заказа на Ozon.ru: https://ozon.ru/product/gravitatsiya-strasti-707068219/?oos_search=false&sh=XPu-9TbW9Q
, на Litres.ru: https://www.litres.ru/stepan-dupliy/gravitaciya-strasti-stihotvoreniya , за рубежом на AliExpress: https://aliexpress.com/item/1005004722134442.html , и в электронном виде Kindle file на Amazon.com: https://amazon.com/dp/B0BDFTT33W .
We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.
In book: S. Duplij, "Polyadic Algebraic Structures", 2022, IOP Publishing (Bristol), Section 1.5. See https://iopscience.iop.org/book/978-0-7503-2648-3
https://arxiv.org/abs/2206.14840
The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories.
Suitable for university students of advanced level algebra courses and mathematical physics courses.
Key features
• Provides a general, unified approach
• Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
https://arxiv.org/abs/2201.08479
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be "entangled" such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields, which can lead to a new category of polyadic fields. Many illustrative concrete examples are presented.
This document proposes a new mechanism for "deforming" or breaking commutativity in algebras called "membership deformation". It involves taking the underlying set of an algebra to be an "obscure/fuzzy set" with elements having membership functions between 0 and 1 rather than a crisp set. The membership functions are incorporated into the commutation relations such that elements with equal membership functions commute, while others do not. This provides a continuous way to deform commutativity. The approach is then generalized to ε-commutative algebras and n-ary algebras. Projective representations of n-ary algebras are also studied in relation to this new type of deformation.
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.
This document discusses generalizing the concept of regularity for semigroups in two ways: higher regularity and higher arity (polyadic semigroups).
For binary semigroups, higher n-regularity is defined such that each element has multiple inverse elements rather than a single inverse. However, for binary semigroups this reduces to ordinary regularity. For polyadic semigroups, several definitions of regularity and higher regularity are introduced to account for the higher arity operations. Idempotents and identities are also generalized for polyadic semigroups. It is shown that the definitions of regularity for polyadic semigroups cannot be reduced in the same
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.
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A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
PPT on Sustainable Land Management presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Microbial interaction
Microorganisms interacts with each other and can be physically associated with another organisms in a variety of ways.
One organism can be located on the surface of another organism as an ectobiont or located within another organism as endobiont.
Microbial interaction may be positive such as mutualism, proto-cooperation, commensalism or may be negative such as parasitism, predation or competition
Types of microbial interaction
Positive interaction: mutualism, proto-cooperation, commensalism
Negative interaction: Ammensalism (antagonism), parasitism, predation, competition
I. Mutualism:
It is defined as the relationship in which each organism in interaction gets benefits from association. It is an obligatory relationship in which mutualist and host are metabolically dependent on each other.
Mutualistic relationship is very specific where one member of association cannot be replaced by another species.
Mutualism require close physical contact between interacting organisms.
Relationship of mutualism allows organisms to exist in habitat that could not occupied by either species alone.
Mutualistic relationship between organisms allows them to act as a single organism.
Examples of mutualism:
i. Lichens:
Lichens are excellent example of mutualism.
They are the association of specific fungi and certain genus of algae. In lichen, fungal partner is called mycobiont and algal partner is called
II. Syntrophism:
It is an association in which the growth of one organism either depends on or improved by the substrate provided by another organism.
In syntrophism both organism in association gets benefits.
Compound A
Utilized by population 1
Compound B
Utilized by population 2
Compound C
utilized by both Population 1+2
Products
In this theoretical example of syntrophism, population 1 is able to utilize and metabolize compound A, forming compound B but cannot metabolize beyond compound B without co-operation of population 2. Population 2is unable to utilize compound A but it can metabolize compound B forming compound C. Then both population 1 and 2 are able to carry out metabolic reaction which leads to formation of end product that neither population could produce alone.
Examples of syntrophism:
i. Methanogenic ecosystem in sludge digester
Methane produced by methanogenic bacteria depends upon interspecies hydrogen transfer by other fermentative bacteria.
Anaerobic fermentative bacteria generate CO2 and H2 utilizing carbohydrates which is then utilized by methanogenic bacteria (Methanobacter) to produce methane.
ii. Lactobacillus arobinosus and Enterococcus faecalis:
In the minimal media, Lactobacillus arobinosus and Enterococcus faecalis are able to grow together but not alone.
The synergistic relationship between E. faecalis and L. arobinosus occurs in which E. faecalis require folic acid
SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆Sérgio Sacani
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4
PPT on Alternate Wetting and Drying presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Anti-Universe And Emergent Gravity and the Dark UniverseSérgio Sacani
Recent theoretical progress indicates that spacetime and gravity emerge together from the entanglement structure of an underlying microscopic theory. These ideas are best understood in Anti-de Sitter space, where they rely on the area law for entanglement entropy. The extension to de Sitter space requires taking into account the entropy and temperature associated with the cosmological horizon. Using insights from string theory, black hole physics and quantum information theory we argue that the positive dark energy leads to a thermal volume law contribution to the entropy that overtakes the area law precisely at the cosmological horizon. Due to the competition between area and volume law entanglement the microscopic de Sitter states do not thermalise at sub-Hubble scales: they exhibit memory effects in the form of an entropy displacement caused by matter. The emergent laws of gravity contain an additional ‘dark’ gravitational force describing the ‘elastic’ response due to the entropy displacement. We derive an estimate of the strength of this extra force in terms of the baryonic mass, Newton’s constant and the Hubble acceleration scale a0 = cH0, and provide evidence for the fact that this additional ‘dark gravity force’ explains the observed phenomena in galaxies and clusters currently attributed to dark matter.
Anti-Universe And Emergent Gravity and the Dark Universe
S. Duplij, Polyadic integer numbers and finite (m,n)-fields (Journal version, P-Adic Num Ultrametr Anal Appl (2017) v. 9, n.4, p. 267-291, DOI:10.1134/S2070046617040033)
1. ISSN 2070-0466, p-Adic Numbers, Ultrametric Analysis and Applications, 2017, Vol. 9, No. 4, pp. 267–291. c Pleiades Publishing, Ltd., 2017.
RESEARCH ARTICLES
Polyadic Integer Numbers and Finite (m, n)-Fields∗
Steven Duplij**
Mathematisches Institute, Universit ¨at M ¨unster,
Einsteinstr. 62, D-48149 M ¨unster, Deutschland
Received July 14, 2017
Abstract—The polyadic integer numbers, which form a polyadic ring, are representatives of a
fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers,
the polyadic Euler function, polyadic division with a remainder, etc. are introduced. Secondary
congruence classes of polyadic integer numbers, which become ordinary residue classes in the
"binary limit", and the corresponding finite polyadic rings are defined. Polyadic versions of (prime)
finite fields are introduced. These can be zeroless, zeroless and nonunital, or have several units; it is
even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields
of the same arity shape and order. None of the above situations is possible in the binary case. It is
conjectured that a finite polyadic field should contain a certain canonical prime polyadic field, defined
here, as a minimal finite subfield, which can be considered as a polyadic analogue of GF (p).
DOI: 10.1134/S2070046617040033
Key words: finite field, polyadic ring, (m.n)-field, polyadic integer numbers, Galois field,
congruence class.
1. INTRODUCTION
The theory of finite fields [1] plays a very important role. From one side, it acts as a “gluing particle”
connecting algebra, combinatorics and number theory (see, e.g. [2]), and from another it has numerous
applications to “reality”: in coding theory, cryptography and computer science [3]. Therefore, any
generalization or variation of its initial statements can lead to interesting and useful consequences for
both of the above. There are two principal peculiarities of finite fields: 1) Uniqueness - they can have
only special numbers of elements (the order is any power of a prime integer pr) and this fully determines
them, in that all finite fields of the same order are isomorphic; 2) Existence of their “minimal” (prime)
finite subfield of order p, which is isomorphic to the congruence class of integers Z pZ. Investigation of
the latter is a bridge to the study of all finite fields, since they act as building blocks of the extended (that
is, all) finite fields.
We propose a special - polyadic - version of the (prime) finite fields in such a way that, instead
of the binary ring of integers Z, we consider a polyadic ring. The concept of the polyadic integer
numbers Z(m,n) as representatives of a fixed congruence class, which form the (m, n)-ring (with m-
ary addition and n-ary multiplication), was introduced in [4]. Here we analyze Z(m,n) in more detail,
by developing elements of a polyadic analog of binary arithmetic: polyadic prime numbers, polyadic
division with a remainder, the polyadic Euler totient function, etc. ... It is important to stress that the
polyadic integer numbers are special variables (we use superscripts for them) which in general have
no connection with ordinary integers (despite the similar notation used in computations), because the
former satisfy different relations, and coincide with the latter in the binary case only. Next we will define
new secondary congruence classes and the corresponding finite (m, n)-rings Z(m,n) (q) of polyadic
integer numbers, which give Z qZ in the “binary limit”. The conditions under which these rings become
fields are given, and the corresponding “abstract” polyadic fields are defined and classified using their
idempotence polyadic order. They have unusual properties, and can be zeroless, zeroless-nonunital or
∗
The text was submitted by the author in English.
**
E-mail: duplijs@math.uni-muenster.de
267
2. 268 DUPLIJ
have several units, and it is even possible for all elements to be units. The subgroup structure of their
(cyclic) multiplicative finite n-ary group is analyzed in detail. For some zeroless finite polyadic fields
their multiplicative n-ary group is a non-intersecting union of subgroups. It is shown that there exist
non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is
possible in the binary case.
Some general properties of polyadic rings and fields were given in [5–8], but their concrete examples
using integers differ considerably from our construction here, and the latter leads to so called nonderived
(proper) versions which have not been considered before.
We conjecture that any (m, n)-field with m > n contains as a subfield one of the prime polyadic fields
constructed here, which can be considered as a polyadic analog of GF (p).
2. PRELIMINARIES
We use the notations and definitions from [4, 9] (see, also, references therein). We recall (only for
self-consistency) some important elements and facts about polyadic rings, which will be needed below.
Informally, a polyadic (m, n)-ring is Rm,n = R | νm, μn , where R is a set, equipped with m-ary
addition νm : Rm → R and n-ary multiplication μn : Rn → R which are connected by the polyadic
distributive law, such that R | νm is a commutative m-ary group and R | μn is a semigroup. A
commutative (cancellative) polyadic ring has a commutative (cancellative) n-ary multiplication μn.
A polyadic ring is called derived, if νm and μn are equivalent to a repetition of the binary addition and
multiplication, while R | + and R | · are commutative (binary) group and semigroup respectively. If
only one operation νm (or μn) has this property, we call such a Rm,n additively (or multiplicatively)
derived (half-derived).
In distinction to binary rings, an n-admissible “length of word (x)” should be congruent to
1 mod (n − 1), containing μ (n − 1) + 1 elements ( μ is a “number of multiplications”) μ
( μ)
n [x] (x ∈
R μ(n−1)+1), so called ( μ (n − 1) + 1)-ads, or polyads. An m-admissible “quantity of words (y)” in
a polyadic “sum” has to be congruent to 1 mod (m − 1), i.e. consisting of ν (m − 1) + 1 summands ( ν
is a “number of additions”) ν
( ν)
m [y] (y ∈ R ν (m−1)+1). Therefore, a straightforward “polyadization” of
any binary expression (m = n = 2) can be introduced as follows: substitute the number of multipliers
μ + 1 → μ (n − 1) + 1 and number of summands ν + 1 → ν (m − 1) + 1, respectively.
An example of “trivial polyadization” is the simplest (m, n)-ring derived from the ring of integers Z
as the set of ν (m − 1) + 1 “sums” of n-admissible ( μ (n − 1) + 1)-ads (x), where x ∈ Z μ(n−1)+1 [6].
The additive m-ary polyadic power and the multiplicative n-ary polyadic power are defined by
(inside polyadic products we denote repeated entries by
k
x, . . . , x as xk)
x ν +m = ν( ν)
m x ν(m−1)+1
, x μ ×n = μ
( μ)
n x μ(n−1)+1
, x ∈ R, (2.1)
such that the polyadic powers and ordinary powers differ by one: x ν +2 = x ν+1, x μ ×2 = x μ+1.
The polyadic idempotents in Rm,n satisfy
x ν +m = x, x μ ×n = x, (2.2)
and are called the additive ν-idempotent and the multiplicative μ-idempotent, respectively.
The additive 1-idempotent, the zero z ∈ R, is (if it exists) defined by
νm [x, z] = z, ∀x ∈ Rm−1
. (2.3)
An element x ∈ R is called (polyadic) nilpotent, if x 1 +m = z, and all higher powers of a nilpotent
element are nilpotent, as follows from (2.3) and associativity.
The unit e of Rm,n is a multiplicative 1-idempotent which is defined (if it exists) as
μn en−1
, x = x, ∀x ∈ R, (2.4)
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
3. POLYADIC INTEGER NUMBERS 269
where (in case of a noncommutative polyadic ring) x can be on any place. An element x ∈ R is called a
(polyadic) μ-reflection, if x μ ×n = e (multiplicative analog of a nilpotent element).
Polyadic rings with zero or unit(s) are called additively or multiplicatively half-derived, and derived
rings have a zero and unit(s) simultaneously. There are polyadic rings which have no unit and no zero,
or with several units and no zero, or where all elements are units. But if a zero exists, it is unique. If a
polyadic ring contains no unit and no zero, we call it a zeroless nonunital polyadic ring. It is obvious
that zeroless nonunital rings can contain other idempotents of higher polyadic powers.
So, in polyadic rings (including the zeroless nonunital ones) invertibility can be governed in a way
which is not connected with unit and zero elements. For a fixed element x ∈ R its additive querelement
˜x and multiplicative querelement ¯x are defined by
νm xm−1
, ˜x = x, μn xn−1
, ¯x = x, (2.5)
where in the second equation, if the n-ary multiplication μn is noncommutative, ¯x can be on any place.
Because R | νm is a commutative group, each x ∈ R has its additive querelement ˜x (and is querable or
“polyadically invertible”). The n-ary semigroup R | μn can have no multiplicatively querable elements
at all. However, if every x ∈ R has its unique querelement, then R | μn is an n-ary group. Obviously,
that n-ary group cannot have nilpotent elements, but can have μ-reflections. Denote R∗ = R {z}, if
the zero z exists. If R∗ | μn is the n-ary group, then Rm,n is a (m, n)-division ring.
Definition 2.1. A commutative (m, n)-division ring Rm,n is a (m, n)-field Fm,n.
The simplest example of a (m, n)-field derived from R is the set of ν (m − 1) + 1 “sums” of admissible
( μ (n − 1) + 1)-ads (x), where x ∈ R μ(n−1)+1. Some nonderived (m, n)-fields are in
Example 2.2. a) The set iR with i2 = −1 is a (2, 3)-field with a zero and no unit (operations are
made in C), but the multiplicative querelement of ix is −i x (x = 0).
b) The set of fractions ix/y | x, y ∈ Zodd, i2 = −1 is a (3, 3)-field with no zero and no unit
(operations are in C), while the additive and multiplicative querelements of ix/y are −ix/y and
−iy/x, respectively.
c) The set of antidiagonal 2 × 2 matrices over R is a (2, 3)-field with zero z =
⎛
⎝
0 0
0 0
⎞
⎠ and two
units e = ±
⎛
⎝
0 1
1 0
⎞
⎠, but the unique querelement of
⎛
⎝
0 x
y 0
⎞
⎠ is
⎛
⎝
0 1/y
1/x 0
⎞
⎠.
3. RING OF POLYADIC INTEGER NUMBERS
Recall the notion of the ring of polyadic integer numbers Z(m,n) which was introduced in [4], where
its difference from the (m, n)-ring of integers from [6] was outlined.
Let us consider a congruence class (residue class) of an integer a modulo b
[[a]]b = {{a + bk} | k ∈ Z, a ∈ Z+, b ∈ N, 0 ≤ a ≤ b − 1} . (3.1)
We denote a representative element by xk = x
[a,b]
k = a + bk, where obviously {xk} is an infinite set.
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
4. 270 DUPLIJ
3.1. External and Internal Operations for Congruence Classes
Informally, there are two ways to equip (3.1) with operations:
1) The “External” way is to define (binary) operations between the congruence classes. Let us define
on the finite underlying set of b congruence classes {[[a]]b}, a = 0, 1, . . . , b − 1 the following new
binary operations (here, if b is fixed, and we denote the binary class representative by an integer
with one prime [[a]]b ≡ a , as well as the corresponding binary operations + , · between classes)
a1 + a2 = (a1 + a2) , (3.2)
a1 · a2 = (a1a2) . (3.3)
Then, the binary residue class ring is defined by
Z bZ = a | + , · , 0 , 1 . (3.4)
In the case of prime b = p, the ring Z pZ becomes a binary finite field having p elements.
2) The “Internal” way is to introduce (polyadic) operations inside a given class [[a]]b (with both a
and b fixed). We introduce the commutative m-ary addition and commutative n-ary multiplication
of representatives xki
of the fixed congruence class by
νm [xk1 , xk2 , . . . , xkm ] = xk1 + xk2 + . . . + xkm , (3.5)
μn [xk1 , xk2 , . . . , xkn ] = xk1 xk2 . . . xkn , xki
∈ [[a]]b , ki ∈ Z. (3.6)
In general, the binary sums xk1 + xk2 and products xk1 xk2 are not in [[a]]b.
Proposition 3.1 ([4]). The polyadic operations νm and μn become closed in [[a]]b, if the
arities (m, n) have the minimal values satisfying
ma ≡ a (mod b) , (3.7)
an
≡ a (mod b) . (3.8)
Polyadic distributivity is inherited from that of Z, and therefore we have
Definition 3.2 ([4]). The congruence class [[a]]b equipped with a structure of nonderived
infinite commutative polyadic ring is called a (m, n)-ring of polyadic integer numbers
Z(m,n) ≡ Z
[a,b]
(m,n) = {[[a]]b | νm, μn} . (3.9)
Obviously, Z(m,n) (as in the binary case) cannot become a polyadic field with any choice of
parameters.
Example 3.3. In the residue class
[[3]]4 = {. . . − 25, −21, −17, −13, −9, −5, −1, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39 . . .} (3.10)
we can add only 4 ν + 1 representatives and multiply 2 μ + 1 representatives ( ν, μ are “numbers”
of m-ary additions and n-ary multiplications respectively) to retain the same class, e.g., take
ν = 2, μ = 3 to get (7 + 11 + 15 + 19 + 23) − 5 − 9 − 13 − 1 = 47 ∈ [[3]]4, ((7 · 3 · 11) · 19 · 15) · 31 ·
27 = 55 103 895 ∈ [[3]]4. Obviously, we cannot add and multiply arbitrary quantities of numbers
in [[3]]4, only the admissible ones. This means that [[3]]4 is the polyadic (5, 3)-ring Z(5,3) = Z
[3,4]
(5,3).
Remark 3.4. After imposing the operations (3.5)–(3.6) the representatives x
[a,b]
k become abstract
variables (elements of the corresponding (m, n)-ring or polyadic integer numbers) which are
not ordinary integers (forming a (2, 2)-ring), but have the latter as their “binary limit”. So in
computations the integral numbers (denoting representatives) should carry their arity shape
(m, n) as additional indices. Indeed, the representative, e.g. 3 = 3(5,3) ∈ Z
[3,4]
(5,3) is different from
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
5. POLYADIC INTEGER NUMBERS 271
3 = 3(3,2) ∈ Z
[1,2]
(3,2), i.e. properly speaking 3(5,3) = 3(3,2), since their operations (multiplication
and addition) are different, because they belong to different polyadic rings, Z
[3,4]
(5,3) and Z
[1,2]
(3,2),
respectively. For conciseness, we omit the indices (m, n), if their value is clear from the context.
Thus, at first sight it seems that one can obtain a polyadic field only in the “external” way, i.e.
using the “trivial polyadization” of the binary finite field Z pZ (just a repetition of the binary group
operations).This leads to the derived polyadic finite fields, which have a very simple structure, in which
the admissible binary sums and binary products of the congruence classes are used [6]. However, in the
next section we propose a new approach, and thereby construct the nonderived finite (m, n)-fields of
polyadic integer numbers Z(m,n).
Remark 3.5. If n = b = p is prime, then (3.8) is valid for any a ∈ N, which is another formulation
of Fermat’s little theorem.
3.2. Prime Polyadic Integer Numbers
Let us introduce a polyadic analog of prime numbers in Z(m,n). First we need
Definition 3.6. A polyadically composite (reducible) number is xk ∈ Z(m,n), such that the expan-
sion
xk = μ( )
n xk1 , xk2 , , xk (n−1)+1
, xki
∈ Z(m,n), (3.11)
is unique, where is a number of n-ary multiplications, and there exist at least one xki
= xk
and xki
= e (i.e. is not equal to unit of Z(m,n), if it exists). Denote the set of such numbers
{xki
} = D (xk) which is called the composition set of xk.
Definition 3.7. An irreducible polyadic number is xk ∈ Z(m,n) cannot be expressed as any (long)
polyadic product (3.11).
Proposition 3.8. In the polyadic ring Z
[a,b]
(m,n) without the unit the elements satisfying
− |a − b|n
< xk < |a − b|n
(3.12)
are irreducible.
Proof. Since 0 ≤ a ≤ b − 1, the minimal absolute value of an element xk ∈ Z
[a,b]
(m,n) is |a − b|. The
minimum of its n-ary product is |a − b|n
, and therefore smaller elements cannot be decomposed.
Example 3.9. In the (6, 5)-ring Z
[8,10]
(6,5)
all polyadic integer numbers are even, and there is no unit,
and so they are binary composite
Z
[8,10]
(6,5) = {. . . − 72, −62, −52, −42, −32, −22, −12, −2, 8, 18, 28, 38, 48, 58, . . .} (3.13)
Nevertheless, the lowest elements, e.g. {−22, −12, 8, 18, 28}, are irreducible, while the smallest
(by absolute value) polyadically composite element is (−32) = μ5 (−2)5
.
Definition 3.10. A range in which all elements are indecomposable is called a polyadic irreducible
gap.
Remark 3.11. We do not demand positivity, as in the binary case, because polyadic integer
numbers Z
[a,b]
(m,n) (3.9) are “symmetric” not with respect to x = 0, but under x = xk=0 = a.
The polyadic analog of binary prime numbers plays an intermediate role between composite and
irreducible elements.
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
6. 272 DUPLIJ
Definition 3.12. A polyadic prime number is xkp ∈ Z(m,n), such that it obeys only the unique
expansion
xkp = μ( )
n xkp , e (n−1)
, (3.14)
where e a polyadic unit of Z(m,n) (if exists).
So, the polyadic prime numbers can appear only in those polyadic rings Z
[a,b]
(m,n) which contain units.
In [4] (Proposition 6.15) it was shown that such rings correspond to the limiting congruence classes
[[1]]b and [[b − 1]]b, and indeed only for them can a + bk = 1 mod b, and e (n−1) can be a neutral sequence
(for e = 1 always, while for e = −1 only when (n − 1) is even).
Proposition 3.13. The prime polyadic numbers can exist only in the limiting polyadic rings
Z
[1,b]
(b+1,2) and Z
[b−1,b]
(b+1,3).
Proof. The equation a + bk = 1 mod b (for 0 ≤ a ≤ b − 1) has two solutions: a = 1 and a = b − 1
corresponding for two limiting congruence classes [[1]]b and [[b − 1]]b, which correspond to
x+
k = bk + 1, (3.15)
x−
k = b (k + 1) − 1, k ∈ Z. (3.16)
The parameters-to-arity mapping (3.41) fixes their multiplication arity to n = 2 and n = 3 respectively,
which gives manifestly
μ2 x+
k1
, x+
k2
= x+
k , k = bk1k2 + k1 + k2, (3.17)
μ3 x−
k1
, x−
k2
, x−
k3
= x−
k , k, ki ∈ Z, i = 1, 2, 3, b ∈ N,
k = b2
k1k2k3 + (b − 1) [b (k1k2 + k2k3 + k1k3) + (b − 1) (k1 + k2 + k3) + (b − 2)] . (3.18)
Therefore, for Z
[1,b]
(b+1,2) we have the unit e = x+
k=0 = 1 (which is obvious for the binary multiplication),
while in Z
[b−1,b]
(b+1,3) the unit is e = x−
k=−1 = −1 (b ≥ 3), and the sequence is e2 is evidently neutral.
Denote the set of ordinary binary prime numbers in the interval 1 ≤ k ≤ kmax by P (kmax), kmax ∈ N.
The set of prime polyadic numbers for the polyadic ring Z
[a,b]
(m,n) in the interval x−kmax ≤ xk ≤ xkmax
is denoted by P
[a,b]
(m,n) (kmax). Obviously, in the binary limit P
[0,1]
(2,2) (kmax) = P (kmax) ∪ {−P (kmax)}.
Nevertheless, prime polyadic numbers can be composite as binary numbers.
Assertion 3.14. The set of prime polyadic numbers in the interval x−kmax ≤ xk ≤ xkmax for Z
[a,b]
(m,n)
can contain composite binary numbers, i.e.
ΔP
[a,b]
(m,n) (kmax) = P
[a,b]
(m,n) (kmax) {{P (xkmax ) ∪ {P (−x−kmax )}} ∩ [[a]]b} = ∅. (3.19)
Definition 3.15. 1) The cardinality of the set of ordinary binary prime numbers P (kmax) is called
a prime-counting function and denoted by π (kmax) = |P (kmax)|.
2) The cardinality of the set of prime polyadic numbers P
[a,b]
(m,n) (kmax) is called a polyadic
prime-counting function and denoted by
π
[a,b]
(m,n) (kmax) = P
[a,b]
(m,n) (kmax) . (3.20)
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
7. POLYADIC INTEGER NUMBERS 273
Example 3.16. 1) Consider Z
[43,44]
(45,3) and kmax = 2, then
P
[43,44]
(45,3) (2) = {−45, −1, 43, 87, 131} , (3.21)
ΔP
[43,44]
(45,3) (2) = {−45} , π
[43,44]
(45,3) (2) = 5. (3.22)
2) For Z
[50,51]
(52,3) and kmax = 5 we have
P
[50,51]
(52,3)
(5) = {−205, −154, −103, −52, −1, 50, 101, 152, 203, 254, 305}, (3.23)
ΔP
[50,51]
(52,3) (5) = {−205, −154, −52, 50, 152, 203, 254, 305}, π
[50,51]
(52,3) (5) = 11. (3.24)
Remark 3.17. This happens because in Z
[a,b]
(m,n) the role of “building blocks” (prime polyadic
numbers) is played by those xk which cannot be presented as a (long) ternary product of other
polyadic integer numbers from the same Z
[a,b]
(m,n) as in (3.11), but which satisfy (3.14) only.
Nevertheless, such prime polyadic numbers can be composite binary prime numbers.
In general, for the limiting cases, in which polyadic prime numbers exist, we have
Proposition 3.18. 1) In Z
[1,b]
(b+1,2) the “smallest” polyadic integer numbers satisfying
−b < kp < b + 2, (3.25)
1 − b2
< xkp < (b + 1)2
, (3.26)
are not decomposable, and therefore such xkp ∈ Z
[1,b]
(b+1,2) are all polyadic prime numbers.
2) For another limiting case Z
[b−1,b]
(b+1,3) the polyadic integer numbers satisfying
1 − b < kp < b − 1, (3.27)
− (b − 1)2
< xkp < b2
− 1, (3.28)
are not ternary decomposable and so all such xkp ∈ Z
[b−1,b]
(b+1,2) are polyadic prime numbers.
Proof. This follows from determining the maximum of the negative values and the minimum of the
positive values of the functions x+
k and x−
k in (3.17)–(3.18).
Definition 3.19. The range in which all elements are polyadically prime numbers is called the
polyadic primes gap, and for the two limiting cases it is given by (3.26) and (3.28), respectively.
For instance, in Z
[50,51]
(52,3) for the polyadic primes gap we have −2500 < xkp < 2600: all such polyadic
integer numbers are polyadically prime, but there are many composite binary numbers among them.
In the same way we can introduce a polyadic analog of the Euler (totient) function which in
the binary case counts the number of coprimes to a given natural number. Denote the set of ordinary
binary numbers k > 1 which are coprime to kmax ∈ N by S (kmax) (named totatives of kmax). Then, the
cardinality of S (kmax) is defined as Euler function ϕ (kmax) = |S (kmax)|. Obviously, if kmax = p is prime,
then ϕ (p) = p − 1. The notion of coprime numbers is based on the divisors: the coprime numbers k1 and
k2 have the greatest common divisor gcd (k1, k2) = 1. In the polyadic case it is not so straightforward,
and we need to start from the basic definitions.
First, we observe that in a (commutative) polyadic ring Rm,n = {R | νm, μn} the analog of the
division operation is usually not defined uniquely, which makes it useless for real applications. Indeed, y
divides x, where x, y ∈ R, if there exists a sequence z ∈ Rn−1 of length (n − 1), such that x = μn [y, z].
To be consistent with the ordinary integer numbers Z, we demand in the polyadic number ring Z(m,n): 1)
Uniqueness of the result; 2) i.e. only one polyadic number (not a sequence) as the result. This naturally
leads to
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8. 274 DUPLIJ
Definition 3.20. A polyadic number (quotient) xk2 polyadically divides a polyadic number
(dividend) xk1 , if there exists xkq := xk1 ÷p xk2 , called the (unique) result of division, such that
xk1 = μn xk2 , xkq
n−1
, xk1 , xk2 , xkq ∈ Z(m,n). (3.29)
Remark 3.21. For polyadic prime numbers (3.14) the only possibility for the quotient is xk2 = xk1
such that xk1 = μn xk1 , (e)n−1
or xk1 ÷p xk1 = e, where e is the unit of Z(m,n).
Assertion 3.22. Polyadic division is distributive from the left
νm [xk1 , xk2 , . . . , xkm ] ÷p xk = νm [(xk1 ÷p xk) , (xk2 ÷p xk) , . . . , (xkm ÷p xk)] , (3.30)
but not distributive from the right.
Proof. This follows from the polyadic distributivity in the (m, n)-ring Z(m,n).
Example 3.23. 1) In the polyadic ring Z
[4,9]
(10,4) we have uniquely x28 ÷p x4 = x4 or 256 ÷p 4 = 4.
2) For the limiting ring Z
[3,4]
(5,3), we find x43 ÷p x1 = x−2 or 175 ÷p 7 = −5.
In the same way we define a polyadic analog of division with a remainder.
Definition 3.24. A polyadic division with a remainder is defined, if for a polyadic dividend xk1
and divisor xk2 there exists a polyadic remainder xkr such that
xk1 = νm μn xk2 , xkq
n−1
, (xkr )m−1
, xk1 , xk2 , xkq , xkr ∈ Z(m,n), (3.31)
which is denoted by xkr = xk1 modp xk2 , and (3.31) can be presented in the following binary form
xk1 = xk2 p xkq p xkr . (3.32)
The distributivity of these operations is governed by distributivity in the polyadic ring Z(m,n).
Example 3.25. In the polyadic ring Z
[8,10]
(6,5) we can have different divisions for the same dividend
as 38 = ((−22) p (−2)) p 78 = ((−92) p (−2)) p 238.
Secondly, because divisibility in the polyadic case is not symmetric with respect to dividend and
divisor (3.29), we define polyadically coprime numbers using the definition of compositeness (3.11).
Definition 3.26. The s polyadic integer numbers xk1 , . . . , xks ∈ Z(m,n) are polyadically coprime, if
their composition sets do not intersect D (xk1 ) ∩ D (xk1 ) ∩ . . . ∩ D (xks ) = ∅.
It is important that this definition does not imply the existence of a unit in Z(m,n), as opposed to the
definition of the polyadic prime numbers (3.14) in which the availability of a unit is crucial.
Assertion 3.27. Polyadically coprime numbers can exist in any polyadic ring Z(m,n), and not only
in the limiting cases with unit (see Proposition 3.13).
Corollary 3.28. All elements in the polyadic irreducible gap are polyadically coprime.
Example 3.29. In Z
[8,10]
(6,5) (3.13) the polyadic integer numbers (−32) = μ5 (−2)5
and 32768 =
μ5 (8)5
are both composed, but polyadically coprime, because the composition sets D (−32) =
{−2} and D (32768) = {8} do not intersect. Alternatively, not polyadically coprime numbers
here are, e.g., (−3072) = μ5 (−12) , (−2)2
, (8)2
and (−64512) = μ5 −2, (8)2
, 18, 28 , because
D (−3072) ∩ D (−64512) = {−2, 8} = ∅. We cannot multiply these two numbers, because the arity
of multiplication is 5.
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
9. POLYADIC INTEGER NUMBERS 275
Remark 3.30. For polyadic integer numbers Z(m,n) we cannot measure the property “to be
coprime” in terms of a single element, as in binary case, by their gcd, because the n-ary
multiplication is only allowed for admissible sequences. Therefore, we need to consider the
intersection of the composition sets.
In the polyadic ring Z
[a,b]
(m,n) for a given kmax ∈ Z+ we denote by S
[a,b]
(m,n) (kmax) the set of polyadic
prime numbers xk ∈ Z
[a,b]
(m,n) which are polyadically coprime to xkmax and x−kmax in the open interval
x−kmax < xk < xkmax (we assume that, if two numbers are coprime, then their opposite numbers are
also coprime). For the binary limit, obviously, S
[0,1]
(2,2) (kmax) = S (kmax) ∪ {−S (kmax)}.
Definition 3.31. The cardinality of S
[a,b]
(m,n) (kmax) is called a polyadic Euler function denoted by
ϕ
[a,b]
(m,n) (kmax) = S
[a,b]
(m,n) (kmax) . (3.33)
In the binary case ϕ
[0,1]
(2,2) (kmax) = 2ϕ (kmax). Because of Remark 3.30, the computation of the
polyadic Euler function requires for each element in the range x−kmax < xk < xkmax a thorough con-
sideration of composition sets.
Example 3.32. 1) For the limiting polyadic ring Z
[1,29]
(30,2) and kmax = 10 we have x−10 = −289,
x10 = 291 and
S
[1,29]
(30,2) (10) = {−260, −202, −173, −115, −86, −28, 1, 59, 88, 146, 175, 233, 262} , (3.34)
ϕ
[1,29]
(30,2) (10) = 13. (3.35)
2) In another limiting case with ternary multiplication Z
[31,32]
(33,3) we get x−5 = −129, x5 = 191
and
S
[31,32]
(33,3) (5) = {−97, −65, −1, 31, 95, 127} , (3.36)
ϕ
[1,29]
(30,2) (5) = 6. (3.37)
3) In the non-limiting case Z
[7,10]
(11,5) and kmax = 10 we have x−10 = −93, x10 = 107 with
S
[7,10]
(11,5) (10) = {−83, −73, −53, −43, −23, −13, 7, 17, 37, 47, 67, 77, 97} , (3.38)
ϕ
[7,10]
(11,5) (10) = 13. (3.39)
4) For the polyadic Euler function in some other non-limiting cases we have
ϕ
[27,49]
(50,15) (7) = 6, ϕ
[17,38]
(39,10) (20) = 21, ϕ
[16,28]
(8,4) (30) = ϕ
[46,50]
(26,6) (15) = 0. (3.40)
3.3. The Parameters-To-Arity Mapping
Let us consider the connection between congruence classes and arities in more detail.
Remark 3.33. a) Solutions to (3.7) and (3.8) do not exist simultaneously for all a and b; b) The
pair a, b determines m, n uniquely; c) It can occur that for several different pairs a, b there can be
the same arities m, n.
Therefore, we have
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
10. 276 DUPLIJ
Assertion 3.34. The parameters-to-arity mapping
ψ : (a, b) −→ (m, n) (3.41)
is a partial surjection.
Here we list the lowest arities which can be obtained with different choices of (a, b).
m = 3
n = 2
⎫
⎪⎬
⎪⎭
:
⎛
⎜
⎝
a = 1
b = 2
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 3
b = 6
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 5
b = 10
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 7
b = 14
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 9
b = 18
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 11
b = 22
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 13
b = 26
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 15
b = 30
⎞
⎟
⎠ ;
m = 4
n = 2
⎫
⎪⎬
⎪⎭
:
⎛
⎜
⎝
a = 1
b = 3
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 4
b = 6
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 4
b = 12
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 10
b = 15
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 7
b = 21
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 16
b = 24
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 10
b = 30
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 22
b = 33
⎞
⎟
⎠ ;
m = 4
n = 3
⎫
⎪⎬
⎪⎭
:
⎛
⎜
⎝
a = 2
b = 3
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 2
b = 6
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 8
b = 12
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 14
b = 21
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 8
b = 24
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 20
b = 30
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 11
b = 33
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 14
b = 42
⎞
⎟
⎠ ;
m = 5
n = 2
⎫
⎪⎬
⎪⎭
:
⎛
⎜
⎝
a = 1
b = 4
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 9
b = 12
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 5
b = 20
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 21
b = 28
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 9
b = 36
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 33
b = 44
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 13
b = 52
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 45
b = 60
⎞
⎟
⎠ ;
m = 5
n = 3
⎫
⎪⎬
⎪⎭
:
⎛
⎜
⎝
a = 3
b = 4
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 3
b = 12
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 15
b = 20
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 7
b = 28
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 27
b = 36
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 39
b = 52
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 15
b = 60
⎞
⎟
⎠ ,
⎛
⎜
⎝
a = 51
b = 68
⎞
⎟
⎠ .
Although it has not been possible to derive a general formula for ψ (a, b), this can be done in some
particular cases
Proposition 3.35. 1) In the limiting cases (a = 1, b − 1) we have
ψ (1, b) = (b + 1, 2) , ψ (b − 1, b) = (b + 1, 3) . (3.42)
2) If a | b, then
ψ (a, ad) = d + 1, min
l
loga (ld + 1) + 1 , (3.43)
where l is the smallest integer for which log takes its minimal integer value.
3) If gcd (a, b) = d, then
ψ (a, b) = ψ (a0d, b0d) = b0 + 1, min
l
loga l
b0
a0
+ 1 + 1 , (3.44)
with the same l.
Proof. All the statements follow directly from (3.7)–(3.8).
In our approach, the concrete choice of operations (3.5)–(3.6) inside a conguence class [[a]]b gives
Assertion 3.36. The number of additions in the polyadic ring Z
[a,b]
(m,n) is greater that the number of
mutiplications m > n with any choice of (a, b).
Also, not all pairs (a, b) are allowed due to (3.7)–(3.8). We list the forbidden (a, b) for b ≤ 20
b = 4 : a = 2;
b = 8 : a = 2, 4, 6;
b = 9 : a = 3, 6;
b = 12 : a = 2, 6, 10;
b = 16 : a = 2, 4, 6, 8, 12, 14;
b = 18 : a = 3, 6, 12, 15;
b = 20 : a = 2, 6, 10, 14, 18. (3.45)
p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS Vol. 9 No. 4 2017
11. POLYADIC INTEGER NUMBERS 277
The characterization of the fixed congruence class [[a]]b and the corresponding (m, n)-ring of
polyadic integer numbers Z
[a,b]
(m,n) can be done in terms of the shape invariants I, J ∈ Z+ defined
uniquely by (TABLE 3 in [4])
I = I[a,b]
m = (m − 1)
a
b
, J = J[a,b]
n =
an − a
b
. (3.46)
Obviously, in the binary case, when m = n = 2 (a = 0, b = 1) both shape invariants vanish, I = J =
0. Nevertheless, there exist “partially” binary cases, when only n = 2 and m = 2, while J is nonzero, for
instance in Z
[6,10]
(6,2) we have I = J = 3. In Example 3.3 for Z
[3,4]
(5,3) we have I = 3, J = 6. In the limiting
cases (3.15)–(3.16) we have, in general, for a fixed b
I
[1,b]
b+1 = 1, J
[1,b]
2 = 0, (3.47)
I
[b−1,b]
b+1 = b − 1, J
[b−1,b]
3 = (b − 1) (b − 2) . (3.48)
Thus, one can classify and distinguish the limiting cases of the congruence classes in terms of the
invariants and their manifest form (3.47)–(3.48).
4. FINITE POLYADIC RINGS
Now we present a special method of constructing a finite nonderived polyadic ring by combining the
“external” and “internal” methods. Let us “apply” 1) to 2), such that instead of (3.4), we introduce
the finite polyadic ring Z(m,n) cZ, where Z(m,n) is defined in (3.9). However, if we directly consider the
“double” class {a + bk + cl} and fix a and b, then the factorization by cZ will not give closed operations
for arbitrary c.
Assertion 4.1. If the finite polyadic ring Z
[a,b]
(m,n)
cZ has q elements, then
c = bq. (4.1)
Proof. It follows from (4.1), that the “double” class remains in [[a]]b.
Remark 4.2. The representatives x
[a,b]
k = a + bk belong to a (m, n)-ring with the polyadic opera-
tions (3.5)–(3.6), while the notions of subtraction, division, modulo and remainder are defined
for binary operations. Therefore, we cannot apply the standard binary modular arithmetic to x
[a,b]
k
directly, but we can define the equivalence relations and corresponding class operations in terms
of k.
4.1. Secondary Congruence Classes
On the set of the “double” classes {a + bk + bql}, k, l ∈ Z and fixed b ∈ N and a = 0, . . . , b − 1 we
define the equivalence relation
k
∼ by
{a + bk1 + bql1}
k
∼ {a + bk2 + bql2} =⇒ k1 − k2 = ql, l, l1, l2 ∈ Z. (4.2)
Proposition 4.3. The equivalence relation
k
∼ is a congruence.
Proof. It follows from the binary congruency of k’s such that k1 ≡ k2 (mod q) which is just a rewritten
form of the last condition of (4.2).
So now we can factorize Z
[a,b]
(m,n) by the congruence
k
∼ and obtain
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12. 278 DUPLIJ
Definition 4.4. A secondary (equivalence) class of a polyadic integer x
[a,b]
k = a + bk ∈ Z
[a,b]
(m,n)
“modulo” bq (with q being the number of representatives x
[a,b]
k , for fixed b ∈ N and 0 ≤ a ≤ b − 1)
is
x
[a,b]
k
bq
= {{(a + bk) + bql} | l ∈ Z, q ∈ N, 0 ≤ k ≤ q − 1} . (4.3)
Remark 4.5. If the binary limit is given by a = 0, b = 1 and Z
[0,1]
(2,2) = Z, then the secondary class
becomes the ordinary class (3.1).
If the values of the parameters a, b, q are clear from the context, we denote the secondary class
representatives by an integer with two primes, as follows x
[a,b]
k
bq
≡ xk ≡ x .
Example 4.6. a) Let a = 3, b = 5, then for q = 4 elements we have the secondary classes with
k = 0, 1, 2, 3 (the corresponding binary limits are in brackets)
x
[3,5]
k
20
= 3 , 8 , 13 , 18 =
⎧
⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
3 = {. . . − 17, 3, 23, 43, 63, . . .} ,
8 = {. . . − 12, 8, 28, 48, 68, . . .} ,
13 = {. . . − 7, 13, 33, 53, 73, . . .} ,
18 = {. . . − 2, 18, 38, 58, 78, . . .} ,
(4.4)
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
[[k]]4 = 0 , 1 , 2 , 3 =
⎧
⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
0 = {. . . − 4, 0, 4, 8, 12, . . .} ,
1 = {. . . − 3, 1, 5, 9, 13, . . .} ,
2 = {. . . − 2, 2, 6, 10, 14, . . .} ,
3 = {. . . − 1, 3, 7, 11, 15, . . .} .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(4.5)
b) For a = 3, b = 6 and for 4 elements and k = 0, 1, 2, 3
x
[3,6]
k
24
= 3 , 9 , 15 , 21 , [[k]]4 = 0 , 1 , 2 , 3 . (4.6)
c) If a = 4, b = 5, for 3 elements and k = 0, 1, 2 we get
x
[4,5]
k
15
= 4 , 9 , 14 , [[k]]3 = 0 , 1 , 2 . (4.7)
The crucial difference between these sets of classes are: 1) they are described by rings of
different arities determined by (3.7) and (3.8); 2) some of them are fields.
4.2. Finite Polyadic Rings of Secondary Classes
Now we determine the operations between secondary classes. The most significant difference with
the binary class operations (3.2)–(3.3) is the fact that secondary classes obey nonderived polyadic
operations.
Proposition 4.7. The set {xk} of q secondary classes k = 0, . . . , q − 1 (with the fixed a, b) can be
endowed with the following commutative m-ary addition
xkadd
= νm xk1
, xk1
, . . . , xkm
, (4.8)
kadd ≡ (k1 + k2 + . . . + km) + I[a,b]
m (mod q) (4.9)
and commutative n-ary multiplication
xkmult
= μn xk1
, xk1
, . . . , xkn
, (4.10)
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13. POLYADIC INTEGER NUMBERS 279
kmult ≡ an−1
(k1 + k2 + . . . + kn) + an−2
b (k1k2 + k2k3 + . . . + kn−1kn) + . . .
+bn−1
k1 . . . kn + J[a,b]
n (mod q) , (4.11)
which satisfy the polyadic distributivity, and the shape invariants I
[a,b]
m , J
[a,b]
n are defined in
(3.46).
Proof. This follows from the definition of the secondary class (4.3) and manifest form of the “underlying”
polyadic operations (3.5)–(3.6), which are commutative and distributive.
Remark 4.8. The binary limit is given by a = 0, b = 1 and m = n = 2, I
[a,b]
m = J
[a,b]
n = 0, such
that the secondary class becomes the ordinary congruence class xk → k , obeying the standard
binary class operations (3.2)–(3.3), which in terms of k are kadd ≡ (k1 + k2) (mod q), kmult ≡
(k1k2) (mod q).
Definition 4.9. The set of secondary classes (4.3) equipped with operations (4.8), (4.10) is
denoted by
Z(m,n) (q) ≡ Z
[a,b]
(m,n) (q) = Z
[a,b]
(m,n) (bq) Z = xk | νm, μn , (4.12)
and is a finite secondary class (m, n)-ring of polyadic integer numbers Z(m,n) ≡ Z
[a,b]
(m,n). The value
q (the number of elements) is called its order.
Informally, Z(m,n) = Z(m,n) (∞). First, note that the constructed finite (m, n)-rings (4.12) have a
much richer structure and exotic properties which do not exist in the binary finite residue class rings
(3.4), and, in general, they give many concrete examples for possible different kinds polyadic rings. One
of such “non-binary” properties is the availability of several units (for odd multiplicative arity n), and
moreover sometimes all ring elements are units (such rings are “automatically” fields, see below).
Example 4.10. a) In (5, 3)-ring Z
[3,4]
(4,3) (2) with 2 secondary classes both elements are units
(we mark units by subscript e) e1 = 3e = 3 , e2 = 7e = 7 , because they are both multiplicative
idempotents and satisfy the following ternary multiplication (cf. (2.4))
μ3 3 , 3 , 3 = 3 , μ3 3 , 3 , 7 = 7 , μ3 3 , 7 , 7 = 3 , μ3 7 , 7 , 7 = 7 . (4.13)
b) In the same way the ring Z
[5,6]
(7,3) (4) consists of only 4 units e1 = 5e , e2 = 11e , e3 = 17e , e4 =
23e , and no zero.
c) Equal arity rings of the same order may be not isomorphic. For instance, Z
[1,3]
(4,2) (2) consists
of unit e = 1e = 1 and zero z = 4z = 4 only, satisfying
μ2 1 , 1 = 1 , μ2 1 , 4 = 4 , μ2 4 , 4 = 4 , (4.14)
and therefore Z
[1,3]
(4,2) (2) is a field, because {1 , 4z } 4z is a (trivial) binary group, consisting of one
element 1e . However, Z
[4,6]
(4,2) (2) has the zero z = 4z = 4 , 10 and has no unit, because
μ2 4 , 4 = 4 , μ2 4 , 10 = 4 , μ2 10 , 10 = 4 , (4.15)
so that Z
[4,6]
(4,2) (2) is not a field, because of the last relation (nilpotency of 10 ). Their additive 4-ary
groups are also not isomorphic (which is easy to show). However, Z
[1,3]
(4,2) (2) and Z
[4,6]
(4,2) (2) have the
same arity and order.
Recalling Assertion 3.34, we conclude more concretely:
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14. 280 DUPLIJ
Assertion 4.11. For a fixed arity shape (m, n), there can be non-isomorphic secondary class
polyadic rings Z(m,n) (q) of the same order q, which describe different binary residue classes [[a]]b.
A polyadic analog of the characteristic can be introduced, when there exist both a unit and zero in a
finite ring. Recall, that if R is a finite binary ring with unit1 and zero 0, then its characteristic is defined
as a smallest integer χ, such that
⎛
⎝
χ
1 + 1+, . . . , +1
⎞
⎠ = χ · 1 = 0. (4.16)
This means that the “number” of unit additions being χ − 1 produces zero. The same is evident for any
other element x ∈ R, because x = x · 1.
Definition 4.12. For the finite polyadic ring Z(m,n) (q) which contains both the unit e and the zero
z, a polyadic characteristic χp is defined as a smallest additive polyadic power (2.2) of e which
is equal to zero
e χp +m = z. (4.17)
In the binary limit, obviously, χp = χ − 1. A polyadic analog of the middle term in (2.7) can be
obtained by using the polyadic distributivity and (3.5) as
e χp +m = e χp(m−1)+1 ×n . (4.18)
In TABLE 1 we present the parameters-to-arity mapping ψ
[a,b]
(m,n) (3.41) together with the polyadic
characteristics of those finite secondary class rings Z
[a,b]
(m,n) (q) which contain both unit(s) and zero, and
which have order less or equal than 10 for b ≤ 6.
Now we turn to the question of which secondary classes can be described by polyadic finite fields.
5. FINITE POLYADIC FIELDS
Let us consider the structure of the finite secondary class rings Z
[a,b]
(m,n) (q) in more detail and determine
which of them are polyadic fields.
Proposition 5.1. A finite polyadic ring Z
[a,b]
(m,n)
(q) is a secondary class finite (m, n)-field F
[a,b]
(m,n)
(q)
if all its elements except z (if it exists) are polyadically multiplicative invertible having a unique
querelement.
Proof. In both cases {{xk} | μn} and {{xk z} | μn} are commutative and cancellative n-ary groups,
which follows from the concrete form of multiplication (4.10). Therefore, according to Definition 2.1, in
such a case Z
[a,b]
(m,n)
(q) becomes a polyadic field F
[a,b]
(m,n)
(q).
5.1. Abstract Finite Polyadic Fields
In the binary case [1] the residue (congruence) class ring (3.4) with q elements Z qZ is a
congruence class (non-extended) field, if its order q = p is a prime number, such that F (p) =
[[a]]p | + , · , 0 , 1 , a = 0, 1, . . . , p − 1. Because all non-extended binary fields of a fixed prime
order p are isomorphic each other and, in tern, isomorphic to the congruence class field F (p), it is natural
to study them in a more “abstract” way, i.e. without connection to a specific congruence class structure.
This can be achieved by consideration of the one-to-one onto mapping from the congruence class to its
representative which preserves the field (ring) structure and provides operations (binary multiplication
and addition with ordinary 0 and 1) by modulo p. In other words, the mapping Φp [[a]]p = a is an
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15. POLYADIC INTEGER NUMBERS 281
Table 1. Polyadic characteristics χp for the finite secondary class (m, n)-rings Z
[a,b]
m,n (q) of order 2 ≤ q ≤ 10
for 2 ≤ b ≤ 6. The orders q which do not give fields are slanted.
a b 2 3 4 5 6
1
m = 3
n = 2
q=3, χp = 1
q=5, χp = 2
q=7, χp = 3
q=9 , χp = 4
m = 4
n = 2
q=2, χp = 1
q=4 , χp = 1
q=5, χp = 3
q=7, χp = 2
q=8 , χp = 5
q=10 , χp = 3
m = 5
n = 2
q=3, χp = 2
q=5, χp = 1
q=7, χp = 5
q=9 , χp = 2
m = 6
n = 2
q=2, χp = 1
q=3, χp = 1
q=4 , χp = 3
q=6 , χp = 1
q=7, χp = 4
q=8 , χp = 3
q=9 , χp = 7
m = 7
n = 2
q=5, χp = 4
q=7, χp = 1
2
m = 4
n = 3
q=2, χp = 1
q=4 , χp = 1
q=5, χp = 3
q=7, χp = 2
q=8 , χp = 5
q=10 , χp = 3
m = 6
n = 5
q=2, χp = 1
q=3, χp = 1
q=4 , χp = 3
q=6 , χp = 1
q=7, χp = 4
q=8 , χp = 3
q=9 , χp = 7
m = 4
n = 3
q=5, χp = 3
q=7, χp = 2
q=10 , χp = 3
3
m = 5
n = 3
q=3, χp = 2
q=5, χp = 1
q=7, χp = 5
q=9 , χp = 2
m = 6
n = 5
q=2, χp = 1
q=3, χp = 1
q=4 , χp = 3
q=6 , χp = 1
q=7, χp = 4
q=8 , χp = 3
m = 3
n = 2
q=5, χp = 2
q=7, χp = 3
4
m = 6
n = 3
q=2, χp = 1
q=3, χp = 1
q=4 , χp = 3
q=6 , χp = 1
q=7, χp = 4
q=8 , χp = 3
q=9 , χp = 7
m = 4
n = 2
q=5, χp = 3
q=7, χp = 2
q=10 , χp = 3
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16. 282 DUPLIJ
isomorphism of binary fields Φp : F (p) → F (p), where F (p) = {{a} | +, ·, 0, 1}mod p is an “abstract”
non-extended (prime) finite field of order p (or Galois field GF (p)).
In a similar way, we introduce a polyadic analog of the “abstract” binary non-extended (prime) finite
fields. Let us consider the set of polyadic integer numbers {xk} ≡ x
[a,b]
k = {a + bk} ∈ Z
[a,b]
(m,n), b ∈ N
and 0 ≤ a ≤ b − 1, 0 ≤ k ≤ q − 1, q ∈ N, which obey the operations (3.5)–(3.6). The polyadic version
of the prime finite field F (p) of order p (or Galois field GF (p)) is given by
Definition 5.2. The “abstract” non-extended (prime) finite (m, n)-field of order q is
F(m,n) (q) ≡ F
[a,b]
(m,n) (q) = {{a + bk} | νm, μn}mod bq , (5.1)
if {{xk} | νm}mod bq is an additive m-ary group, and {{xk} | μn}mod bq (or, when zero z exists,
{{xk z} | μn}mod bq) is a multiplicative n-ary group.
Then we define a one-to-one onto mapping from the secondary congruence class to its representative
by Φ
[a,b]
q x
[a,b]
k
bq
= x
[a,b]
k and arrive at the following
Proposition 5.3. The mapping Φ
[a,b]
q : F
[a,b]
(m,n) (q) → F
[a,b]
(m,n) (q) is a polyadic ring homomorphism
(being, in fact, an isomorphism) and satisfies (here we use the “prime” notations)
Φ[a,b]
q νm x1, x2, . . . , xm = νm Φ[a,b]
q (x1) , Φ[a,b]
q (x2) , . . . , Φ[a,b]
q (xm) , (5.2)
Φ[a,b]
q μn x1, x2, . . . , xn = μn Φ[a,b]
q (x1) , Φ[a,b]
q (x2) , . . . , Φ[a,b]
q (xn) . (5.3)
Proof. This follows directly from (3.5)–(3.6) and (4.8)–(4.10). Obviously, a mapping defined in this way
governs the polyadic distributivity, and therefore Φ
[a,b]
q is a ring homomorphism, or, more exactly, a 1-
place heteromorphism for m-ary addition together with n-ary multiplication (see [4]). Because xk → xk
is one-to-one for any fixed 0 ≤ k ≤ q − 1, Φ
[a,b]
q is an isomorphism.
In TABLE 2 we present the “abstract” non-extended polyadic finite fields F
[a,b]
(m,n) (q) of lowest arity
shape (m, n) and orders q. The forbidden pairs (a, b) for F
[a,b]
(m,n) (q) coincide with ones for polyadic rings
Z
[a,b]
(m,n) listed in (3.45).
5.2. Multiplicative Structure
In the multiplicative structure the following crucial differences between the binary finite fields and
F(m,n) (q) can be outlined.
Remark 5.4. The order of a non-extended finite polyadic field may not be prime (e.g., F
[1,2]
(3,2) (4),
F
[3,4]
(5,3) (8), F
[2,6]
(4,3) (9)), and may not even be a power of a prime binary number (e.g. F
[5,6]
(7,3) (6),
F
[3,10]
(11,5) (10)), see TABLE 3.
Remark 5.5. The polyadic characteristic χp of a non-extended finite polyadic field can have
values such that χp + 1 (corresponding in the binary case to the ordinary characteristic χ) can
be nonprime (TABLE 1).
Assertion 5.6. If a secondary class x
[a,b]
k
bq
contains no zero, it can be isomorphic to the
“abstract” zeroless finite polyadic field F
[a,b]
(m,n) (q).
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17. POLYADIC INTEGER NUMBERS 283
Table 2. Content and arities of the secondary class finite polyadic rings Z
[a,b]
(m,n) (q) and the corresponding
simple finite polyadic (m, n)-fields F
[a,b]
m,n (q) of order 2 ≤ q ≤ 4 (framed) for 2 ≤ b ≤ 6. The subscripts e and z
mark those secondary classes (two primes are omitted) which play the roles of polyadic unit and polyadic zero
respectively. The double frames denote the finite polyadic fields containing both a unit and zero. The last line
in a cell (corresponding to a fixed congruence class [[a]]b) gives the allowed orders of finite polyadic fields for
5 ≤ q ≤ 10, and bold numbers mark the orders of such fields which contain both unit(s) and zero.
a b 2 3 4 5 6
1
m = 3
n = 2
1e,3
1e,3z,5
1e,3,5,7
q=5,7,8
m = 4
n = 2
1e,4z
1e,4,7
1e,4z,7,10
q=5,7,9
m = 5
n = 2
1e,5
1e,5,9z
1e,5,9,13
q=5,7,8
m = 6
n = 2
1e,6z
1e,6z,11
1e,6,11,16z
q=5,7
m = 7
n = 2
1e,7
1e,7,13
1e,7,13,19
q=5,6,7,8,9
2
m = 4
n = 3
2z,5e
2,5,8e
2,5e,8z,11e
q=5,7,9
m = 6
n = 5
2z,7e
2e,7,12z
2,7e,12z,17e
q=5,7
m = 4
n = 3
2,8z
2,8e,14
2,8z,14,20
q=5,7,9
3
m = 5
n = 3
3e,7e
3z,7e,11e
3,7e,11,15e
q=5,6,7,8
m = 6
n = 5
3e,8z
3z,8e,13e
3e,8z,13e,18
q=5,7
m = 3
n = 2
3,9e
3,9z,15
3,9e,15,21
q=5,7,8
4
m = 6
n = 3
4z,9e
4e,9z,14e
4z,9e,14,19e
q=5,7
m = 4
n = 2
4z,10
4,10e,16
4,10,16z,22
q=5,7,9
5
m = 7
n = 3
5e,11e
5,11,17e
5e,11e,17e,23e
q=5,6,7,8,9
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18. 284 DUPLIJ
Zeroless fields are marked by one frame in TABLE 2. There exist finite polyadic fields with more than
one unit, and also all elements can be units. Such cases are marked in TABLE 3 by subscripts which
indicate the number of units.
Denote the Abelian finite multiplicative n-ary group of F
[a,b]
(m,n) (q) by G
[a,b]
n (q).
Example 5.7. 1) The finite (6, 3)-field F
[4,5]
(6,3) (3) of order 3 has two units {4, 14} ≡ {4e, 14e} and zero
9 ≡ 9z, and its multiplicative 3-ary group G
[4,5]
3 (2) is
μ3 [4, 4, 4] = 4, μ3 [4, 4, 14] = 14, μ3 [4, 14, 14] = 4, μ3 [14, 14, 14] = 14.
2) In F
[5,6]
(7,3) (4) of order 4 all the elements {5, 11, 17, 23} ≡ {5e, 11e, 17e, 23e} are units (see (2.4)),
because for its 3-ary group G
[5,6]
3 (4) we have
μ3 [5, 5, 5] = 5, μ3 [11, 11, 11] = 11, μ3 [17, 17, 17] = 17, μ3 [23, 23, 23] = 23,
μ3 [5, 5, 11] = 11, μ3 [5, 5, 17] = 17, μ3 [5, 5, 23] = 23,
μ3 [11, 11, 5] = 5, μ3 [11, 11, 17] = 17, μ3 [11, 11, 23] = 23,
μ3 [17, 17, 5] = 5, μ3 [17, 17, 11] = 11, μ3 [17, 17, 23] = 23,
μ3 [23, 23, 5] = 5, μ3 [23, 23, 11] = 11, μ3 [23, 23, 17] = 17.
In general, n-ary groups may contain no units (and multiplicative idempotents) at all, and invertibility
is controlled in another way, by querelements: each element of any n-ary group should be (uniquely)
“quereable” (2.5). In case of (m, n)-fields both m-ary additive group and n-ary multiplicative group
G
[a,b]
n (q) can be of this kind. By analogy with zeroless-nonunital rings we have
Definition 5.8. A polyadic field F(m,n) is called zeroless-nonunital, if it contains no zero and no
unit.
Assertion 5.9. The zeroless-nonunital polyadic fields are totally (additively and multiplica-
tively) nonderived.
Proposition 5.10. 1) If gcd (bq) = 1, then F
[a,b]
(m,n) (q) is zeroless. 2) The zero can exist only, if
gcd (bq) = 1 and the field order q = p is prime.
Proof. It follows directly from the definition of the polyadic zero (2.3) and (4.9).
Let us consider examples of zeroless-nonunital finite fields of polyadic integer numbers F
[a,b]
(m,n) (q).
Example 5.11. 1) The zeroless-nonunital polyadic finite fields having lowest |a + b| are, e.g.,
F
[3,8]
(9,3) (2), F
[3,8]
(9,3) (4), F
[5,8]
(9,3) (4), F
[5,8]
(9,3) (8), also F
[4,9]
(10,4) (3), F
[4,9]
(10,4) (9), and F
[7,9]
(10,4) (3), F
[7,9]
(10,4) (9).
2) The multiplication of the zeroless-nonunital (9, 3)-field F
[5,8]
(9,3) (2) is
μ3 [5, 5, 5] = 13, μ3 [5, 5, 13] = 5, μ3 [5, 13, 13] = 13, μ3 [13, 13, 13] = 5.
Using (2.2) we find the (unique) multiplicative querelements ¯5 = 13, 13 = 5. The addition of
F
[5,8]
(9,3) (2) is
ν9 59
= 13, ν9 58
, 13 = 5, ν9 57
, 132
= 13, ν9 56
, 133
= 5, ν9 55
, 134
= 13,
ν9 54
, 135
= 5, ν9 53
, 136
= 13, ν9 52
, 137
= 5, ν9 5, 138
= 13, ν9 139
= 5.
The additive (unique) querelements are ˜5 = 13, 13 = 5. So all elements are additively and
multiplicatively querable (polyadically invertible), and therefore ν9 is 9-ary additive group
operation and μ3 is 3-ary multiplicative group operation, as it should be for a field. Because it
contains no unit and no zero, F
[5,8]
(9,3) (2) is actually a zeroless-nonunital finite (9, 3)-field of order
2.
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19. POLYADIC INTEGER NUMBERS 285
Other zeroless-nonunital finite polyadic fields are marked by frames in TABLE 3.
Remark 5.12. The absence of zero does not guarantee that a (m, n)-ring R
[a,b]
(m,n) (q) is a field. For
that, both [[a]]b | νm and [[a]]b | μn have to be polyadic groups.
Example 5.13. The (4, 3)-ring R
[2,3]
(4,3) (6) is zeroless, and [[3]]4 | ν4 is its 4-ary additive group
(each element has a unique additive querelement). Despite each element of [[2]]3 | μ3 having
a querelement, it is not a multiplicative 3-ary group, because for the two elements 2 and 14 we
have nonunique querelements
μ3 [2, 2, 5] = 2, μ3 [2, 2, 14] = 2, μ3 [14, 14, 2] = 14, μ3 [14, 14, 11] = 14. (5.4)
The conditions on the congruence classes [[a]]b and the invariants I, J (3.46), which give the same
arity structure are given in [4]. Note, that there exist polyadic fields of the same arities (m, n) and the
same order q which are not isomorphic (in contrast with what is possible in the binary case).
Example 5.14. The polyadic (9, 3)-fields corresponding to the congruence classes [[5]]8 and [[7]]8
are not isomorphic for orders q = 2, 4, 8 (see TABLE 3). Despite both being zeroless, the first
F
[5,8]
(9,3) (q) are nonunital, while the second F
[7,8]
(9,3) (q) has two units, which makes an isomorphism
impossible.
Recall [1], that in a (non-extended, prime) finite binary field F (p), the order of an element x ∈ F (p)
is defined as a smallest integer λ such that xλ = 1. Obviously, the set of fixed order elements forms a
cyclic subgroup Gλ of the multiplicative binary group of F (p), and λ | (p − 1). If λ = p − 1, such an
element is called a primitive (root), it generates all elements, and these exist in any finite binary field.
Moreover, any element of F (p) is idempotent xp = x, while all its nonzero elements satisfy xp−1 = 1
(Fermat’s little theorem). A non-extended (prime) finite field is fully determined by its order p (up to
isomorphism), and, moreover, any F (p) is isomorphic to Z pZ.
In the polyadic case, the situation is more complicated. Because the related secondary class structure
(5.1) contains parameters in addition to the number of elements q, the order of (non-extended) polyadic
fields may not be prime, or nor even a power of a prime integer (e.g. F
[5,6]
(7,3) (6) or F
[3,10]
(11,5) (10)). Also, as
was shown above, finite polyadic fields can be zeroless, nonunital and have many (or even all) units (see
TABLE 3). Therefore, we cannot use units in the definition of the element order. Instead, we propose an
alternative:
Definition 5.15. If an element of the finite polyadic field x ∈ F(m,n) (q) satisfies
x λp ×n = x, (5.5)
then the smallest such λp is called the idempotence polyadic order and denoted ord x = λp.
Obviously, λp = λ (see (2.1)).
Definition 5.16. The idempotence polyadic order λ
[a,b]
p of a finite polyadic field F
[a,b]
(m,n) (q) is
the maximum λp of all its elements, and we call such field λ
[a,b]
p -idempotent and denote
ord F
[a,b]
(m,n) (q) = λ
[a,b]
p .
In TABLE 3 we present the idempotence polyadic order λ
[a,b]
p for the (allowed) finite polyadic fields
F
[a,b]
(m,n) (q) (5.1) with 2 ≤ b ≤ 10 and order q ≤ 10.
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Definition 5.17. Denote by q∗ the number of nonzero distinct elements in F(m,n) (q)
q∗ =
⎧
⎨
⎩
q − 1, if ∃z ∈ F(m,n) (q)
q, if z ∈ F(m,n) (q) ,
(5.6)
which is called a reduced (field) order.
The second choice of (5.6) in the binary case is absent, because any commutative binary group (as
the additive group of a field) contains a zero (the identity of this group), and therefore any binary field has
a zero, which does not always hold for the m-ary additive group of F(m,n) (see Example 5.11).
Theorem 5.18. If a finite polyadic field F(m,n) (q) has an order q, such that q∗ = qadm
∗ = (n − 1) +
1 is n-admissible, then (for n ≥ 3 and one unit):
1) A sequence of the length q∗ (n − 1) built from any fixed element y ∈ F(m,n) (q) is neutral
μ(q∗)
n x, yq∗(n−1)
= x, ∀x ∈ F(m,n) (q) . (5.7)
2) Any element y satisfies the polyadic idempotency condition
y q∗ ×n = y, ∀y ∈ F(m,n) (q) . (5.8)
Proof. 1) Take a long n-ary product of the q∗ distinct nonzero elements x0 = μ
( )
n x1, x2, . . . , xq∗
,
such that q∗ can take only multiplicatively n-admissible values qadm
∗ , where ∈ N is a “number” of n-
ary multiplications. Then polyadically multiply each xi by a fixed element y ∈ F(m,n) (q) such that all q∗
elements μn xi, yn−1 will be distinct as well. Therefore, their product should be the same x0. Using
commutativity and associativity, we obtain
x0 = μ( )
n x1, x2, . . . , xq∗
= μ( )
n μn x1, yn−1
, μn x2, yn−1
, . . . , μn xq∗, yn−1
= μ(q∗)
n μ( )
n x1, x2, . . . , xq∗
, yq∗(n−1)
= μ(q∗)
n x0, yq∗(n−1)
. (5.9)
2) Insert into the formula obtained above x0 = y, then use (2.2) to get (5.8).
Finite polyadic fields F
[a,b]
(m,n) (q) having n-admissible reduced order q∗ = qadm
∗ = (n − 1) + 1 ( ∈ N)
(underlined in TABLE 3) are closest to the binary finite fields F (p) in their general properties: they are
half-derived, while if they contain a zero, they are fully derived. If q∗ = qadm
∗ , then F
[a,b]
(m,n)
(q) can be
nonunital or contain more than one unit (subscripts in TABLE 3).
Assertion 5.19. The finite fields F
[a,b]
(m,n) (q) of n-admissible reduced order q∗ = qadm
∗ cannot have
more than one unit and cannot be zeroless-nonunital.
Assertion 5.20. If q∗ = qadm
∗ , and F
[a,b]
(m,n) (q) is unital zeroless, then the reduced order q∗ is the
product of the idempotence polyadic field order λ
[a,b]
p = ord F
[a,b]
(m,n) (q) and the number of units κe
(if a b and n ≥ 3)
q∗ = λ[a,b]
p κe. (5.10)
Let us consider the structure of the multiplicative group G
[a,b]
n (q∗) of F
[a,b]
(m,n) (q) in more detail. Some
properties of commutative cyclic n-ary groups were considered for particular relations between orders
and arity. Here we have: 1) more parameters and different relations between these, the arity and order; 2)
the (m, n)-field under consideration, which leads to additional restrictions. In such a way exotic polyadic
groups and fields arise which have unusual properties that have not been studied before.
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22. 288 DUPLIJ
Definition 5.21. An element xprim ∈ G
[a,b]
n (q∗) is called n-ary primitive, if its idempotence order
is
λp = ord xprim = q∗. (5.11)
Then, all λp polyadic powers x
1 ×n
prim , x
2 ×n
prim , . . . , x
q∗ ×n
prim ≡ xprim generate other distinct elements,
and so G
[a,b]
n (q∗) is a finite cyclic n-ary group generated by xprim, i.e. G
[a,b]
n (q∗) = x
i ×n
prim | μn . We
denote a number primitive elements in F
[a,b]
(m,n) (q) by κprim.
Assertion 5.22. For zeroless F
[a,b]
(m,n) (q) and prime order q = p, we have λ
[a,b]
p = q, and G
[a,b]
n (q) is
indecomposable (n ≥ 3).
Example 5.23. The smallest 3-admissible zeroless polyadic field is F
[2,3]
(4,3) (3) with the unit e = 8
and two 3-ary primitive elements 2, 5 having 3-idempotence order ord 2 = ord 5 = 3, so κprim = 2
, because
2 1 ×3 = 8, 2 2 ×3 = 5, 2 3 ×3 = 2, 5 1 ×3 = 8, 5 2 ×3 = 2, 5 3 ×3 = 5, (5.12)
and therefore G
[2,3]
3 (3) is a cyclic indecomposable 3-ary group.
Assertion 5.24. If F
[a,b]
(m,n) (q) is zeroless-nonunital, then every element is n-ary primitive, κprim =
q, also λ
[a,b]
p = q (the order q can be not prime), and G
[a,b]
n (q) is a indecomposable commutative
cyclic n-ary group without identity (n ≥ 3).
Example 5.25. The (10, 7)-field F
[5,9]
(10,7) (9) is zeroless-nonunital, each element (has λp = 9) is
primitive and generates the whole field, and therefore κprim = 9, thus the 7-ary multiplicative
group G
[5,9]
7 (9) is indecomposable and without identity.
The structure of G
[a,b]
n (q∗) can be extremely nontrivial and may have no analogs in the binary case.
Assertion 5.26. If there exists more than one unit, then:
1) If G
[a,b]
n (q∗) can be decomposed on its n-ary subroups, the number of units κe coincides with
the number of its cyclic n-ary subgroups G
[a,b]
n (q∗) = G1 ∪ G2 . . . ∪ Gke which do not intersect
Gi ∩ Gj = ∅, i, j = i = 1, . . . , κe, i = j.
2) If a zero exists, then each Gi has its own unit ei, i = 1, . . . , κe.
3) In the zeroless case G
[a,b]
n (q) = G1 ∪ G2 . . . ∪ Gke ∪ E (G), where E (G) = {ei} is the split-off
subgroup of units.
Example 5.27. 1) In the (9, 3)-field F
[5,8]
(9,3) (7) there is a single zero z = 21 ≡ 21z and two units
e1 = 13 ≡ 13e, e2 = 29 ≡ 29e, and so its multiplicative 3-ary group G
[5,8]
3 (6) = {5, 13, 29, 37, 45, 53}
consists of two nonintersecting (which is not possible in the binary case) 3-ary cyclic subgroups
G1 = {5, 13e, 45} and G2 = {29e, 37, 53} (for both λp = 3)
G1 = 5 1 ×3 = 13e, 5 2 ×3 = 45, 5 3 ×3 = 5 , ¯5 = 45, 45 = 5,
G2 = 37 1 ×3 = 29e, 37 2 ×3 = 53, 37 3 ×3 = 37 , 37 = 53, 53 = 37.
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23. POLYADIC INTEGER NUMBERS 289
All nonunital elements in G
[5,8]
3 (6) are (polyadic) 1-reflections, because 5 1 ×3 = 45 1 ×3 = 13e and
37 1 ×3 = 53 1 ×3 = 29e, and so the subgroup of units E (G) = {13e, 29e} is unsplit E (G) ∩ G1,2 = ∅.
2) For the zeroless F
[7,8]
(9,3) (8), its multiplicative 3-group G
[5,8]
3 (6) = {7, 15, 23, 31, 39, 47, 55, 63}
has two units e1 = 31 ≡ 31e, e2 = 63 ≡ 63e, and it splits into two nonintersecting nonunital cyclic
3-subgroups (λp = 4 and λp = 2) and the subgroup of units
G1 = 7 1 ×3 = 23, 7 2 ×3 = 39, 7 3 ×3 = 55, 7 4 ×3 = 4 , ¯7 = 55, 55 = 7, 23 = 39, 39 = 23,
G2 = 15 1 ×3 = 47, 15 2 ×3 = 15 , 15 = 47, 47 = 15,
E (G) = {31e, 63e} .
There are no μ-reflections, and so E (G) splits out E (G) ∩ G1,2 = ∅.
If all elements are units E (G) = G
[a,b]
n (q), then, obviously, this group is 1-idempotent, and λp = 1.
Assertion 5.28. If F
[a,b]
(m,n) (q) is zeroless-nonunital, then there no n-ary cyclic subgroups in
G
[a,b]
n (q).
The subfield structure of F
[a,b]
(m,n) (q) can coincide with the corresponding subgroup structure of the
multiplicative n-ary group G
[a,b]
n (q∗), only if its additive m-ary group has the same subgroup structure.
However, additive m-ary groups of all polyadic fields F
[a,b]
(m,n) (q) have the same structure: they are cyclic
and have no proper m-ary subgroups, each element generates all other elements, i.e. it is a primitive
root. Therefore, we arrive at
Theorem 5.29. The polyadic field F
[a,b]
(m,n) (q), being isomorphic to the (m, n)-field of polyadic
integer numbers Z
[a,b]
(m,n) (q), has no any proper subfield.
In this sense, F
[a,b]
(m,n) (q) can be named a prime polyadic field.
6. CONCLUDING REMARKS
Recall that any binary finite field has an order which is a power of a prime number q = pr (its
characteristic), and all such fields are isomorphic and contain a prime subfield GF (p) of order p which
is isomorphic to the congruence (residue) class field Z pZ [1].
Conjecture 6.1. A finite (m, n)-field (with m > n) should contain a minimal subfield which is
isomorphic to one of the prime polyadic fields constructed above, and therefore F
[a,b]
(m,n) (q) can be
interpreted as a polyadic analog of GF (p).
This conjecture opens the promising possibility that the presented unusual properties of the polyadic
finite fields (which do not exist in binary fields) could have non-standard applications (e.g. in number
theory, cryptography and coding theory) and could lead to a specific polyadic version of the Galois theory.
ACKNOWLEDGEMENTS
The author would like to express his sincere thankfulness to Joachim Cuntz, Christopher Deninger,
Mike Hewitt, Maurice Kibler, Grigorij Kurinnoj, Daniel Lenz, Jim Stasheff, Alexander Voronov, and
Wend Werner for fruitful discussions.
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24. 290 DUPLIJ
APPENDIX. MULTIPLICATIVE PROPERTIES OF EXOTIC FINITE POLYADIC FIELDS
Here we list concrete examples of finite polyadic fields which have properties that are not possible in
the binary case (see TABLE 3). Only the multiplication of fields will be shown, because their additive part
is huge (many pages) for higher arities, and does not carry so much distinctive information.
1) The first exotic finite polyadic field which has a number of elements which is not a prime
number, or prime power (as it should be for a finite binary field) is F
[5,6]
(7,3) (6), which consists of 6
elements {5, 11, 17, 23, 29, 35}, q = 6. It is zeroless and contains two units {17, 35} ≡ {17e, 35e}, κe =
2, and each element has the idempotence polyadic order λp = 3, i.e. μ3 x7 = x, ∀x ∈ F
[5,6]
(7,3) (6). The
multiplication is
μ3 [5, 5, 5] = 17, μ3 [5, 5, 11] = 23, μ3 [5, 5, 17] = 29, μ3 [5, 5, 23] = 35, μ3 [5, 5, 29] = 5,
μ3 [5, 5, 35] = 11, μ3 [5, 11, 11] = 29, μ3 [5, 11, 17] = 35, μ3 [5, 11, 23] = 5, μ3 [5, 11, 29] = 11,
μ3 [5, 11, 35] = 17, μ3 [5, 17, 17] = 5, μ3 [5, 17, 23] = 11, μ3 [5, 17, 29] = 17, μ3 [5, 17, 35] = 23,
μ3 [5, 23, 23] = 17, μ3 [5, 23, 29] = 23, μ3 [5, 23, 35] = 29, μ3 [5, 29, 29] = 29, μ3 [5, 29, 35] = 35,
μ3 [5, 35, 35] = 5, μ3 [11, 11, 11] = 35, μ3 [11, 11, 17] = 5, μ3 [11, 11, 23] = 11, μ3 [11, 11, 29] = 17,
μ3 [11, 11, 35] = 23, μ3 [11, 11, 17] = 5, μ3 [11, 17, 17] = 11, μ3 [11, 17, 23] = 17, μ3 [11, 17, 29] = 23,
μ3 [11, 17, 35] = 29, μ3 [11, 17, 23] = 17, μ3 [11, 17, 29] = 23, μ3 [11, 17, 35] = 29, μ3 [11, 23, 23] = 23,
μ3 [11, 23, 29] = 29, μ3 [11, 23, 23] = 23, μ3 [11, 23, 35] = 35, μ3 [11, 29, 29] = 35, μ3 [11, 29, 35] = 5,
μ3 [11, 35, 35] = 11, μ3 [17, 17, 17] = 17, μ3 [17, 17, 23] = 23, μ3 [17, 17, 29] = 29, μ3 [17, 17, 35] = 35,
μ3 [17, 23, 23] = 29, μ3 [17, 23, 29] = 35, μ3 [17, 29, 29] = 5, μ3 [17, 29, 35] = 11, μ3 [17, 35, 35] = 17,
μ3 [23, 23, 23] = 35, μ3 [23, 23, 29] = 5, μ3 [23, 23, 35] = 11, μ3 [23, 29, 29] = 11, μ3 [23, 29, 35] = 17,
μ3 [23, 35, 35] = 23, μ3 [29, 29, 29] = 17, μ3 [29, 29, 35] = 23, μ3 [29, 35, 35] = 29, μ3 [35, 35, 35] = 35.
The multiplicative querelements are ¯5 = 29, 29 = 5, 11 = 23, 23 = 11. Because
5 1 ×3 = 17e, 5 2 ×3 = 29, 5 3 ×3 = 5, 29 1 ×3 = 17e, 29 2 ×3 = 5, 29 3 ×3 = 29,
11 1 ×3 = 35e, 11 2 ×3 = 23, 11 3 ×3 = 11, 23 1 ×3 = 35e, 23 2 ×3 = 11, 23 3 ×3 = 23,
the multiplicative 3-ary group G
[5,6]
(7,3) (6) consists of two nonintersecting cyclic 3-ary subgroups
G
[5,6]
(7,3) (6) = G1 ∪ G2, G1 ∩ G2 = ∅,
G1 = {5, 17e, 29} ,
G2 = {11, 23, 35e} ,
which is impossible for binary subgroups, as these always intersect in the identity of the binary group.
2) The finite polyadic field F
[5,6]
(7,3) (4) = {{5, 11, 17, 23} | ν7, μ3} which has the same arity shape as
above, but with order 4, has the exotic property that all elements are units, which follows from its
multiplication
μ3 [5, 5, 5] = 5, μ3 [5, 5, 11] = 11, μ3 [5, 5, 17] = 17, μ3 [5, 5, 23] = 23, μ3 [5, 11, 11] = 5,
μ3 [5, 11, 17] = 23, μ3 [5, 11, 23] = 17, μ3 [5, 17, 17] = 5, μ3 [5, 17, 23] = 11, μ3 [5, 23, 23] = 5,
μ3 [11, 11, 11] = 11, μ3 [11, 11, 17] = 17, μ3 [11, 11, 23] = 23, μ3 [11, 17, 17] = 11, μ3 [11, 17, 23] = 5,
μ3 [11, 23, 23] = 11, μ3 [17, 17, 17] = 17, μ3 [17, 17, 23] = 23, μ3 [17, 23, 23] = 17, μ3 [23, 23, 23] = 23.
3) Next we show by construction, that (as opposed to the case of binary finite fields) there exist non-
isomorphic finite polyadic fields of the same order and arity shape. Indeed, consider these two (9, 3)-fields
of order 2, that are F
[3,8]
(9,3) (2) and F
[7,8]
(9,3) (2). The first is zeroless-nonunital, while the second is zeroless
with two units, i.e. all elements are units. The multiplication of F
[3,8]
(9,3) (2) is
μ3 [3, 3, 3] = 11, μ3 [3, 3, 11] = 3, μ3 [3, 11, 11] = 11, μ3 [11, 11, 11] = 3,
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25. POLYADIC INTEGER NUMBERS 291
having the multiplicative querelements ¯3 = 11, 11 = 3. For F
[7,8]
(9,3) (2) we get the 3-group of units
μ3 [7, 7, 7] = 7, μ3 [7, 7, 15] = 15, μ3 [7, 15, 15] = 7, μ3 [15, 15, 15] = 15.
They have different idempotence polyadic orders ord F
[3,8]
(9,3) (2) = 2 and ord F
[7,8]
(9,3) (2) = 1. Despite their
additive m-ary groups being isomorphic, it follows from the above multiplicative structure, that it is not
possible to construct an isomorphism between the fields themselves.
4) The smallest exotic finite polyadic field with more than one unit is
F
[2,3]
(4,3)
(5) = {{2, 5, 8, 11, 14} | ν4, μ3} of order 5 with two units {11, 14} ≡ {11e, 14e} and the zero 5 ≡
5z. The presence of zero allows us to define the polyadic characteristic (4.17) which is 3 (see TABLE 1),
because the 3rd additive power of all elements is equal to zero
2 3 +4 = 8 3 +4 = 11
3 +4
e = 14
3 +4
e = 5z.
The additive querelements are
˜2 = 11e, ˜8 = 14e, 11e = 8, 14e = 2.
The idempotence polyadic order is ord F
[2,3]
(4,3) (5) = 2, because for nonunit and nonzero elements
2 2 ×3 = 2, 8 2 ×3 = 8,
and their multiplicative querelements are ¯2 = 8, ¯8 = 2. The multiplication is given by the cyclic 3-ary
group G
[2,3]
3 (4) = {{2, 8, 11, 14} | μ3} as:
μ3 [2, 2, 2] = 8, μ3 [2, 2, 8] = 2, μ3 [2, 2, 11] = 14, μ3 [2, 2, 14] = 11, μ3 [2, 8, 8] = 8,
μ3 [2, 8, 11] = 11, μ3 [2, 8, 14] = 14, μ3 [2, 11, 11] = 2, μ3 [2, 11, 14] = 8, μ3 [2, 14, 14] = 2,
μ3 [8, 8, 8] = 2, μ3 [8, 8, 11] = 14, μ3 [8, 8, 14] = 11, μ3 [8, 11, 11] = 8, μ3 [8, 11, 14] = 2,
μ3 [8, 14, 14] = 8, μ3 [11, 11, 11] = 11, μ3 [11, 11, 14] = 14, μ3 [11, 14, 14] = 11, μ3 [14, 14, 14] = 14.
We observe that, despite having two units, the cyclic 3-ary group G
[2,3]
3 (4) has no decomposition into
nonintersecting cyclic 3-ary subgroups.One of the reasons is that the polyadic field F
[5,6]
(7,3) (6) is zeroless,
while F
[2,3]
(4,3) (5) has a zero (see Assertion 5.26).
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