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
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
1) The document presents two lemmas showing that when dividing Fibonacci numbers or Lucas numbers by other numbers in their respective sequences, the quotients round to Lucas numbers and the remainders are Fibonacci or Lucas numbers.
2) It provides proofs of the lemmas using known properties of the Fibonacci and Lucas sequences.
3) The document notes that this quotient property does not hold for general recursive sequences defined similarly to the Fibonacci sequence.
This report summarizes recent work proving the fundamental lemma, which is an important step in Langlands' endoscopy theory. The fundamental lemma relates orbital integrals of a reductive group to those of its endoscopic groups. The report provides examples of how orbital integrals arise in counting problems for lattices and abelian varieties over finite fields. It also discusses how stable orbital integrals and their κ-sisters are used in the stable trace formula to relate traces of automorphic representations to orbital integrals.
1. Finite fields are algebraic structures that are both fields and finite sets. They have important applications in computer science, coding theory, and cryptography.
2. For each prime p and positive integer n, there exists a unique finite field of order pn, denoted as GF(pn).
3. GF(pn) contains subfields of order pm for each divisor m of n. The only subfields are those of order pm.
This document discusses finite fields and Galois fields. Some key points:
- A finite field, also called a Galois field, is denoted as GF(q) where the elements can take q different values. It has defined addition and multiplication operations with certain properties.
- Finite fields can be constructed with a prime number p of elements by performing arithmetic modulo-p. Larger finite fields GF(pm) can be constructed using irreducible polynomials over smaller finite fields.
- Polynomials over a finite field can be added, multiplied, and divided. Irreducible polynomials play an important role in constructing finite field extensions.
- The Galois field GF(2m) with 2m elements is constructed by
On Fractional Fourier Transform Moments Based On Ambiguity FunctionCSCJournals
The fractional Fourier transform can be considered as a rotated standard Fourier transform in general and its benefit in signal processing is growing to be known more. Noise removing is one application that fractional Fourier transform can do well if the signal dilation is perfectly known. In this paper, we have computed the first and second order of moments of fractional Fourier transform according to the ambiguity function exactly. In addition we have derived some relations between time and spectral moments with those obtained in fractional domain. We will prove that the first moment in fractional Fourier transform can also be considered as a rotated the time and frequency gravity in general. For more satisfaction, we choose five different types signals and obtain analytically their fractional Fourier transform and the first and second-order moments in time and frequency and fractional domains as well.
The document compares the low field electron transport properties in compounds of groups III-V semiconductors by solving the Boltzmann equation using an iterative technique. It calculates the temperature and doping dependencies of electron mobility in InP, InAs, GaP and GaAs. The electron mobility decreases with increasing temperature from 100K to 500K for each material due to increased electron-phonon scattering. Electron mobility also increases significantly with higher doping concentration at low temperatures. The iterative results show good agreement with other calculations and experiments. Electron mobility is highest in InAs and lowest in GaP at 300K, due to differences in their effective masses.
1) Fourier developed a theory of infinite series of sine and cosine functions to solve differential equations describing vibratory motion that had no solutions in terms of elementary functions. These series, now known as Fourier series, can represent any periodic function.
2) A Fourier series decomposes a periodic function into an infinite sum of sines and cosines of integer multiples of the fundamental frequency. The coefficients of the sines and cosines are determined from the function itself.
3) Under certain conditions on the periodic function and its derivatives, the Fourier series converges to the value of the function at points of continuity and averages the left and right limits at points of discontinuity.
This summary provides the key details from the document in 3 sentences:
The document investigates the structure of unital 3-fields, which are fields where addition requires 3 summands rather than the usual 2. It is shown that unital 3-fields are isomorphic to the set of invertible elements in a local ring R with Z2Z as the residual field. Pairs of elements in the 3-field are used to define binary operations that allow reducing the arity and connecting the 3-field to binary algebra. The structure of finite 3-fields is examined, proving properties like the number of elements being a power of 2.
1) The document presents two lemmas showing that when dividing Fibonacci numbers or Lucas numbers by other numbers in their respective sequences, the quotients round to Lucas numbers and the remainders are Fibonacci or Lucas numbers.
2) It provides proofs of the lemmas using known properties of the Fibonacci and Lucas sequences.
3) The document notes that this quotient property does not hold for general recursive sequences defined similarly to the Fibonacci sequence.
This report summarizes recent work proving the fundamental lemma, which is an important step in Langlands' endoscopy theory. The fundamental lemma relates orbital integrals of a reductive group to those of its endoscopic groups. The report provides examples of how orbital integrals arise in counting problems for lattices and abelian varieties over finite fields. It also discusses how stable orbital integrals and their κ-sisters are used in the stable trace formula to relate traces of automorphic representations to orbital integrals.
1. Finite fields are algebraic structures that are both fields and finite sets. They have important applications in computer science, coding theory, and cryptography.
2. For each prime p and positive integer n, there exists a unique finite field of order pn, denoted as GF(pn).
3. GF(pn) contains subfields of order pm for each divisor m of n. The only subfields are those of order pm.
This document discusses finite fields and Galois fields. Some key points:
- A finite field, also called a Galois field, is denoted as GF(q) where the elements can take q different values. It has defined addition and multiplication operations with certain properties.
- Finite fields can be constructed with a prime number p of elements by performing arithmetic modulo-p. Larger finite fields GF(pm) can be constructed using irreducible polynomials over smaller finite fields.
- Polynomials over a finite field can be added, multiplied, and divided. Irreducible polynomials play an important role in constructing finite field extensions.
- The Galois field GF(2m) with 2m elements is constructed by
On Fractional Fourier Transform Moments Based On Ambiguity FunctionCSCJournals
The fractional Fourier transform can be considered as a rotated standard Fourier transform in general and its benefit in signal processing is growing to be known more. Noise removing is one application that fractional Fourier transform can do well if the signal dilation is perfectly known. In this paper, we have computed the first and second order of moments of fractional Fourier transform according to the ambiguity function exactly. In addition we have derived some relations between time and spectral moments with those obtained in fractional domain. We will prove that the first moment in fractional Fourier transform can also be considered as a rotated the time and frequency gravity in general. For more satisfaction, we choose five different types signals and obtain analytically their fractional Fourier transform and the first and second-order moments in time and frequency and fractional domains as well.
The document compares the low field electron transport properties in compounds of groups III-V semiconductors by solving the Boltzmann equation using an iterative technique. It calculates the temperature and doping dependencies of electron mobility in InP, InAs, GaP and GaAs. The electron mobility decreases with increasing temperature from 100K to 500K for each material due to increased electron-phonon scattering. Electron mobility also increases significantly with higher doping concentration at low temperatures. The iterative results show good agreement with other calculations and experiments. Electron mobility is highest in InAs and lowest in GaP at 300K, due to differences in their effective masses.
1) Fourier developed a theory of infinite series of sine and cosine functions to solve differential equations describing vibratory motion that had no solutions in terms of elementary functions. These series, now known as Fourier series, can represent any periodic function.
2) A Fourier series decomposes a periodic function into an infinite sum of sines and cosines of integer multiples of the fundamental frequency. The coefficients of the sines and cosines are determined from the function itself.
3) Under certain conditions on the periodic function and its derivatives, the Fourier series converges to the value of the function at points of continuity and averages the left and right limits at points of discontinuity.
Master Thesis on Rotating Cryostats and FFT, DRAFT VERSIONKaarle Kulvik
This thesis studied the vibrational and rotational aspects of the Cryo I Helsinki cryostat using Fourier analysis methods. Extensive Fourier analysis was performed to model the vibrations mathematically. The goal was to lower noise levels to improve cryostat operations. The second part of the work tested the homogeneity of a superconducting magnet. Significant preparation was required to build testing equipment for evaluating the magnet.
Fourier Series of Music by Robert FusteroRobertFustero
The document discusses the Fourier series expansion of a mathematical function that describes musical harmony and the Pythagorean tuning system. It shows that the ratios between notes in the Pythagorean scale can be expressed as a linear sequence of powers of 2 and 3. This allows the ratios to be written as terms of a Fourier series expansion, with a periodic nature where the exponential resets every time the integer value increases by the period. The Fourier coefficients are then calculated through integration to express the function in its Fourier form.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
The document verifies intermediate value theorem (IVT) problems for different functions over specified intervals.
It contains 4 problems:
1) Verifies IVT for f(x)=4+3x-x^2 from 2 to 5. Finds c=3/√21 such that f(c)=1.
2) Verifies IVT for f(x)=25-x^2 from -4.5 to 3. Finds c=-4 such that f(c)=3.
3) Explains IVT cannot be used for discontinuous functions like f(x)=4/(x+2) from -3 to 1.
4) Uses IVT to show the function y
N. Bilić: AdS Braneworld with Back-reactionSEENET-MTP
- A 3-brane moving in an AdS5 background of the Randall-Sundrum model behaves like a tachyon field with an inverse quartic potential.
- When including the back-reaction of the radion field, the tachyon Lagrangian is modified by its interaction with the radion. As a result, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
- The dynamical brane causes two effects of back-reaction: 1) the geometric tachyon affects the bulk geometry, and 2) the back-reaction qualitatively changes the tachyon by forming a composite substance with the radion and a modified equation of state.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
This document discusses permutation groups and related algebraic structures. It begins by defining permutations as bijective functions on a set and provides examples of permutations on finite sets. It then defines permutation groups and notes that the symmetric group Sn is the group of all permutations of a set of size n. The document covers key topics such as the composition of permutations, expressing permutations as products of disjoint cycles including transpositions, and classifying permutations as even or odd. It also introduces the alternating group An as the subgroup of Sn containing all even permutations.
This document summarizes the topological structure of ternary residue curve maps, which describe the dynamics of ternary distillation processes. It introduces differential equations that model ternary distillation and place a meaningful structure on ternary phase diagrams. By recognizing this structure is subject to the Poincaré-Hopf index theorem, the authors obtained a topological relationship between azeotropes and pure components in ternary mixtures. This relationship provides useful information about ternary mixture distillation behavior and predicts situations where ternary azeotropes cannot occur.
This document introduces modal tableau calculus as an alternative to Hilbert calculus for proving formulas in modal logic. It defines modal tableau rules and prefixes that represent worlds in a Kripke structure. Examples are provided to construct tableau proofs for formulas in the modal logic K. Exercises are included to prove or disprove formulas in K and S4 using the tableau method.
This document provides an introduction to Galois theory and fields. It discusses how Galois theory originated from studying roots of polynomials and determining which polynomials are solvable by radicals. The document introduces some key concepts from Galois theory, including field extensions, the Galois group of a field extension, and Galois' theorem relating the solvability of a polynomial to the solvability of its Galois group. It also provides background on rings, algebras, and polynomial rings to set up the foundations of Galois theory.
This document discusses Fourier transforms and their application to Fourier transform infrared radiation (FTIR). It begins by introducing Fourier transforms and how they are used in fields like chemistry and physics. It then provides mathematical definitions and theorems to establish complex vector spaces and inner products, which are necessary foundations for further discussion. The document focuses on using Fourier transforms to analyze infrared radiation spectra from materials in order to detect their atomic composition through absorption characteristics.
(1) The document discusses products of LF-topologies and separation concepts in LF-topological spaces.
(2) It defines what a GL-monoid is and introduces uniform structures on GL-monoids to characterize arbitrary products of elements in a GL-monoid.
(3) The paper then builds the LF-topology product of a family of LF-topological spaces and shows that Kolmogoroff and Hausdorff properties are inherited by the product LF-topology from the factor spaces.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
This document provides an overview of Fourier series and Fourier transforms. It discusses the history of Fourier analysis and how Fourier introduced Fourier series to solve heat equations. It defines Fourier series and covers topics like odd and even functions, half-range Fourier series, and the complex form of Fourier series. The document also discusses the relationship between Fourier transforms and Laplace transforms. It concludes by listing some applications of Fourier analysis in fields like electrical engineering, acoustics, optics, and more.
This document provides an overview of field theory concepts related to constructing a toy Standard Model using an SU(2) x U(1) gauge symmetry. It discusses how scalar and spinor fields can be incorporated into a Lagrangian that respects this symmetry. It describes how the gauge fields transform under subgroups like the diagonal subgroup, and how this relates to the masses of the Z and W bosons. It also discusses the Higgs potential and how it gives mass to the Higgs boson while the Goldstone bosons are eliminated via a gauge condition.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This presentation provides an introduction to Galois fields, which are finite fields with a prime number of elements. The objectives are to discuss preliminaries like sets and groups, introduce Galois fields and provide examples, discuss related theorems, and describe the computational approach. A sample computation in FORTRAN verifies the theorem that any element in a Galois field can be expressed as the sum of two squares.
The document provides an overview and history of the wavelet transform. It can be summarized as follows:
1. The wavelet transform was developed to address limitations of the Fourier transform and short-time Fourier transform in analyzing signals both in time and frequency. It uses wavelets of limited duration that can be scaled and translated.
2. The history of the wavelet transform began in 1909 with Haar wavelets. The concept of wavelets was then proposed in 1981 and the term was coined in 1984. Important developments included the construction of additional orthogonal wavelets in 1985, the proposal of the multiresolution concept in 1988, and the fast wavelet transform algorithm in 1989, enabling numerous applications.
3.
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
More Related Content
Similar to S. Duplij, W. Werner, "Extensions of special 3-fields", https://arxiv.org/abs/2212.08606
Master Thesis on Rotating Cryostats and FFT, DRAFT VERSIONKaarle Kulvik
This thesis studied the vibrational and rotational aspects of the Cryo I Helsinki cryostat using Fourier analysis methods. Extensive Fourier analysis was performed to model the vibrations mathematically. The goal was to lower noise levels to improve cryostat operations. The second part of the work tested the homogeneity of a superconducting magnet. Significant preparation was required to build testing equipment for evaluating the magnet.
Fourier Series of Music by Robert FusteroRobertFustero
The document discusses the Fourier series expansion of a mathematical function that describes musical harmony and the Pythagorean tuning system. It shows that the ratios between notes in the Pythagorean scale can be expressed as a linear sequence of powers of 2 and 3. This allows the ratios to be written as terms of a Fourier series expansion, with a periodic nature where the exponential resets every time the integer value increases by the period. The Fourier coefficients are then calculated through integration to express the function in its Fourier form.
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
The document verifies intermediate value theorem (IVT) problems for different functions over specified intervals.
It contains 4 problems:
1) Verifies IVT for f(x)=4+3x-x^2 from 2 to 5. Finds c=3/√21 such that f(c)=1.
2) Verifies IVT for f(x)=25-x^2 from -4.5 to 3. Finds c=-4 such that f(c)=3.
3) Explains IVT cannot be used for discontinuous functions like f(x)=4/(x+2) from -3 to 1.
4) Uses IVT to show the function y
N. Bilić: AdS Braneworld with Back-reactionSEENET-MTP
- A 3-brane moving in an AdS5 background of the Randall-Sundrum model behaves like a tachyon field with an inverse quartic potential.
- When including the back-reaction of the radion field, the tachyon Lagrangian is modified by its interaction with the radion. As a result, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
- The dynamical brane causes two effects of back-reaction: 1) the geometric tachyon affects the bulk geometry, and 2) the back-reaction qualitatively changes the tachyon by forming a composite substance with the radion and a modified equation of state.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
This document discusses permutation groups and related algebraic structures. It begins by defining permutations as bijective functions on a set and provides examples of permutations on finite sets. It then defines permutation groups and notes that the symmetric group Sn is the group of all permutations of a set of size n. The document covers key topics such as the composition of permutations, expressing permutations as products of disjoint cycles including transpositions, and classifying permutations as even or odd. It also introduces the alternating group An as the subgroup of Sn containing all even permutations.
This document summarizes the topological structure of ternary residue curve maps, which describe the dynamics of ternary distillation processes. It introduces differential equations that model ternary distillation and place a meaningful structure on ternary phase diagrams. By recognizing this structure is subject to the Poincaré-Hopf index theorem, the authors obtained a topological relationship between azeotropes and pure components in ternary mixtures. This relationship provides useful information about ternary mixture distillation behavior and predicts situations where ternary azeotropes cannot occur.
This document introduces modal tableau calculus as an alternative to Hilbert calculus for proving formulas in modal logic. It defines modal tableau rules and prefixes that represent worlds in a Kripke structure. Examples are provided to construct tableau proofs for formulas in the modal logic K. Exercises are included to prove or disprove formulas in K and S4 using the tableau method.
This document provides an introduction to Galois theory and fields. It discusses how Galois theory originated from studying roots of polynomials and determining which polynomials are solvable by radicals. The document introduces some key concepts from Galois theory, including field extensions, the Galois group of a field extension, and Galois' theorem relating the solvability of a polynomial to the solvability of its Galois group. It also provides background on rings, algebras, and polynomial rings to set up the foundations of Galois theory.
This document discusses Fourier transforms and their application to Fourier transform infrared radiation (FTIR). It begins by introducing Fourier transforms and how they are used in fields like chemistry and physics. It then provides mathematical definitions and theorems to establish complex vector spaces and inner products, which are necessary foundations for further discussion. The document focuses on using Fourier transforms to analyze infrared radiation spectra from materials in order to detect their atomic composition through absorption characteristics.
(1) The document discusses products of LF-topologies and separation concepts in LF-topological spaces.
(2) It defines what a GL-monoid is and introduces uniform structures on GL-monoids to characterize arbitrary products of elements in a GL-monoid.
(3) The paper then builds the LF-topology product of a family of LF-topological spaces and shows that Kolmogoroff and Hausdorff properties are inherited by the product LF-topology from the factor spaces.
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU (2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU (2, 1) is the
intersection of two quadrics in P5 . In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU (2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
This document provides an overview of Fourier series and Fourier transforms. It discusses the history of Fourier analysis and how Fourier introduced Fourier series to solve heat equations. It defines Fourier series and covers topics like odd and even functions, half-range Fourier series, and the complex form of Fourier series. The document also discusses the relationship between Fourier transforms and Laplace transforms. It concludes by listing some applications of Fourier analysis in fields like electrical engineering, acoustics, optics, and more.
This document provides an overview of field theory concepts related to constructing a toy Standard Model using an SU(2) x U(1) gauge symmetry. It discusses how scalar and spinor fields can be incorporated into a Lagrangian that respects this symmetry. It describes how the gauge fields transform under subgroups like the diagonal subgroup, and how this relates to the masses of the Z and W bosons. It also discusses the Higgs potential and how it gives mass to the Higgs boson while the Goldstone bosons are eliminated via a gauge condition.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This presentation provides an introduction to Galois fields, which are finite fields with a prime number of elements. The objectives are to discuss preliminaries like sets and groups, introduce Galois fields and provide examples, discuss related theorems, and describe the computational approach. A sample computation in FORTRAN verifies the theorem that any element in a Galois field can be expressed as the sum of two squares.
The document provides an overview and history of the wavelet transform. It can be summarized as follows:
1. The wavelet transform was developed to address limitations of the Fourier transform and short-time Fourier transform in analyzing signals both in time and frequency. It uses wavelets of limited duration that can be scaled and translated.
2. The history of the wavelet transform began in 1909 with Haar wavelets. The concept of wavelets was then proposed in 1981 and the term was coined in 1984. Important developments included the construction of additional orthogonal wavelets in 1985, the proposal of the multiresolution concept in 1988, and the fast wavelet transform algorithm in 1989, enabling numerous applications.
3.
Similar to S. Duplij, W. Werner, "Extensions of special 3-fields", https://arxiv.org/abs/2212.08606 (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
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 .
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.
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.
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.
PPT on Direct Seeded Rice 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.
Travis Hills of MN is Making Clean Water Accessible to All Through High Flux ...Travis Hills MN
By harnessing the power of High Flux Vacuum Membrane Distillation, Travis Hills from MN envisions a future where clean and safe drinking water is accessible to all, regardless of geographical location or economic status.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
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.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
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When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
S. Duplij, W. Werner, "Extensions of special 3-fields", https://arxiv.org/abs/2212.08606
1. EXTENSIONS OF SPECIAL 3-FIELDS
STEVEN DUPLIJ AND WEND WERNER
Abstract. 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.
1. Introduction
Algebraic structure which is based on composing more than two elements can be traced
back to early work of Dörnte [8] and Post [18] and has later shown an increase in interest,
see for instance [3, 4, 5, 6, 9, 11, 12, 15, 16], and especially in physical model building
[1, 2, 7, 17, 13, 14]. On many occasions, such a theory substantially profit from embedding
objects into a larger structure in which such seemingly unconventional algebraic structure
can be reduced to more conventional concepts. For a typical example see the brief discussion
of ternary commutative groups in [10].
In cases where multiple algebraic operations of this kind are present (for example, the 3-
fields investigated here) such an approach, however, is less successful, and the theory requires
rather novel techniques. Our principal definition in the following is
Definition 1.1. A set R equipped with two operations R3
Ñ R, called ternary addition
and ternary multiplication, is a called a 3-ring, iff for each r, r1,2,3, s1,2 P R
(1) there are additive and, respectively, multiplicative querelements r P R and p
r so that
r ˆ
`r ˆ
`r1 “ r1 as well as rp
rr1 “ r1,
(2) s1s2pr1 ˆ
`r2 ˆ
`r3q “ s1s2r1 ˆ
`s1s2r2 ˆ
`s1s2r3, and
(3) both operations are associative (i.e. no brackets are needed for multiple applications
of these operations)
R is called commutative, if the order of factors and summands can be permuted in any
possible way, and unital iff there is an element 1 P R with 11r “ r for all r P R.
A (unital) 3-field K is a (unital) 3-ring iff for each k P K there is p
k P K so that kp
kk1 “ k1
for all k1 P K.
In a 3-ring (or, more generally, in a commutative 3-group), the presence of a multiplicative
unit alone allows to introduce a binary product of commutative groups
x ¨ y “ x1y, (1.1)
which has the property that applying it twice on three factors results in the original ternary
product. Similarly, for a 3-ring R, a zero element 0 is defined by requiring that
0xy “ x0y “ xy0 “ 0 for all x, y P R. (1.2)
arXiv:2212.08606v1
[math.RA]
15
Oct
2022
2. 2
Such an element is uniquely determined and as in the case of a 1-element for the multipli-
cation, it allows for reducing ternary addition of R to a binary one.
The fields we will investigate in the following will come equipped with a ternary addition
(and no zero element), and a multiplication that possesses a unit so that, right from the
outset, we will assume multiplication to be binary. We then have
Theorem 1.2 ([10] Theorems 3.4, 6.1). A unital 3-field F embeds into a binary field K (with
its inherited ternary addition and binary multiplication) if for each y P Fzt1u the equation
x ` y ´ xy “ 1 (1.3)
has the only solution x “ 1. Whenever F is finite this happens if and only iff F “ t1u.
Let us explain some of the basic features of the theory that will be used in the following
(details are in [10]).
To each unital 3-field F belongs a uniquely determined (binary) local ring UpFq into
which it embeds as the subset UpFq˚
of units in such a way that the ternary sum of F
coincides with the binary sum of UpFq applied twice. (And, conversely, the units of any
local ring U with residual field F2 form a unital 3-field, with its inherited structure.) We
have UpFq “ QpFq Y F, where the binary non-unital ring QpFq consists of all mappings
(pairs) qa,b : f ÞÝÑ f ˆ
`aˆ
`b, a, b P F, with addition and multiplication coming from pointwise
operations on the set of mappings F Ñ F. We will frequently make use of the fact that each
q P QpFq has a unique representation in the form q “ q1,f , f P F. As we will consider every
unital 3-field F to be naturally embedded into the binary ring UpFq so that the querelement
f from F becomes ´f.
Remark 1.3. It is for this reason that we no longer will formally distinguish between the
binary and ternary sums and write + throughout.
Any ternary morphism φ : F1 Ñ F2 canonically extends to a binary morphism Qpφq :
QpF1q Ñ QpF2q (and hence to a binary morphism Upφq : UpF1q Ñ UpF2q) via φpqa,bq “
qφpaq,φpbq. The kernel of Qpφq is a binary ideal in UpF1q, and we will address these binary
ideals of UpF1q as the (ternary) ideals of F1. We also will formally write a short exact
sequence of unital 3-fields as
0 ÝÑ J ÝÑ F ÝÑ F0 ÝÑ 0, (1.4)
with the understanding that we actually are dealing with unital 3-fields F1, F and F0, a short
exact sequence of binary rings,
0 ÝÑ UpF1q ÝÑ UpFq ÝÑ UpF0q ÝÑ 0, (1.5)
and the ideal J Ď QpF1q arising as the kernel of the second arrow.
Using analogous definitions for unital 3-rings, the quotient of a unital 3-ring by an ideal J
of UpRq is a unital 3-field iff for any proper ideal J0 of UpRq for which J Ď J0 it follows that
J0 X R “ H. Note that this condition is automatically satisfied when R already is a 3-field.
The basic examples are the prime fields TFpnq “ t2k ´ 1 | k “ 1, . . . , 2n´1
u Ď Z{2n
and
TFp8q, the ternary field of quotients of the unital 3-ring 2Z ` 1. As unital 3-fields, they
are generated by the element 1. Since the unit of each unital 3-field F generates a uniquely
determined prime field Pn inside F, the number of elements in this subfield, the characteristic
χpFq “ 2n´1
of F, is well-defined.
Extensions of 3-fields are a much more complicated subject than its binary counterpart.
The present investigation is a first attempt at a deeper understanding. The next section
3. 3
deals with products which exist thanks to the absence of a zero element. In some cases, they
provide 3-field extensions. The final section is devoted to 3-field extensions of t1u, which are
numerous and follow only in some cases the paths of Galois theory.
2. Products of 3-fields
2.1. Cartesian Products. Since unital 3-fields are supposed to be proper, for each pair
of unital 3-fields F1,2, all elements of the Cartesian product F1 ˆ F2 posses a multiplicative
inverse so that F1 ˆ F2 is itself a unital 3-field, under pointwise operations. For the same
reason, however, there is no canonical embedding of one of these fields into F1 ˆ F2. (There
is an exception in characteristic 1, though: In this case
ι1 : F1 Ñ F1 ˆ F2, ι1pfq “ pf, 12q (2.1)
is an embedding with π1ι1 “ IdF1 .)
Theorem 2.1. Let F be a unital 3-field. Then the following are equivalent:
(1) F is the Cartesian product of two 3-fields F1,2.
(2) There are Q1,2 Ď QpFq so that as a binary ring, QpFq “ Q1 ‘ Q2.
Proof. If F “ F1 ˆF2 then qpa1,a2q,pb1,b2q P QpF1 ˆF2q acts by qpa1,a2q,pb1,b2qpf1, f2q “ pa1 `b1 `
f1, a2 ` b2 ` f2q establishing the binary ring isomorphism
Z : QpF1 ˆ F2q Ñ QpF1q ‘ QpF2q, qpa1,a2q,pb1,b2q ÞÑ pqa1,b1 , qa2,b2 q . (2.2)
Assuming QpFq “ Q1 ‘ Q2 as rings, both Qi are ideals of QpFq and hence yield 3-fields
Fi “ F{Qi with corresponding quotient maps πi. Then πpfq “ pπ1pfq, π2pfqq defines an
injective morphism F Ñ F1 ˆ F2 of 3-fields. Because π´1
1 pf1q “ f0
1 ` Q1 for some f0
1 P F
and π2pf0
1 ` Q1q “ F2, π is surjective.
The above result and its proof can easily be extended to infinite products
ś
iPI Fi of 3-
fields, and this is the product in the categorical sense: Whenever, for a 3-field G, there are
morphisms ψi : G Ñ Fi there is a unique mapping ψ : G Ñ
ś
iPI Fi which can be shown to
be a 3-field morphism.
Note also that QpFq “ QpF1q ‘ QpF2q is not related to a similar decomposition of UpFq,
since
rQpF1q ‘ QpF2qs Y rF1 ˆ F2s ‰ rQpF1q Y F1s ‘ rQpF2q Y F2s (2.3)
Observe that modifying a construction from [10] in order to create direct sums is futile:
Letting, for an odd number of finite 3 fields F1, . . . , Fn (odd, in order to have a unit and,
possibly, invertibility),
n
à
i“1
Fi “
#
pfiq P
n
à
i“1
UpFiq
ˇ
ˇ
ˇ
ˇ
ˇ
n
ÿ
i“1
B pfiq “ 1
+
(2.4)
where B : UpFq Ñ F2 denotes the natural grading for the unital 3-field F, it turns out that
pfiq possesses an inverse iff pfiqN
“ p1, . . . , 1q and it follows that the set of invertible elements
within
Àn
i“1 Fi equals
Śn
i“1 Fi.
4. 4
2.2. Semi-direct Products and Unitization of Algebras. We will base the notion of
a semi-direct products of 3-fields on the concept of a split short exact sequence of unital
3-fields 0 Ñ J Ñ F Ñ G Ñ 0.
Definition 2.2. The unital 3-field F is the (internal) semi-direct product of the ideal J Ď
QpFq and the subfield G iff G is the image of an epimorphism π : F Ñ G with Ker π “ J
and right inverse ι : G Ñ F.
More abstractly, semi-direct products are connected to algebras over 3-fields. Recall [10]
Definition 2.3. Let A be a binary ring and F a unital 3-field. We call A a binary algebra
over F, iff F acts on A in such a way that, for all f, f1,2,3 P F and a, a1,2 P A,
(1) fpa1 ` a2q “ fa1 ` fa2
(2) pf1 ` f2 ` f3qa “ f1a ` f2a ` f3a
(3) pf1f2qa “ f1pf2aq
(4) 1F a “ a
Similarly, we will call A a 3-algebra over F iff A is a 3-ring (with addition coming from an
underlying commutative 3-group) equipped with a binary product so that, for all f, f1,2,3 P F
and a, a1,2 P A,
fpa1 ` a2 ` a3q “ fa1 ` fa2 ` fa3 (2.5)
while axioms (2)-(4) for a binary algebra are left untouched.
Note that we do not assume A to be unital.
Example 2.4. QpFq is an F-algebra where the action of F is given by f1q1,f “ qf1,f1f ,
whenever q1,f P QpFq and f1 P F.
Example 2.5. More generally, if the morphism of unital 3-fields π : F Ñ G possesses the
right inverse σ : G Ñ F, the ideal J “ Ker Qpπq is a G-algebra with G-action g.j “ σpgqj.
We will also need the fact that a binary F-algebra A becomes a binary module over the
local ring UpFq by letting
q1,f a “ a ` fa, f P F, a P A. (2.6)
In case A is a 3-algebra, QpFq can only act on QpAq via
q1,f qa,b “ qa,b`fa`fb. (2.7)
Semi-direct products for unital 3-fields will be related to the unitization of an algebra over
a unital 3-field. There are two variants, one with purely binary operations, and another one
for which addition becomes ternary.
Definition 2.6. Let A be an algebra over the unital 3-field F. Then the binary unitization
of A is defined on the additive direct sum
A``
F “ UpFq ‘ A (2.8)
with multiplication
pu1, a1qpu2, a2q “ pu1u2, u1a2 ` u2a1 ` a1a2q. (2.9)
Ternary unitization is based on the additively written commutative 3-group A`
F “ F ‘ A
and a binary product which is equal to the one in the binary case.
It is straightforward to check that A``
F is a binary unital ring, and A`
F is a unital 3-ring.
5. 5
Theorem 2.7. The unitization A`
of a nilpotent algebra A over the unital 3-field F is a
unital 3-field, and it follows that A`
is a semi-direct product with canonical quotient map
A`
Ñ F and natural split F Ñ F ‘ 0.
Proof. Since A`
always is a unital 3-ring we still have to show that each element pf, aq has
an inverse. But, due to nilpotency of A,
p1, aq´1
“
˜
1,
N
ÿ
ν“1
p´1qν
aν
¸
(2.10)
for N large enough, and hence pf, aq´1
“ f´1
p1, f´1
aq´1
.
Corollary 2.8. For a finite unital 3-field F0 a subfield F of F0, and an ideal J Ď QpF0q
equipped with the natural action of F, A`
F is a unital 3-field.
There are infinite unital 3-fields F for which QpFq is not nilpotent so in order to find
3-fields among unitizations we need to define invertibility before unitization.
Definition 2.9. An algebra A over a unital 3-field F is called a Q-algebra iff for each a P A
there is a#
P A such that aa#
“ a ` a#
.
Example 2.10. The simplest example is an algebra A over the 3-field TFp0q “ t1u. Equiv-
alently, A is a binary ring such that a ` a “ 0 for all a P A and, at the same time, a
(binary) algebra over GFp2q. If also a2
“ 0 for all a P A, A`
TFp0q is a unital 3-field in which
p1, aq´1
“ p1, aq for all a P A. If the product of A vanishes identically and if we select a basis
B for the vector space A, we find
A`
TFp0q – TFp1q|B|
(2.11)
This example also covers the case in which F acts trivially on A, i.e. fa “ a, for all f P F,
a P A.
Example 2.11. Each nilpotent algebra A over the unital 3-field F is a Q-algebra with
a#
“ ´
N
ÿ
ν“1
aν
, (2.12)
where aN`1
“ 0.
Lemma 2.12. An algebra A over a unital 3-field F is a Q-algebra if and only if A`
F is a
unital 3-field.
Proof. Suppose A is a Q-algebra. We must prove that each element is invertible, and, as in
the proof of Theorem 2.7, it suffices to show p1, qq´1
exists. But it follows from the definition
of q#
that p1, qqp1, q#
q “ p1, 0q.
Conversely, if p1, qq has inverse pf, qq it follows that f “ 1 and q ` q ` qq “ 0 so that q
provides the #-element, providing A with the structure of a Q-algebra.
Theorem 2.13. Let F be a unital 3-field. There exists a bijective correspondence between
the unitization of Q-algebras A over F and semi-direct products of 3-fields J ¸ F:
(1) For each Q-algebra A over F, the mapping πA,F : A`
F Ñ F, pf, aq ÞÑ f establishes
the short exact sequence of unital 3-fields
0 ÝÑ A ÝÑ A`
F ÝÑ F ÝÑ 0 (2.13)
which is split by σA,F : f ÞÑ pf, 0q and so, A`
F “ A ¸ F.
6. 6
(2) Conversely, each split exact sequence 0 Ñ J Ñ F0 Ñ F Ñ 0 naturally defines the
structure of a Q-algebra over F on J and it follows that F0 “ J ¸ F is isomorphic
to J`
F .
Proof. By the definition of A`
F and Lemma 2.12, πA,F and σA,F are morphisms of unital
3-fields giving rise to the split exact sequence 0 Ñ A Ñ A`
F Ñ F Ñ 0.
In order to prove the second statement, denote by σ : F Ñ F0 the right inverse to the
quotient morphism π : F0 Ñ F. Example 2.5 shows J is a Q-algebra over F, and the
mapping
J`
F ÝÑ F0, pf, qq ÞÝÑ σpfq ` q (2.14)
is an isomorphism of unital 3-fields.
Example 2.14. Let F be a subfield of F0 and J an ideal of UpF0q. For each q “ q1,f P J,
q#
“ q1,f´1 is in J since q#
“ ´qq#
´q. It follows that J is a Q-algebra over F and so F ˙J
is a unital 3-field.
Example 2.15. Let F0 be a unital 3-field with unital subfield F1 and let F “ F0 ˆ F1. Then
QpFq “ QpF0q ‘ QpF1q and π2pf0, f1q Ñ f1 is a surjective morphism with kernel
J “ q “ qp1,1q,pg0,g1q P QpFq
ˇ
ˇ 0 “ Qpπ2qpqq “ q1,f1
(
“ QpF0q ‘ t0u (2.15)
π2 has the left inverse σ : F1 Ñ F, σpf1q “ pf1, f1q which produces an action of diag F1ˆF1 –
F1 on J given by
pf1, f1qq1,f0 “ f1q1,f0 “ q1,f1`f1f0´1 (2.16)
J is a Q-algebra over F1 with q#
1,f0
“ q1,f´1
0
, and the morphism Φ : J`
F1
Ñ F0 ˆ F1,
Φpf1 ‘ q1,f0 q “ pf1, f1 ` q1,f0 q “ pf1, 1 ` f0 ` f1q (2.17)
is an isomorphism.
2.3. Creating a 3-field action. We complement the above with an intrinsic characteriza-
tion of the binary rings QpFq, this time without using the action of a unital 3-field. First an
observation: The ring structure of QpFq uniquely determines the underlying unital 3-field.
Proposition 2.16. Fix a unital 3-field F1,2 are unital 3-fields and suppose Ψ : QpF1q Ñ
QpF2q is a binary ring morphism. Define Φ : F1 Ñ F2 through Ψpq1,f q “ q1,Φpfq. Then Φ is
a unital 3-field morphism which is an automorphism whenever Ψ is.
Proof. Using the involved definitions, one checks that Φ respects addition and multiplication
of F. Furthermore, 0 “ Ψpq1,1q “ q1,Φp1q, and so Φ is unital. If Ψ is an automorphism and
Ψ´1
pq1,aq “ qp1, p
Φpaqq then, necessarily p
Φ “ Φ´1
.
Definition 2.17. A commutative (binary) ring Q is called a Q-ring iff
(1) There exists a 2-unit τ P Q so that τq “ q ` q for all q P Q.
(2) For each q P Q exists a unique #-element q#
P Q with q#
q “ q ` q#
.
The morphisms between Q-rings are those ring morphisms that map the respective τ-
elements onto each other and respect the #-involution.
We note some consequences of these axioms:
(1) A Q-ring is never unital, since we would have 1#
“ 1 ` 1#
and so 1 “ 0.
(2) τ#
“ τ and τn
“ 2n´1
τ
7. 7
(3) There might be more than one 2-element: two such elements τ1,2 have to satisfy
pτ1 ´ τ2qx “ 0 for all x P Q.
(4) Similarly, q#
1,2 P Q are #-elements for q P Q iff pq#
1 ´ q#
2 qpq ´ 1q “ 0, within the
unitization of Q. As we will see below, this determines the element q#
uniquely.
Theorem 2.18. Q is a Q-ring, iff there is a unital 3-field F such that Q “ QpFq.
Proof. Suppose Q “ QpFq. Then, letting τ “ q1,1 and q#
1,f “ q1,f´1 , we have turned QpFq
into a Q-ring. Conversely, whenever F is a Q-ring, define ternary addition ˆ
` as well as
binary multiplication ˆ
ˆ by
f1 ˆ
`f2 ˆ
`f3 “ f1 ` f2 ` f3 ´ τ, f ˆ
ˆg “ τ ´ f ´ g ` fg (2.18)
Then the querelement of f P F for ˆ
` is f “ τ ´ f, the distributive law holds since
f ˆ
ˆpf1 ˆ
`f2 ˆ
`f3q “ 2τ ´ 3f ´ f1 ´ f2 ´ f3 ` ff1 ` ff2 ` ff3 “ f ˆ
ˆf1 ˆ
`f ˆ
ˆf2 ˆ
`f ˆ
ˆf3, (2.19)
also τ ˆ
ˆf “ τ ` f ´ τf “ f “ f, and f# ˆ
ˆf “ f ` f# ´ ff# “ 0 “ τ.
Corollary 2.19. For a nilpotent ring Q there exists a unital 3-field F with Q “ QpFq iff Q
contains a 2-unit τ.
Example 2.20. In the simplest case, when the product on a binary ring R of characteristic
2 vanishes, each element τ P R can serve as a 2-element, while r#
“ ´r. For the resulting
unital 3-field FpRqτ we have
r ˆ
`sˆ
`t “ r ` s ` t ` τ r ˆ
ˆs “ r ` s ` τ, (2.20)
and whatever the choice of τ P R, the mapping Φτ : r ÞÝÑ r ` τ provides an isomorphism
between FpRq0 and FpRqτ .
Example 2.21. The rings of pairs Qpnq “ t2k P Z{2n
| k “ 0, . . . , 2n´1
´ 1u for the 3-fields
TFpnq “ t2k ´ 1 P Z{2n
| k “ 1, . . . , 2n´1
u are Q-rings, with τ “ 2 and, taking the inverse
inside Z{2n
,
q#
“
q
q ´ 1
. (2.21)
Similarly, Qp8q “ tp{q | p “ 2r, q “ 2s ` 1, r, s P Zu is a Q-ring with τ “ 2 and
pp{qq#
“
p
p ´ q
. (2.22)
Note that in these examples, τ is unique.
Example 2.22. If A is a Q-algebra over the unital 3-field F, the ideal QpFq‘A of the binary
unitization A``
F is (still a Q-algebra over F but also) a Q-ring, with τ “ pq1,1, 0q. The map
Ψ : QpA`
F q ÝÑ QpFq ‘ A, qp1,0q,pf,aq ÞÝÑ pq1,f , aq , (2.23)
establishes an isomorphism.
8. 8
3. 3-field extensions of t1u
It could be argued that an extension of a 3-field F0 should be any unital 3-field F arising
in a short exact sequence
0 ÝÑ J ÝÑ F ÝÑ F0 ÝÑ 0, (3.1)
where J is (identified with) an ideal of UpFq. In this case, F “ F0 ˆ F1 would qualify as an
extension of F0, and the projection onto F0 leads to the short exact sequence
0 ÝÑ QpF1q ˆ 0 ÝÑ F0 ˆ F1 ÝÑ F0 ÝÑ 0. (3.2)
We will nonetheless follow the established notion and call the unital 3-field F an extension
of the unital 3-field F0 in case F0 embeds into F.
In the present situation the structure of all subfields is more complex: Since t1u being
a subfield of a unital 3-field F is equivalent to χpFq “ 1, these extensions coincide with
all unital 3-fields of characteristic 1. The extensions we will consider here belong to the
following class.
Definition 3.1. Fix a unital 3-field F as well as natural numbers n1, . . . , nk P N and put
Fpn1, . . . , nkq “
#
1 `
ÿ
0‰α
εαpx ´ 1qα
ˇ
ˇ
ˇ
ˇ
ˇ
εα P F2, px ´ 1qnκ
“ 0, κ “ 1, . . . , k
+
. (3.3)
Note that by using semi-direct sums we may reduce the number of variables in this exam-
ple: Consider the map xi ÞÝÑ 1 and extend it to an epimorphism πi on Fpn1, . . . , nkq,
1 `
ÿ
0‰α
εαpx ´ 1qα
ÞÝÑ 1 `
ÿ
0‰α, αi“0
εαpx ´ 1qα
(3.4)
with image Fpn1, . . . , ni´1, ni`1, . . . nkq and kernel consisting of the ideal
Ji “ pxi ´ 1q
ÿ
α
εαpx ´ 1qα
. (3.5)
The elements of Fpn1, . . . , ni´1, ni`1, . . . nkq embed naturally into Fpn1, . . . , nkq and act on
Ji by multiplication so that
Fpn1, . . . , nkq “ Fpn1, . . . , ni´1, ni`1, . . . nkq ˙ Ji (3.6)
A slightly more sophisticated way of writing down these 3-field extensions is obtained in
the following way.
Definition 3.2. Fix a finite Abelian binary group G as well as a local (binary) ring R with
residual field F2 “ t0, 1u. Consider the group algebra RG over R. Then
FG “ tf P RG | fp0q P R˚
u (3.7)
is called the ternary group algebra of G over the 3-field R˚
.
Theorem 3.3. FG is a 3-field, extending R˚
, and, in case R “ t0, . . . , 2n
´ 1u “
Up1, 3, . . . , 2n
´ 1q, each finite unital field extensions of the unital 3-field t1, 3, . . . , 2n
´ 1u
which is contained in an extension generated by a single elements is isomorphic to one of
these fields.
9. 9
3.1. Extensions of characteristic 0, generated by a single element. We are going to
consider extensions of F0 “ TFp0q “ t1u, with prime field identical to TFp0q, and generated
by a single element.
Theorem 3.4. Any finite unital 3-field F of characteristic 1 which is generated by a single
element x is isomorphic to F0pnP q where nP is the smallest natural number with p1´xqn
“ 0.
If pPq “ min ti | ηi ‰ 0u then nP “ rn{ks.
Proof. Consider the polynomial 3-ring F0rxs “ t1 `
řn
ν“1 ενpx ´ 1qν
| εν P F2u for which
UF0rxs is the principal domain F2rxs. Consequently, whenever F is generated by a single
element, there is a polynomial P P QF0rxs “ t
řn
ν“0 ενpx ´ 1qν
| εν P F2u such that F “
F0rxs{xPy. The polynomial P cannot be of the form
P “ px ´ 1qn
˜
1 ` px ´ 1qk
`
N
ÿ
ν“k`1
πνpx ´ 1qν
¸
, πν P F2, (3.8)
because then x1 ` px ´ 1qk
`
řN
ν“k`1 πνpx ´ 1qν
y would be strictly larger than xPy, intersect
F0rxs and thus contradict [10][Theorem 1]. So, P “ px´1qn
for the smallest n with px´1qn
“
0 within the 3-field F.
Any attempt at creating a theory related to classical Galois theory has to deal with a
number of obstacles, one of them, for example, the less useful factorization of polynomials:
In F0p4q, for example, the polynomial 1 ` px ´ 1q ` px ´ 1q3
“ r1 ` px ´ 1qs r1 ` px ´ 1q3
s
vanishes at 1 when using the right hand representation, while it takes the value 1 when 1 is
plugged into 1 ` px ´ 1q ` px ´ 1q3
.
Proposition 3.5. The ideals of F0pnq are
Ik “
#
ÿ
νěk
ενpx ´ 1qν
ˇ
ˇ
ˇ
ˇ
ˇ
εν P F2
+
, k “ 1, . . . , n ´ 1 (3.9)
Consequently, F0pnq never is a Cartesian product of 3-fields.
Proof. Since the ring UF0pnq “
řn´1
ν“0 ενpx ´ 1qν
ˇ
ˇ εi P F2
(
is principal, for each proper
ideal I in QF0pnq the ideal
ă I, F0pnq ą“
!ÿ
qip1 ` riq
ˇ
ˇ
ˇ qi P I, ri P QF0pnq
)
Ď I ` I (3.10)
is the same as I and so I “ă P0 ą, with P0 P QF0pnq. Since each P P QF0pnq can be written
P “ px ´ 1qs
p1 ` Rq, p1 ` Rq invertible, 1 ď s ď n ´ 1, it follows that the ideals of QF0pnq
are precisely those of the form Ik “ă px ´ 1qk
ą. But Is X It “ Imaxts,tu, Is ` It “ Imints,tu
and, by applying Theorem 2.1, we conclude that F0pnq never is a proper Cartesian product
of 3-fields.
Whether or not F0pnq can be written as a semi-direct product is a slightly more delicate
question, and we will return to it elsewhere.
Lemma 3.6. Denote by ϕn : F0pnq Ñ F0pnq the Frobenius morphism P ÞÝÑ P2
, by F0pnq2
its image, and, for k ě 2 let µn,k : F0pnq Ñ F0pkq be defined by
µn,k
˜
1 `
n´1
ÿ
i“1
εipx ´ 1qi
¸
“ 1 `
k´1
ÿ
i“1
εipx ´ 1qi
. (3.11)
10. 10
(1) ϕn and µn,k are morphisms,
Ker ϕn “ Irn{2s, and Ker µn,k “ Ik (3.12)
(2) The product on Ker ϕn vanishes identically so that p1 ` P1qp1 ` P2q “ 1 ` P1 ` P2.
Proof. (1) Both maps are well-known (and easily seen to be) morphisms. Furthermore, the
image of %n can naturally be embedded into F0pnq.
The additive structure of F0pnq is most transparent on UF0pnq, which is a (binary) vector
space with basis px ´ 1qk
ˇ
ˇ k “ 0, . . . , n ´ 1
(
. The multiplicative structure of these fields
is the following.
Theorem 3.7. Each element f P F0pnq has a uniquely defined factorization f “ γα0
0 . . . γαK
K
where
γk “ 1 ` px ´ 1q2k`1
0 ď 2k ` 1 ď n ´ 1, (3.13)
and it follows that the multiplicative group underlying F0pnq is isomorphic to the direct prod-
uct Cn,0 ˆ . . . ˆ Cn,Kn of the cycles Cn,k of the γk, with Kn “ max tk | 2k ` 1 ď n ´ 1u, and
Cn,k – Z{2spn,kq
, spn, kq “ min t2t
| 2t
k ě nu. Consequently, with respect to its multiplicative
structure,
F0pnq –
ź
0ď2k`1ďn
`
Z{2spn,kq
˘rpn,kq
, (3.14)
Proof. The cases n “ 2 is trivial, and the elements of F0p3q are γk
0 , k “ 0, . . . , 3. If the
statement is true for F0pnq, f P F0pn`1q has a uniquely defined factorization f “ gγα0
0 . . . γαK
K
where g P t1 ` εpx ´ 1qn
| ε “ 0, 1u, the multiplicative kernel of the canonical morphism
π : F0pn ` 1q Ñ F0pnq. If n is odd, this already proves the claim; in case n is odd and
g ‰ 1 we must have g “ p1 ` px ´ 1q`
q2m
, ` odd, and the statement of the result follows for
n ` 1.
3.2. Intermediate Fields. We start with slightly generalizing Corollary 2.19.
Theorem 3.8. Suppose Q is a nilpotent ring with 2-unit τ and let F be the unital 3-field
with QpFq “ Q.
(1) Q1
ù 1 ` Q1
Ď F establishes a one-to-one correspondence between the subrings Q1
of Q with τ P Q1
and the unital 3-subfields F1
of F.
(2) The unital 3-subfield F0 is generated by elements 1 ` q1, . . . , 1 ` qn iff QpF0q is gen-
erated by τ “ q1,1 and q1, . . . , qn.
(3) The automorphisms Φ of F leaving the subfield F1
pointwise fixed correspond to the
automorphisms QΦ, leaving Q1
“ QpF1
q pointwise fixed.
Proof. (1) If F0 is a unital 3-subfield of F, the unit of F0 must equal the one in F, and QpF0q
is a subring containing ζ “ q1,1 which has the property that ζq “ q ` q for all q P QpF0q.
Note that 1 ` QpF0q “ F0.
Conversely, assume that Q Ď QpFq with τ P Q and define F1 “ tf P F | q1,f P Qu. We
have 1 P F1 since τ P Q and ´1 P F1 because 0 P Q. It follows similarly that for all f, g P F1,
1 ` f ` g P F1, f ` g ` gf P F1, (3.15)
and that there is ˘
f with 1 ` f ` ˘
f “ ´1. From this, and the nilpotency of Q it follows that
F1 is unital 3-field such that F1 “ 1 ` Q.
11. 11
(2) The subring Q0 generated by τ and q1, . . . , qn consists of the elements kτ `
ř
α εαqα
,
k P N0. Note that due to nilpotency, this ring is of finite characteristic. Then 1 ` Q0 is
a unital 3-field, consisting of elements k `
ř
α εαqα
, k odd, and resolving the brackets in
k `
ř
α εαp1`qqα
shows that 1`Q0 is the unital 3-field generated by 1 and 1`q1, . . . , 1`qn.
Conversely, if F0 is generated by 1, 1 ` q1, . . . , 1 ` qn, QpF0q contains τ, q1, . . . , qn and thus
the binary ring Q0, generated by these elements. By the first part of this proof, F0 “ 1`Q0,
and so Q0 “ QpF0q.
(3) This follows from the involved definitions.
Remark 3.9. The last theorem can be easily generalized to Q-rings and their Q-subrings,
where a Q-subring Q1 of the Q-ring Q is defined by the requirement that τ P Q1 and that
Q1 is invariant under the #-operation.
Corollary 3.10. For each ideal J of the finite unital 3-field F the subalgebra 1 ` J a unital
subfield.
Proof. For ζ “ q1,1 P QpFq we have ζ “ qpq ´ 1q´1
P J.
Corollary 3.11. For a unital 3-subfield F of F0pnq which is generated by polynomials
P1, . . . , Pg there exist natural numbers n1, . . . , ng such that F – F0pn1, . . . , ngq.
Example 3.12. The smallest subfields, those of order 2, are generated by polynomials
1 ` px ´ 1qk
p1 ` P1q “
`
1 ` px ´ 1qk
˘
p1 ` P1q ´ P1 (3.16)
with 2k ě n. We call k the lower degree of the subfield. As we will see, the automorphisms
which are constant on such a field are of the form ΨP with P0 “ px ´ 1q ` px ´ 1q`
p1 ` P1q
with ` ě n´k `1 so that Aut F0pnq cannot distinguish between subfields of the same degree
in terms of fixed point subgroups. On the other hand, all elements of Aut F0pnq preserve k,
and, since
Ψp1 ` P1q “ pΨpPq ´ 1q Ψpx ´ 1q´k
´ 1 “ px ´ 1qk
p1 ` Q1q, (3.17)
Example 3.13. The subfield of squares, F0pnq2
“ f P F0pnq | f “ 1 `
ř
νě1 ηνpx ´ 1q2ν
(
is
isomorphic to all subfields generated by a polynomial of the form f0 “ 1 ` px ´ 1q2
f1,
We will need
Lemma 3.14. Fix a natural number α, write α “
řnα
ν“0 αν2ν
, αν “ 0, 1, and let
Nα “
#
nα
ÿ
ν“0
εναν2ν
ˇ
ˇ
ˇ
ˇ
ˇ
εν “ 0, 1
+
. (3.18)
Then “
1 ` px ´ 1qk
‰α
“ 1 `
ÿ
nPNα
px ´ 1qkn
. (3.19)
Conversely, given N Ď t1, . . . , n ´ 1u, then PN “ 1 `
ř
νPN px ´ 1qν
“ p1 ` px ´ 1qk
qα
iff
N Ď kN and
N “
#
ÿ
ν
εν2ν
ˇ
ˇ
ˇ
ˇ
ˇ
k2ν
P N, εν “ 0, 1
+
X t1, . . . , n ´ 1u. (3.20)
Proof. We have
p1 ` px ´ 1qk
qα
“
nα
ź
ν“0
`
1 ` px ´ 1qk2ν ˘αν
, (3.21)
12. 12
and each subset N1
Ď tν | αν “ 1u corresponds to exactly on of the summands px ´ 1qkn
of
1 `
ř
nPNα
px ´ 1qkn
, where n “
ř
νPN1 2ν
(and with n “ 0 for the empty set).
We start with a ‘local’ version Theorem 3.4.
Lemma 3.15. The unital 3-subfield generated by P “ 1 ` px ´ 1qk
p1 ` Pkq P F0pnq is given
by
xPy “
#
1 `
ÿ
0ăkiďn´1
ηipx ´ 1qki
p1 ` Pkqi
ˇ
ˇ
ˇ
ˇ
ˇ
ηi P F2
+
“: F1 (3.22)
and is isomorphic to
x1 ` px ´ 1qk
y “
$
%
1 `
ÿ
0ăiďpn´1q{k
ηipx ´ 1qki
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ηi P F2
,
.
-
. (3.23)
Proof. By (a slight modification of) Lemma 3.14 the elements of xPy belong to F1, and
since F1 is a unital 3-field, xPy Ď F1. By Theorem 3.4, the isomorphism class of xPy is
determined by the minimal number nP with pP ´ 1qnP
“ 0. This number is the same for P
and 1 ` px ´ 1qk
hence xPy – x1 ` px ´ 1qk
y are both of cardinality 2nP ´1
. Since also F1 is
of this cardinality, F1 “ xPy.
Theorem 3.16. F Ď F0pnq is a subfield iff there are natural numbers g1, . . . , gk ď n ´ 1,
k ď n ´ 1, with
F – F0pn; g1, . . . gkq :“ x1 ` px ´ 1qg1
, . . . , 1 ` px ´ 1qgk
y “
“
#
1 `
ÿ
1ďν1g1`...`νkgkďn´1
εν1ν2...νk
px ´ 1qν1g1`...`νkgk
ˇ
ˇ
ˇ
ˇ
ˇ
εν1...νk
P F2
+
“: Gpn; g1, . . . , gkq.
(3.24)
Proof. In order to see that F0pn; g1, . . . , gkq is unital 3-subfield it suffices to show that it is
a unital 3-subring. But this is obvious, and it follows that F0pn; g1, . . . , gkq is contained in
Gpn; g1, . . . , gkq. By Lemma 3.15,
1 ` px ´ 1qµgi`νgj
“
“ p1 ` px ´ 1qµgi
qp1 ` px ´ 1qνgj
q ´ p1 ` px ´ 1qµgi
q ´ p1 ` px ´ 1qνgj
q (3.25)
is contained in F0pn; g1, . . . gkq, and similarly, 1`px´1qν1g1`...`νkgk P F0pn; g1, . . . , gkq. Using
induction over the number of elements in tεν1,...,νk
“ 1u with the induction step of adjoining
1 ` px ´ 1qxµ,gy
to 1 `
ř
1ďxg,νyďn´1 ενpx ´ 1qxν,gy
being established by
1 ` px ´ 1qxµ,gy
`
ÿ
1ďxg,νyďn´1
ενpx ´ 1qxν,gy
“
“ 1 `
`
1 ` px ´ 1qxµ,gy
˘
`
¨
˝1 `
ÿ
1ďxg,νyďn´1
ενpx ´ 1qxν,gy
˛
‚, (3.26)
one shows that, conversely, Gpn; g1, . . . , gkq Ď F0pn; g1, . . . , gkq.
13. 13
Let F Ď F0pnq be a subfield. We call the semigroup
Ex F “ tα P Z{n | 1 ` px ´ aqα
P Fu (3.27)
the exponents of F. The proof of the following results rest on
Lemma 3.17. Suppose S is a sub-semigroup of the additive semigroup Z{n and N0 one of
its subsets. Then
(1) there are uniquely determined elements s1 ă . . . ă sk P S with
sκ`1 “ min Sz tz1s1 ` . . . ` zκsκ | zi P Z{nu (3.28)
S “ tz1s1 ` . . . ` zksk | zi P Z{nu (3.29)
Equivalently, none of the si divides sj.
(2) The sub-semigroup SpN0q generated by N0 can be constructed inductively by letting
s1 “ min N0 and sn`1 “ min N0z tz1s1 ` . . . znsn | z1, . . . zn P Nu as long as the latter
set is not empty. If M is the index before this happens,
SpN0q “ z tz1s1 ` . . . zM sM | z1, . . . zM P Nu (3.30)
Corollary 3.18. The set of isomorphism classes of subfields of F0pnq is order isomorphic
to the set of sub-semigroups of the additive semigroup Z{n. Furthermore, two subfields F1,2
of F0pnq are isomorphic iff Ex F1 “ Ex F2.
3.3. The group of automorphisms and the global involution. For the following it is
important to observe that the elements of
QF0pnq “
#
n´1
ÿ
i“1
εi px ´ 1qi
ˇ
ˇ
ˇ
ˇ
ˇ
εi P F2
+
(3.31)
are in 1-1 correspondence with unital endomorphisms Φ of F0pnq: Each such Φ is uniquely
determined by the polynomial PΦ “ QΦpx ´ 1q P QF0pnq and, conversely, any polynomial
P “ 1 ` P0 P F0pnq “ 1 ` QF0pnq provides a unital endomorphisms ΦP , defined for Q “
1 ` Q0px ´ 1q P F0pnq by
ΦP p1 ` Q0px ´ 1qq “ 1 ` Q0 ˝ P0 mod px ´ 1qn
. (3.32)
This map is well-defined because the ideal generated by px ´ 1qn
for the ternary polynomial
ring F0rxs within
QF0rxs “
#
N
ÿ
i“1
εi px ´ 1qi
ˇ
ˇ
ˇ
ˇ
ˇ
N P N, εi P F2
+
(3.33)
is invariant under composition with polynomials in px ´ 1q from the right. It is additive and
respects multiplication because
ΦP pp1 ` Q0px ´ 1qq p1 ` R0px ´ 1qqq “
“ 1 ` pQ0px ´ 1q ` R0px ´ 1q ` Q0px ´ 1qR0px ´ 1qq ˝ P0 “
“ ΦP p1 ` Q0px ´ 1qq ΦP p1 ` R0px ´ 1qq (3.34)
Since ΦQ ˝ ΦP “ ΦQ˝P we have shown the first part of
14. 14
Theorem 3.19. The mapping Φ ÞÝÑ QΦpx´1q establishes an isomorphism between the ring
of unital endomorphisms of F0pnq with respect to composition of maps and QF0pnq, equipped
with composition of polynomials in the variable px ´ 1q.
Under this identification, a polynomial P corresponds to an automorphism of F0pnq iff
there is a polynomial Q P F0pnq such that P0 ˝ Q0 “ P0 ˝ Q0 “ px ´ 1q which is equivalent to
QΦpx ´ 1q “ px ´ 1q `
n´1
ÿ
i“2
εkpx ´ 1qk
. (3.35)
Proof. We still have to show the final statement. For arbitrary n “ α0`α12`. . .`αN 2N
P N,
αν P F2 and any polynomial P0 “
řn´1
i“1 εkpx ´ 1qk
Pn
0 “
˜
n´1
ÿ
i“1
εkpx ´ 1qk
¸α0
˜
n´1
ÿ
i“1
εkpx ´ 1q2k
¸α1
. . .
˜
n´1
ÿ
i“1
εkpx ´ 1q2N
¸αN
, (3.36)
with some of the factors of highest order potentially equal to zero. Accordingly, for no
polynomial Q0, Q0 ˝ P0 has px ´ 1q as lowest term if P0 doesn’t, and any polynomial P0
giving rise to an automorphism must have px ´ 1q as lowest term. 1
(This could also be
shown by using the ideal structure of F0pnq.)
For the converse, we represent ΦP with P0 “ px ´ 1qk
p1 ` P1q as a matrix with respect to
the basis px ´ 1qν
, ν “ 1, . . . , n ´ 1, of UpF0pnqq. Writing
ΦP rpx ´ 1qν
s “ px ´ 1qν
p1 ` P1qν
(3.37)
as row vectors, it turns out that the resulting matrix is upper triangular with 1’s on the
main diagonal so that ΦP must be invertible.
Using the multinomial identity
pa1 ` . . . amqn
“
ÿ
i1`...`im“n
n!
i1! . . . im!
ai1
1 . . . aim
m (3.38)
we find for the polynomial P0 “ px ´ 1q `
řn´1
ν“2 ενpx ´ 1qν
Pk
0 “
ÿ
i1`...in´1“k
k!
i1! . . . in´1!
ε2 . . . εn´1px ´ 1qi1`2i2`...`pn´1qin´1
“ (3.39)
“
n´1
ÿ
`“k
¨
˝
ÿ
i1`2i2`...`pn´1qin´1“`
k!
i1! . . . in´1!
ε2 . . . εn´1
˛
‚px ´ 1q`
(3.40)
On the other hand, over F2, and with n “
řN
ν“0 nν2ν
,
pa1 ` . . . amqn
“
N
ź
ν“0
`
a2ν
1 ` . . . ` a2ν
m
˘nν
(3.41)
and
Pk
0 “
N
ź
ν“0
`
px ´ 1q2ν
` . . . ` εµpx ´ 1q2ν µ
. . . ` εmpx ´ 1q2ν m
˘nν
(3.42)
1Note that this is different for polynomials P of the form 1 `
ř
kě1 εkxk
, since large powers of the
polynomial x might contain a constant term. For example, in F0p5q, x5
“ x4
` x ´ 1, and also px ` x2
`
x4
q ˝ p1 ` x ` x2
q “ x.
15. 15
Corollary 3.20. The coefficients of AP “ pαP
k`q are
αP
k` “
ÿ
i1`2i2`...`pn´1qin´1“`
k!
i1! . . . in´1!
ε2 . . . εn´1, (3.43)
for k “ 1, . . . , n ´ 1, ` “ k, . . . , n ´ 1.
If opΦP q “ 2`pΦP q
denotes the order of ΦP P Aut F0pnq then Φ´1
P “ Φ
opΦP q´1
P . An algorithm
of lower complexity is obtained using matrix products
Corollary 3.21. Let P “ 1 ` px ´ 1q ` P1 “
řn
ν“0 ενpx ´ 1qν
be the polynomial that belongs
to ΦP P AutpFq. Then the coefficients of the polynomial p
P “
řn
ν“0 p
ενpx ´ 1qν
for which
Φ´1
P “ Φp
P can be recursively calculated using
p
ε0 “ p
ε1 “ 1 p
ε` “ p
ε1ε
p1q
` ` p
ε2ε
p2q
` ` . . . ` p
ε`´1ε
p`´1q
` (3.44)
Lemma 3.22. Let sn “ ´
řn´1
ν“1px ´ 1qν
P QF0pnq. Then Rn :“ Φsn has order 2 and
Rn
«
1 `
n´1
ÿ
i“1
εipx ´ 1qi
ff
“ 1 `
n´1
ÿ
i“1
εi
`
px ´ 1q#
˘i
. (3.45)
Proof. Since, with respect to multiplication,
sn “ 1 ` rpx ´ 1q ´ 1s´1
(3.46)
it follows that 1 ` psn ´ 1q´1
“ px ´ 1q or, sn ˝ sn “ px ´ 1q (which is equivalent to observing
that q ÞÑ q#
is an involution, Definition 2.9 and Example 2.11). The somewhat more explicit
form for Rn is a consequence of the fact that sn “ px ´ 1q#
.
Lemma 3.23. A group G of order 2n has a representation Z{2¸H iff G contains a normal
subgroup H of order n as well as an involution r P GzH.
Proof. The short exact sequence H Ñ G Ñ Z{2 can be split by mapping the non-unit of
Z{2 to r.
Lemma 3.24. Define
Mn,k : Aut F0pnq Ñ Aut F0pkq, Mn,kΦpx ` Ikq “ Φpxq ` Ik, (3.47)
where 1 `
řn´1
ν“1 ενpx ´ 1qν
` Ik P F0pnq{Ik is identified with 1 `
řk´1
ν“1 ενpx ´ 1qν
P F0pkq.
(1) Identifying F0pkq with F0pnq{Ik, if Φ “ ΦP with P “ px ´ 1q `
řn´1
ν“2 ενpx ´ 1qν
then
Mn,kΦP “ px ´ 1q `
k´1
ÿ
ν“2
ενpx ´ 1qν
, (3.48)
and Ker Mn,k is equal to
Γn,k “
#
ΦP P Aut F0pnq | P0 “ px ´ 1q `
n´1
ÿ
ν“k
ενpx ´ 1qν
, εν P F2
+
(3.49)
(2) Restricting Mn,k`1 to Γn,k yields an exact sequence
0 ÝÑ Γn,k`1 ÝÑ Γn,k ÝÑ Γk`1,k ÝÑ 0 (3.50)
This sequence splits whenever k ą 2 so that in this case,
Γn,k – Γn,k`1 ¸ F2. (3.51)
16. 16
(3) Similarly, ϕn generates the group morphism
%n : Aut F0pnq Ñ Aut
`
F0pnq2
˘
, %npΦq “ Φ|F0pnq2 (3.52)
Identifying the subfield of squares in F0pnq with F0
`Xn
2
˘
, the effect of %n on the
polynomial P representing Φ “ ΦP can be seen as
%n : Aut F0pnq Ñ Aut F0
´Yn
2
]¯
, P ÞÑ P|F0ptn
2
uq
(3.53)
In this picture, ΦP P Ker %n iff P “ px ´ 1q `
ř
νěrn{2s ενpx ´ 1qν
.
(4) Accordingly, for each k with 2k
ď n there is a short exact sequence
Gn,k ÝÑ Aut F0pnq ÝÑ Aut
´
pF0pnq2k
¯
– Aut F0
´Y n
2k
]¯
(3.54)
where Gn,k is the normal subgroup of automorphisms for which 1`px´1q2k
is a fixed
point. The polynomials P0, representing Φ P Gn,k via Φp1 ` px ´ 1qq “ 1 ` P0px ´ 1q,
are characterized by
P0 “ px ´ 1q `
ÿ
iąpn´1q{2k
εipx ´ 1qi
“
px ´ 1q ` px ´ 1qrpn´1q{2ks
n´1´rpn´1q{2ks
ÿ
i“1
εipx ´ 1qi
, εi P F2, (3.55)
and it follows that Gn,1 is commutative.
Proof. (1) Because Ik “ Ker µn,k “
řn´1
i“k`1 εipx ´ 1qi
ˇ
ˇ εi P F2
(
is invariant under substitu-
tion by any polynomial Q0 “ px´1q`
řn
i“2 ηipx´1qi
(which also follows from Lemma 3.5, each
Φ P Aut F0pnq acts on F0pnq{ Ker µn,k “ F0pkq by substitution and an application of µn,k.
The resulting element Mn,kpΦq P Aut F0pkq then maps x´1 to µn,kP0 for Φ “ ΦP P Aut F0pnq.
Consequently, Mn,k : Aut F0pnq Ñ Aut F0pkq is a morphism with kernel consisting of auto-
morphisms ΦP P Aut F0pnq such that for all Q “ 1 `
řk´1
i“1 qipx ´ 1qi
µn,kΦP pQq “ µn,k p1 ` Q0 ˝ P0q “ Q. (3.56)
which shows that P “ 1 ` px ´ 1q `
řn
i“k ηipx ´ 1qi
.
(2) The short exact sequence is established using (1). Polynomials for elements of
Γn,kzΓn,k`1 are of the form P “ px´1q`px´1qk
`
řn´1
i“k`1 εipx´1qi
so that Rn R Γn,kzΓn,k`l
if k ą 2. The claim follows from Lemma 3.22 and Lemma 3.23.
(3) %npΦq is a well-defined morphism. Identifying F0ptn{2uq with F0pnq2
via σnpPq “ P2
,
one has for P “ px ´ 1q `
řn´1
ν“2 ενpx ´ 1qν
σ´1
n %npΦP qσnpx ´ 1q “ σ´1
n P2
“ px ´ 1q `
tn{2u
ÿ
ν“2
ενpx ´ 1qν
(3.57)
The elements ΦP in the kernel Gn,k of %k
n restrict to all polynomials
ř
i2kďn´1 εipx ´ 1qi2k
as
the identity or,
ř
i2kďn´1 εiP0px´1qi2k
“
ř
i2kďn´1 εipx´1qi2k
. Equivalently, P2k
0 “ px´1q2k
,
showing that P0 “ px ´ 1q `
ř
nďi2k εipx ´ 1qi
. The commutativity of Gn,1 is a consequence
17. 17
of the fact that for P0, Q0 P Gn,1 we have
P0 ˝ Q0px ´ 1q “ Q0px ´ 1q `
n´1
ÿ
i“rn{2s
εiQi
0px ´ 1q “ Q0px ´ 1q ` P0px ´ 1q (3.58)
Corollary 3.25. For each n ě 3,
Aut F0pnq – F2 ¸ F2 . . . ¸ F2, (3.59)
an pn ´ 2q-fold semi-direct product of F2.
Proof. By induction, this is a consequence of Lemma 3.6 (2).
This last result by itself does not reveal much of the structure of Aut F0pnq: Analogous
results hold for each nilpotent Lie groups and, in all likelihood, it is possible to obtain
the former by showing that Aut F0pnq is a (nilpotent) Lie group in characteristic 2. More
information can be gained by further investigating the involution that is underlying the proof
of the above result.
Definition 3.26. Let f P F0pnq as well as Φ P Aut F0pnq.
(1) Define conjugations f˚
and Φ˚
by
f˚
“ Rnpfq Φ˚
pfq “ Φpf˚
q˚
, (3.60)
(2) and we denote the 1-eigenspaces of these involutions by
F0pnq1 “ tf P F0pnq | f˚
“ fu , Aut F0pnq1 “ tΦ P Aut F0pnq | Φ˚
“ Φu (3.61)
Example 3.27. All elements of Jk with k ě n{2 are contained in F0pnq1, because if f “ px´1qk
then, for k “
ř
n0
αν2ν
f˚
“ sk
n “
ź
n0
˜
n´1
ÿ
µ“1
px ´ 1qναν 2ν
¸
(3.62)
Appendix A. A representation of the truncated Nottingham groups
Aut F0p3q – Aut F0p7q
The case n “ 2 is trivial, for n “ 3 the polynomials 1 ` px ´ 1q and Ppxq “ 1 ` px ´ 1q `
px ´ 1q2
give rise to identity as well as to the automorphism with representing matrix
AP “
ˆ
1 1
0 1
˙
(A.1)
If n “ 4, besides the identity, P1 “ 1 ` px ´ 1q ` px ´ 1q2
, P2 “ 1 ` px ´ 1q ` px ´ 1q3
and
P3 “ 1 ` px ´ 1q ` px ´ 1q2
` px ´ 1q3
correspond to
A1 “
¨
˝
1 1 0
0 1 0
0 0 1
˛
‚ A2 “
¨
˝
1 0 1
0 1 0
0 0 1
˛
‚ A3 “
¨
˝
1 1 1
0 1 0
0 0 1
˛
‚, (A.2)
20. 20
E A2
A3
A6
A7
A1
A4
A5
conj
conj
Figure 1. Cycle graph for Aut F0p5q “ C4 ¸ C2 “ D4, the Dihedral Group of
order 8, with C2 acting by inversion. The dotted lines indicate conjugation, as
in Definition 3.26.
22. 22
References
[1] R. Ablamowicz, On the structure of ternary Clifford algebras and their irreducible representations, Adv.
Appl. Clifford Algebr. 32 (2022), 39.
[2] J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev.
D77 (2008), 065008.
[3] D. Bohle and W. Werner, A K-theoretic approach to the classification of symmetric spaces, J. Pure and
App. Algebra 219 (2015), 4295–4321.
[4] N. Celakoski, On pF, Gq-rings, God. Zb., Mat. Fak. Univ. Kiril Metodij Skopje 28 (1977), 5–15.
[5] G. Crombez, The Post coset theorem for pn, mq-rings, Ist. Veneto Sci. Lett. Arti, Atti, Cl. Sci. Mat.
Natur. 131 (1973), 1–7.
[6] G. Crombez and J. Timm, On pn, mq-quotient rings, Abh. Math. Semin. Univ. Hamb. 37 (1972), 200–
203.
[7] J. de Azcarraga and J. M. Izquierdo, n-Ary algebras: A review with applications, J. Phys. A43 (2010),
293001.
[8] W. Dörnte, Unterschungen über einen verallgemeinerten Gruppenbegriff, Math. Z. 29 (1929), 1–19.
[9] S. Duplij, Polyadic Algebraic Structures, IOP Publishing, Bristol, 2022.
[10] S. Duplij and W. Werner, Structure of unital 3-fields, Math. Semesterber. 68 (2021), 27–53.
[11] H. A. Elgendy and M. R. Bremner, Universal associative envelopes of pn`1q-dimensional n-Lie algebras,
Commun. Algebra 40 (2012), 1827–1842.
[12] P. M. Higgins, Completely semisimple semigroups and epimorphisms, Proc. Amer. Math. Soc. 96 (1986),
387–390.
[13] R. Kerner, Ternary algebraic structures and their applications in physics, in 23rd International Collo-
quium on Group Theoretical Methods in Physics, (V. Dobrev, A. Inomata, G. Pogosyan, L. Mardoyan,
and A. Sisakyan, eds.), Joint Inst. Nucl. Res., JINR Publishing, Dubna, 2000 (arXiv preprint: math-
ph/0011023).
[14] R. Kerner, A Z3 generalization of Pauli’s principle, quark algebra and the Lorentz invariance, in The
Sixth International School on Field Theory and Gravitation, (J. Alves Rodrigues, Waldyr, R. Kerner,
G. O. Pires, and C. Pinheiro, eds.), Vol. 1483 of AIP Conference Series, 2012, pp. 144–168.
[15] A. G. Kurosh, Multioperator rings and algebras, Russian Math. Surveys 24 (1969), 1–13.
[16] J. J. Leeson and A. T. Butson, On the general theory of pm, nq rings., Algebra Univers. 11 (1980),
42–76.
[17] Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. 7 (1973), 2405–2412.
[18] E. L. Post, Polyadic groups, Trans. Amer. Math. Soc. 48 (1940), 208–350.
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