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.
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.
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.
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.
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.
Existence and Controllability Result for the Nonlinear First Order Fuzzy Neut...ijfls
In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Weighted Analogue of Inverse Maxwell Distribution with ApplicationsPremier Publishers
In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
Existence and Controllability Result for the Nonlinear First Order Fuzzy Neut...ijfls
In this paper, we devoted study the existence and controllability for the nonlinear fuzzy neutral integrodifferential equations with control system in E_N. Moreover we study the fuzzy solution for the normal, convex, upper semicontinuous and compactly supported interval fuzzy number. The results are obtained by using the contraction principle theorem. An example to illustrate the theory.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Weighted Analogue of Inverse Maxwell Distribution with ApplicationsPremier Publishers
In the present study, we established a new statistical model named as weighted inverse Maxwell distribution (WIMD). Its several statistical properties including moments, moment generating function, characteristics function, order statistics, shanon entropy has been discussed. The expression for reliability, mode, harmonic mean, hazard rate function has been derived. In addition, it also contains some special cases that are well known. Moreover, the behavior of probability density function (p.d.f) has been shown through graphs by choosing different values of parameters. Finally, the performance of the proposed model is explained through two data sets. By which we conclude that the established distribution provides better fit.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
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.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012
Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh
Reference Books:
1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004.
2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005.
3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012.
Please comment and feel free to ask anything related. Thanks!
https://arxiv.org/abs/1905.01927
A quantum mechanical model that realizes the Z2xZ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.
This paper reviews the fundamental concepts and
the basic theory of polarization mode dispersion(PMD) in optical
fibers. It introduces a unified notation and methodology to
link the various views and concepts in jones space and strokes
space. The discussion includes the relation between Jones vectors
and Strokes vectors and how they are used in formulating the
jones matrix by the unitary system matrix.
In this paper, the L1 norm of continuous functions and corresponding continuous estimation of regression parameters are defined. The continuous L1 norm estimation problem of one and two parameters linear models in the continuous case is solved. We proceed to use the functional form and parameters of the probability distribution function of income to exactly determine the L1 norm approximation of the corresponding Lorenz curve of the statistical population under consideration.
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
https://arxiv.org/abs/2312.01366.
We introduce a new class of division algebras, hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the proposed earlier matrix polyadization procedure which increases the algebra dimension. The obtained algebras obey the binary addition and 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 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. Then we obtain another series of n-ary algebras corresponding to the binary division algebras which have more dimension, that is proportional to intermediate arities, and they are not isomorphic to those obtained by the previous constructions. Second, we propose a new iterative process (we call it "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half dimension, we call them "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 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 introduced unitless ternary division algebra of imaginary "half-octonions" is ternary alternative.
178 pages, 6 Chapters. DOI: 10.1088/978-0-7503-5281-9. This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers. In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way. Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.Part of IOP Series in Coherent Sources, Quantum Fundamentals, and Applications. Key features • Provides new mathematical methods for quantum computing. • Presents material in a widely accessible way. • Contains methods for unconventional computing where there is computational complexity. • Provides methods to increase speed and efficiency. For the light paperback version use MyPrint service here: https://iopscience.iop.org/book/mono/978-0-7503-5281-9, also PDF, ePub and Kindle. For the libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Innovative-Quantum-Computing/?K=9780750352796. Amazon ordering: https://www.amazon.de/gp/product/0750352795
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.
https://www.mdpi.com/books/book/6455
We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.
https://arxiv.org/abs/2212.08606
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Книга «Поэфизика души» представляет собой полное, на момент издания 2022 г., собрание прозаических произведений автора. Как рассказы, так и миниатюры на полстраницы, пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга поэтическими образами, воплощенными в прозе. Также включены юмористические путевые заметки о поездке в Китай.
Книга "Поэфизика души", Степан Дуплий – полное собрание прозы 2022, 230 стр. вышла в Ridero: https://ridero.ru/books/poefizika_dushi и Kindle Edition file на Амазоне: https://amazon.com/dp/B0B9Y4X4VJ . "Бумажную" книгу можно заказать на Озоне https://ozon.ru/product/poefizika-dushi-682515885/?sh=XPu-9Sb42Q и на ЛитРес: https://litres.ru/stepan-dupliy/poefizika-dushi-emocionalnaya-proza-kitayskiy-shtrih-punktir . Google books: https://books.google.com/books?id=9w2DEAAAQBAJ .
Книгу можно заказать из-за рубежа на AliExpress: https://aliexpress.com/item/1005004660613179.html .
Книга «Гравитация страсти» представляет собой полное собрание стихотворений автора на момент издания (август, 2022). Стихотворения пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга необычными поэтическими образами.
Книга "Гравитация страсти", Степан Дуплий - полное собрание стихотворений 2022, 338 стр. вышла в Ridero: https://ridero.ru/books/gravitaciya_strasti
. Книга в мягкой обложке доступна для заказа на Ozon.ru: https://ozon.ru/product/gravitatsiya-strasti-707068219/?oos_search=false&sh=XPu-9TbW9Q
, на Litres.ru: https://www.litres.ru/stepan-dupliy/gravitaciya-strasti-stihotvoreniya , за рубежом на AliExpress: https://aliexpress.com/item/1005004722134442.html , и в электронном виде Kindle file на Amazon.com: https://amazon.com/dp/B0BDFTT33W .
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.
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/ fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and ?-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" ?-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and ?-commutativity are studied.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the \textquotedblleft product\textquotedblright\ obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the \textquotedblleft Kronecker\textquotedblright\ obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.
Online first: https://www.intechopen.com/online-first/obscure-qubits-and-membership-amplitudes
Abstract: In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Salas, V. (2024) "John of St. Thomas (Poinsot) on the Science of Sacred Theol...Studia Poinsotiana
I Introduction
II Subalternation and Theology
III Theology and Dogmatic Declarations
IV The Mixed Principles of Theology
V Virtual Revelation: The Unity of Theology
VI Theology as a Natural Science
VII Theology’s Certitude
VIII Conclusion
Notes
Bibliography
All the contents are fully attributable to the author, Doctor Victor Salas. Should you wish to get this text republished, get in touch with the author or the editorial committee of the Studia Poinsotiana. Insofar as possible, we will be happy to broker your contact.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Toxic effects of heavy metals : Lead and Arsenicsanjana502982
Heavy metals are naturally occuring metallic chemical elements that have relatively high density, and are toxic at even low concentrations. All toxic metals are termed as heavy metals irrespective of their atomic mass and density, eg. arsenic, lead, mercury, cadmium, thallium, chromium, etc.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
2. 2 of 16
conditions. Thus, further investigations are needed to develop the higher regular and
inverse polyadic semigroups.
2. GENERALIZED n-REGULAR ELEMENTS IN SEMIGROUPS
Here we formulate some basic notions in terms of tuples and the “quantity/number
of multiplications”, in such a way that they can be naturally generalized from the binary to
polyadic case [18].
2.1. Binary n-regular single elements
Indeed, if S is a set, its `th Cartesian product S×` consists of all `-tuples x(`) =
(x1, x2, . . . , x`), xi ∈ S (we omit powers (`) in the obvious cases x(`) ≡ x, ` ∈ N). Let S2 ≡
Sk=2 = hS | µ2, assoc2i be a binary (having arity k = 2) semigroup with the underlying set
S, and the (binary k = 2) multiplication µ2 ≡ µk=2 : S × S → S (sometimes we will denote
µ2[x1, x2] = x1x2). The (binary) associativity assoc2 : (x1x2)x3 = x1(x2x3), ∀xi ∈ S, allows
us to omit brackets, and in the language of tuples x1x2 . . . x` = µ
[`−1]
2
h
x(`)
i
(= x(`) ≡ x,
if the number of elements is not important or evident, and also distinguish between the
product x(`) of ` elements and the `-tuples x(`)), i.e. the product of ` elements contains ` − 1
(binary) multiplications (whereas in the polyadic case this is different [19]). If all elements
in a tuple coincide, we write x` = µ
[`−1]
2
h
x(`)
i
(this form will be important in the polyadic
generalization below). An element xi ∈ S satisfying x` = x (or µ
[`−1]
2
h
x(`)
i
= x) is called
(binary) `-idempotent.
An element x of the semigroup S2 is called (von Neumann) regular, if the equation
xyx = x, or (2.1)
µ
[`−1=2]
k=2 [x, y, x] = x, x, y ∈ S, (2.2)
has a solution y in S2, which need not be unique. An element y is called a regular element
conjugated to x [3] (also, an inverse element [2]). The set of the regular elements conjugated
to x is denoted by Vx [4].
In search of generalizing the regularity condition (2.1) in S2 for the fixed element
x ∈ S, we arrive at the following possibilities:
1. Higher regularity generalization: instead of one element y ∈ S, use an n-tuple y(n) ∈ S×n
and the corresponding product y(n).
2. Higher arity generalization: instead of the binary product µ2, consider the polyadic (or
k-ary semigroup) product with the multiplication µk.
It follows from the generalization (i) that we can write (for the binary case)
Definition 1. An element x of the (binary) semigroup S2 is called higher n-regular, if there exists
(at least one, not necessarily unique) (n − 1)-element solution y(n−1) of
xy(n−1)
x = x, or (2.3)
µ
[n]
k=2
h
x, y(n−1)
, x
i
= x, x, y1, . . . , yn−1 ∈ S. (2.4)
Definition 2. The tuple y(n−1) from (2.4) is called higher n-regular conjugated to x, or y(n−1) is
an n-inverse (tuple) for x.
Without any additional conditions, for instance splitting the tuple y(n−1) for some
reason (as below), the definition (2.3) reduces to the ordinary regularity (2.1), because the
binary semigroup is closed with respect to the multiplication of any number of elements,
and so there exists z ∈ S, such that y(n−1) = z for any y1, . . . , yn−1 ∈ S. Therefore, we have
the following reduction
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 September 2021
3. 3 of 16
Assertion 1. Any higher n-regular element in a binary semigroup is 2-regular (or regular in the
ordinary sense (2.1)).
Indeed, if we consider the higher 3-regularity condition x1x2x3x1 = x1 for a single
element x1, and x2, x3 ∈ S, it obviously coincides with the ordinary regularity condition
for the single element x (2.1). However, if we cycle both the regularity conditions, they
will not necessarily coincide, and new structures can appear. Therefore, using several
different mutually consistent n-regularity conditions (2.3) we will be able to construct
a corresponding (binary) semigroup which will not be reduced to an ordinary regular
semigroup (see below).
2.2. Polyadic n-regular single elements
Following the higher arity generalization (ii), we introduce higher n-regular single
elements in polyadic (k-ary) semigroups and show that the definition of regularity for
higher arities cannot be done similarly to ordinary regularity, but requires a polyadic
analog of n-regularity for a single element of a polyadic semigroup. Before proceeding we
introduce
Definition 3 (Arity invariance principle). In an algebraic universal system with operations of
different arities the general form of expressions should not depend on numerical arity values.
This gives the following prescription for how to generalize expressions from the binary
shape to a polyadic shape:
1. Write down an expression using the operations manifestly.
2. Change arities from the binary values to the needed higher values.
3. Take into account the corresponding changes of tuple lengths according to the concrete
argument numbers of operations.
Example 1. If on a set S one defines a binary operation (multiplication) µ2 : S × S → S, then one
(left) neutral element e ∈ S for x ∈ S is defined by µ2[e, x] = x (usually ex = x). But for 4-ary
operation µ4 : S×4 → S it becomes µ4[e1, e2, e3, x] = x, where e1, e2, e3 ∈ S, so we can have a
neutral sequence e = e(3) = (e1, e2, e3) (a tuple e of the length 3), and only in the particular case
e1 = e2 = e3 = e, do we obtain one neutral element e. Obviously, the composition of two (n = 2)
4-ary (k = 4) operations, denoted by µ
[2]
4 ≡ µ
[`−1=2]
k=4 (as in (2.2)) should act on a word of a length
7, since µ4[µ4[x1, x2, x3, x4], x5, x6, x7], xi ∈ S.
In general, if a polyadic system hS | µki has only one operation (k-ary multiplication),
the length of words is “quantized”
L
(`)
k = `(k − 1) + 1, (2.5)
where ` is a “quantity of multiplications”.
Let Sk = hS | µk, assocki be a polyadic (k-ary) semigroup with the underlying set S, and
the k-ary multiplication µk : S×k → S [19]. The k-ary (total) associativity assock can be treated
as invariance of two k-ary product composition µ
[2]
k with respect to the brackets [18]
assock : µ
[2]
k [x, y, z] = µk[x, µk[y], z] = invariant, ∀xi, yi, zi ∈ S, (2.6)
for all placements of the inner multiplication µk in the r.h.s. of (2.6) with the fixed order-
ing of (2k − 1) elements (see (2.5)), and x, y, z are tuples of the needed length. The total
associativity (2.6) allows us to write a product containing ` multiplications using external
brackets only µ
[`]
k [x] (the long product [20]), where x is a tuple of length `(k − 1) + 1, because
of (2.5). If all elements of the tuple x = x(`) are the same we write it as x(`). A polyadic
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 September 2021
4. 4 of 16
power (reflecting the “number of multiplications”, but not of the number of elements, as in
the binary case) becomes
xh`i
= µ
[`]
k
h
x`(k−1)+1
i
, (2.7)
because of (2.5). So the ordinary (binary) power p (as a number of elements in a product
xp, which is different from the tuple x(`)) is p = ` + 1. An element x from a polyadic
semigroup Sk is called polyadic h`i-idempotent, if xh`i = x, and it is polyadic idempotent, if
[19]
xh`=1i
≡ µk
h
xk
i
= x. (2.8)
Recall also that a left polyadic identity e ∈ S is defined by
µk
h
e(k−1)
, x
i
= x, ∀x ∈ S, (2.9)
and e is a polyadic identity, if x can be on any place. If e = e(x) depends on x, then we call it
a local polyadic identity (see [21] for the binary case). It follows from (2.9) that all polyadic
identities (or neutral elements) are idempotents.
An element x of the polyadic semigroup Sk is called regular [17], if the equation
µ
[2]
k
h
x, y(2k−3)
, x
i
= x, x ∈ S, (2.10)
has 2k − 3 solutions y1, . . . , y2k−3 ∈ S, which need not be unique (since L
(2)
k − 2 = 2k − 3,
see (2.5)). If we have only the relation (2.10), then polyadic associativity allows us to apply
one (internal) polyadic multiplication and remove k − 1 elements to obtain [17]
µk
h
x, z(k−2)
, x
i
= x, x ∈ S, (2.11)
and we call (2.11) the reduced polyadic regularity of a single element, while (2.10) will be
called the full regularity of a single element. Thus, for a single element x ∈ S, by analogy
with Assertion 1, we have
Assertion 2. In a polyadic semigroup Sk, if any single element x ∈ S satisfies regularity (2.10), it
also satisfies reduced regularity (2.11).
From (2.11) it follows
Corollary 1. Reduced regularity exists, if and only if the arity of multiplication has k 2.
Example 2. In the minimal ternary case k = 3 we have the regularity µ
[2]
3 [x1, x2, x3, x4, x1] =
x1, xi ∈ S, and putting z = µ3[x2, x3, x4] gives the reduced regularity
µ3[x1, z, x1] = x1. (2.12)
But these regularities will give different cycles: one is of length 5, and the other is of length 3.
We now construct a polyadic analog of the higher n-regularity condition (2.4) for a
single element.
Definition 4. An element x of a polyadic semigroup Sk is called full higher n-regular, if there
exists (at least one, not necessarily unique) (n(k − 1) − 1)-element solution (tuple) y(n(k−1)−1) of
µ
[n]
k
h
x, y(n(k−1)−1)
, x
i
= x, x, y1, . . . , yn(k−1)−1 ∈ S, n ≥ 2. (2.13)
Definition 5. The tuple y(n(k−1)−1) from (2.13) is called higher n-regular k-ary conjugated to x,
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 September 2021
5. 5 of 16
or y(n(k−1)−1) is n-inverse k-ary tuple for x of the length (n(k − 1) − 1).
Example 3. In the full higher 3-regular ternary semigroup S3 = hS | µ3i we have for each element
x ∈ S the condition
µ
[2]
3 [x, y1, y2, y3, y4, y5, x] = x, (2.14)
and so each element x ∈ S has a 3-inverse ternary 5-tuple y(5) = (y1, y2, y3, y4, y5), or the 5-tuple
y(5) is the higher 3-regular ternary conjugated to x.
Comparing (2.13) with the full regularity condition (2.10) and its reduction (2.11), we
observe that the full n-regularity condition can be reduced several times to get different
versions of reduced regularity for a single element.
Definition 6. An element x of a polyadic semigroup Sk is called m-reduced higher n-
regular, if there exists a (not necessarily unique) ((n − m)(k − 1) − 1)-element solution (tuple)
y((n−m)(k−1)−1) of
µ
[n]
k
h
x, y((n−m)(k−1)−1)
, x
i
= x, x, y1, . . . , y(n−m)(k−1)−1 ∈ S, 1 ≤ m ≤ n − 1, n ≥ 2.
(2.15)
Example 4. In the ternary case k = 3, we have (for a single element x1 ∈ S) the higher 3-regularity
condition
µ
[3]
3 [x1, x2, x3, x4, x5, x6, x1] = x1, xi ∈ S. (2.16)
We can reduce this relation twice, for instance, as follows: put t = µ3[x2, x3, x4] to get
µ
[2]
3 [x1, t, x5, x6, x1] = x1, (2.17)
and then z = µ3[t, x5, x6] and obtain
µ3[x1, z, x1] = x1, t, z ∈ S, (2.18)
which coincides with (2.12). Again we observe that (2.16), (2.17) and (2.18) give different cycles of
the length 7, 5 and 3, respectively.
3. HIGHER n-INVERSE SEMIGROUPS
Here we introduce and study the higher n-regular and n-inverse (binary) semigroups.
We recall the standard definitions to establish notations (see, e.g., [2,3,5]).
There are two different definitions of a regular semigroup:
1. One-relation definition [22].
2. Multi-relation definition [23,24].
According to the one-relation definition (i): A semigroup S ≡ S2 = hS | µ2, assoc2i is
called regular if each element x ∈ S has its regular conjugated element y ∈ S defined by
xyx = x (2.1), and y need not be unique [22].
Elements x1, x2 of S are called inverse to each other (regular conjugated, mutually
regular), if
x1x2x1 = x1, x2x1x2 = x2, (3.1)
µ
[2]
2 [x1, x2, x1] = x1, µ
[2]
2 [x2, x1, x2] = x2, x1, x2 ∈ S. (3.2)
According to the multi-relation definition (ii): A semigroup is called regular, if each
element x1 has its inverse x2 (3.1), and x2 can be not unique.
For a binary semigroup S2 and ordinary 2-regularity these definitions are equivalent
[2,25]. Indeed, if x1 has some regular conjugated element y ∈ S, and x1yx1 = x1, then
choosing x2 = yx1y, we immediately obtain (3.1).
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Definition 7. A 2-tuple X(2) = (x1, x2) ∈ S × S is called 2-regular sequence of inverses, if it
satisfies (3.1).
Thus we can define regular semigroups in terms of regular sequences of inverses.
Definition 8. A binary semigroup is called a regular (or 2-regular) semigroup, if each element
belongs to a (not necessary unique) 2-regular sequence of inverses X(2) ∈ S × S.
3.1. Higher n-regular semigroups
Following [15], in a regular semigroup S2, for each x1 ∈ S there exists x2 ∈ S such
that x1x2x1 = x1, and for x2 ∈ S there exists y ∈ S satisfying and x2yx2 = x2. Denoting
x3 = yx2, then x2 = x2(yx2) = x2x3, and we observe that x1x2x3x1 = x1, i.e. in a regular
semigroup, from 2-regularity there follows 3-regularity of S2 in the one-relation definition
(i) (cf. for a single element Assertion 1). By analogy, in a regular semigroup, 2-regularity of
an element implies its n-regularity in the one-relation definition (i). Indeed, in this point
there is a difference between the definitions (i) and (ii).
Let us introduce higher analogs of the regular sequences (3.1) (see Definition 7).
Definition 9. A n-tuple X(n) = (x1, . . . , xn) ∈ S×n is called an (ordered) n-regular sequence of
inverses, if it satisfies the n relations
x1x2x3 . . . xnx1 = x1, (3.3)
x2x3 . . . xnx1x2 = x2, (3.4)
.
.
.
xnx1x2 . . . xn−1xn = xn, (3.5)
or in the manifest form of the binary multiplication µ2 (which is needed for the higher arity
generalization)
µ
[n]
2 [x1, x2, x3, . . . , xn, x1] = x1, (3.6)
µ
[n]
2 [x2, x3, . . . , xn, x1, x2] = x2, (3.7)
.
.
.
µ
[n]
2 [xn, x1, x2, . . . , xn−1, xn] = xn. (3.8)
Denote for each element xi ∈ S, i = 1, . . . , n, its n-inverse (n − 1)-tuple (see Definition
2) by xi ∈ S×(n−1), where
x1 = (x2, x3, . . . , xn−1, xn), (3.9)
x2 = (x3, x4 . . . , xn−1, xn, x1), (3.10)
.
.
.
xn = (x1, x2, . . . , xn−2, xn−1). (3.11)
Then the definition of an n-regular sequence of inverses (3.3)–(3.8) will take the concise
form (cf. (2.1) and (2.3))
xixixi = xi, (3.12)
µ
[n]
2 [xi, xi, xi] = xi, xi ∈ S, i = 1, . . . , n. (3.13)
Now by analogy with Definition 8 we have the following (multi-relation)
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Definition 10. A binary semigroup S2 is called a (cycled) n-regular semigroup, if each its element
belongs to a (not necessary unique) n-regular sequence of inverses X(n) ∈ S×n satisfying (3.3)–(
3.5).
This definition incorporates the ordinary regular semigroups (3.1) by putting n = 2.
Remark 1. The concept of n-regularity is strongly multi-relational, in that all relations in (3.3)–(
3.5) or (3.12) should hold, which leads us to consider all the elements in the sequence of inverses on
a par and cycled, analogously to the ordinary case (3.1). This can be expressed informally as “one
element of a n-regular semigroup has n − 1 inverses”.
Example 5. In the 3-regularity case we have three relations defining the regularity
x1x2x3x1 = x1, (3.14)
x2x3x1x2 = x2, (3.15)
x3x1x2x3 = x3. (3.16)
Now the sequence of inverses is X(3) = (x1, x2, x3), and we can say in a symmetrical
way: all of them are “mutually inverses one to another”, iff all three relations (3.14)–(3.15) hold
simultaneously. Also, using (3.9)–(3.11) we see that each element x1, x2, x3 has its 3-inverse 2-tuple
(or, informally, the pair of inverses)
x1 = (x2, x3), (3.17)
x2 = (x3, x1), (3.18)
x3 = (x1, x2). (3.19)
Remark 2. In what follows, we will use this 3-regularity example, when providing derivations and
proofs, for clarity and conciseness. This is worthwhile, because the general n-regularity case does
not differ from n = 3 in principle, and only forces us to consider cumbersome computations with
many indices and variables without enhancing our understanding of the structures.
Let us formulate the Thierrin theorem [25] in the n-regular setting, which connects
one-element regularity and multi-element regularity.
Lemma 1. Every n-regular element in a semigroup has its n-inverse tuple with n − 1 elements.
Proof. Let x1 be a 3-regular element x1 of a semigroup S2, then we can write
x1 = x1y2y3x1, x1, y2, y3 ∈ S. (3.20)
We are to find elements x2, x3 which satisfy (3.14)–(3.16). Put
x2 = y2y3x1y2, (3.21)
x3 = y3x1y2y3. (3.22)
Then for the l.h.s. of (3.14)–(3.16) we derive
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x1x2x3x1 = x1y2y3(x1y2y3x1)y2y3x1 = x1y2y3(x1y2y3x1)
= x1y2y3x1 = x1, (3.23)
x2x3x1x2 = y2(y3x1y2y3)(x1y2y3x1)y2y3(x1y2y3x1)y2
= y2x3(x1x2y3x1)y2 = y2x3x1y2 = x2, (3.24)
x3x1x2x3 = y3x1y2y3(x1y2y3x1)y2y3(x1y2y3x1)y2y3
= y3(x1y2y3x1)y2y3 = y3x1y2y3 = x3. (3.25)
Therefore, if we have any 3-regular element (3.20), the relations (3.14)-(3.19) hold.
We now show that in semigroups the higher n-regularity (in the multi-relation formu-
lation) is wider than for 2-regularity.
Lemma 2. If a semigroup S2 is n-regular, such that the n relations (3.12) are valid, then S2 is
2-regular as well.
Proof. In the case n = 3 we use (3.14)-(3.16) and denote x2x3 = y ∈ S. Then x1yx1 = x1,
and multiplying (3.15) and (3.16) by x3 from the right and x2 from the left, respectively, we
obtain
(x2x3x1x2)x3 = (x2)x3 =⇒ (x2x3)x1(x2x3) = (x2x3) =⇒ yx1y = y, (3.26)
x2(x3x1x2x3) = x2(x3) =⇒ (x2x3)x1(x2x3) = (x2x3) =⇒ yx1y = y. (3.27)
This means that x1 and y are mutually 2-inverse (or 2-regularly conjugated), and so S2 is
2-regular.
The converse requires some conditions on the semigroup.
Theorem 1. The following statements are equivalent for a regular (2-regular) semigroup S2:
1. The 2-regular semigroup S2 is n-regular.
2. The regular semigroup S2 is cancellative.
Proof. (i)⇒(ii) Suppose S2 is 2-regular, which means that there are tuples satisfying (3.1),
or in our notation here for each x1 ∈ S there exists y ∈ S such that x1yx1 = x1 and yx1y = y.
Let y be presented as a product of two elements y = x2x3, x2, x3 ∈ S, which is possible
since the underlying set S of the semigroup is closed with respect to multiplication. After
substitution y into the 2-regularity conditions, we obtain
x1yx1 = x1 =⇒ x1(x2x3)x1 = x1 =⇒ x1x2x3x1 = x1, (3.28)
yx1y = y =⇒ (x2x3)x1(x2x3) = (x2x3) =⇒ (x2x3x1x2)x3 = (x2)x3 (3.29)
=⇒ x2(x3x1x2x3) = x2(x3). (3.30)
We observe that the first line coincides with (3.14), but obtaining (3.15) and (3.16) from
the second and third lines here requires right and left cancellativity, respectively.
(ii)⇒(i) After applying cancellativity to (3.29) and (3.30), then equating expressions in
brackets one obtains (3.15) and (3.16), correspondingly. The first line (3.28) coincides with (
3.14) in any case.
It is known that a cancellative regular semigroup is a group [2,7].
Corollary 2. If a regular semigroup S2 is n-regular, it is a group.
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3.2. Idempotents and higher n-inverse semigroups
Lemma 3. In an n-regular semigroup there are at least n (binary higher) idempotents (no summa-
tion)
ei = xixi, i = 1, . . . , n. (3.31)
Proof. Multiply (3.12) by xi from the right to get xixixixi = xixi.
In the case n = 3 the (higher) idempotents (3.31) become
e1 = x1x2x3, e2 = x2x3x1, e3 = x3x1x2, (3.32)
having the following “chain” commutation relations with elements
e1x1 = x1e2 = x1, (3.33)
e2x2 = x2e3 = x2, (3.34)
e3x3 = x3e1 = x3. (3.35)
Corollary 3. Each higher idempotent ei is a left unit (neutral element) for xi and a right unit
(neutral element) for xi−1.
Lemma 4. If in an n-regular semigroup (higher) idempotents commute, then each element has a
unique n-regular sequence of inverses.
Proof. Suppose in a 3-regular semigroup defined by (3.14)–(3.16) the element x1 has two
pairs of inverses (3.9) x1 = (x2, x3) and x0
1 = (x0
2, x3). Then, in addition to (3.14)–(3.16) and
the triple of idempotents (3.32) we have
x1x0
2x3x1 = x1, (3.36)
x0
2x3x1x0
2 = x0
2, (3.37)
x3x1x0
2x3 = x3. (3.38)
and
e0
1 = x1x0
2x3, e0
2 = x0
2x3x1, e0
3 = x3x1x0
2, (3.39)
and also the idempotents (3.32) and (3.39) commute. We derive
x2 = x2x3x1x2 = x2x3 x1x0
2x3x1
x1x2 = x2x3 x1x0
2x3 x1x0
2x3x1
x2
= x2x3x1x0
2 x3x1x0
2
(x3x1x2) = x2x3x1x0
2(x3x1x2) x3x1x0
2
= (x2x3x1) x0
2x3x1
x2x3x1x0
2 = x0
2x3x1
(x2x3x1)x2x3x1x0
2
= x0
2x3(x1x2x3x1)x2x3x1x0
2 = x0
2x3(x1x2x3x1)x0
2 = x0
2x3x1x0
2 = x0
2. (3.40)
A similar derivation can be done for x00
1 = (x2, x00
3 ) and other indices.
Definition 11. A higher n-regular semigroup is called a higher n-inverse semigroup, if each
element xi ∈ S has its unique n-inverse (n − 1)-tuple xi ∈ S×(n−1) (3.9)–(3.11).
Thus, as in the regular (multi-relation 2-regular) semigroups [26–28], we arrive at a
higher regular generalization of the characterization of the inverse semigroups by
Theorem 2. A higher n-regular semigroup is an inverse semigroup, if their idempotents commute.
Further properties of n-inverse semigroups will be studied elsewhere using the stan-
dard methods (see, e.g. [2,6,7,29]).
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4. HIGHER n-INVERSE POLYADIC SEMIGROUPS
We now introduce a polyadic version of the higher n-regular and n-inverse semigroups
defined in the previous section. For this we will use the relations with the manifest form of
the binary multiplication to follow (ii)-(iii) of the arity invariance principle (see Definition
3).
4.1. Higher n-regular polyadic semigroups
First, define the polyadic version of the higher n-regular sequences (3.6)–(3.8) in the
framework of the arity invariance principle by substituting µ
[n]
2 7→ µ
[n]
k and then changing
sizes of tuples, and generalizing to higher n (2.10).
Definition 12. In a polyadic (k-ary) semigroup Sk = hS | µki (see Subsection 2.2) a n(k − 1)-
tuple X(n(k−1)) =
x1, . . . , xn(k−1)
∈ S×n(k−1) is called a polyadic n-regular sequence of in-
verses, if it satisfies the n(k − 1) relations
µ
[n]
k
h
x1, x2, x3, . . . , xn(k−1), x1
i
= x1, (4.1)
µ
[n]
k
h
x2, x3, . . . , xn(k−1), x1, x2
i
= x2, (4.2)
.
.
.
µ
[n]
k
h
xn(k−1), x1, x2, . . . , xn(k−1)−1, xn(k−1)
i
= xn(k−1). (4.3)
Definition 13. A polyadic semigroup Sk is called an n-regular semigroup, if each element belongs
to a (not necessary unique) n-regular sequence of inverses X(n(k−1)) ∈ S×n(k−1).
In a polyadic n-regular semigroup, each element instead of having one inverse (regular
conjugated) element will have a tuple, and we now, instead of (2.1)–(2.2), have
Definition 14. In a polyadic (k-ary) n-regular semigroup each element xi ∈ S, where i =
1, . . . , n(k − 1), has a polyadic n-inverse, as the (n(k − 1) − 1)-tuple x
[k]
i ∈ S×(n(k−1)−1) (cf.
the binary case Definition 2 and (3.9)–(3.11))
x
[k]
1 =
x2, x3, . . . , xn(k−1)−1, xn(k−1)
, (4.4)
x
[k]
2 =
x3, x4 . . . , xn(k−1)−1, xn(k−1), x1
, (4.5)
.
.
.
x
[k]
n(k−1)
=
x1, x2, . . . , xn(k−1)−2, xn(k−1)−1
. (4.6)
In this notation the polyadic (k-ary) n-regular sequence of inverses (4.1)–(4.3) can be
written in the customary concise form (see (3.13) for a binary semigroup)
µ
[n]
k
h
xi, x
[k]
i , xi
i
= xi, xi ∈ S, i = 1, . . . , n(k − 1). (4.7)
Note that at first sight one could consider here also the procedure of reducing the
number of multiplications from any number to one [17], as in the single element regularity
case (see Assertion 2 and Example 2, 4). However, if we consider the whole polyadic
n-regular sequence of inverses (4.1)–(4.3) consisting of n(k − 1) relations (4.1)–(4.3), it will
be not possible in general to reduce arity in all of them simultaneously.
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Example 6. If we consider the ordinary regularity (2-regularity) for a ternary semigroup S3, we
obtain the ternary regular sequence of n(k − 1) = 4 elements and four regularity relations
µ
[2]
3 [x1, x2, x3, x4, x1] = x1, (4.8)
µ
[2]
3 [x2, x3, x4, x1, x2] = x2, (4.9)
µ
[2]
3 [x3, x4, x1, x2, x3] = x3, (4.10)
µ
[2]
3 [x4, x1, x2, x3, x4] = x4, xi ∈ S. (4.11)
Elements x1, x2, x3, x4 have these ternary inverses (cf. the binary 3-regularity (3.17)–(3.19))
x
[3]
1 = (x2, x3, x4), (4.12)
x
[3]
2 = (x3, x4, x1), (4.13)
x
[3]
3 = (x4, x1, x2), (4.14)
x
[3]
4 = (x1, x2, x3). (4.15)
Remark 3. This definition of regularity in ternary semigroups is in full agreement with the arity
invariance principle (Definition 3): it has (the minimum) two (k-ary) multiplications as in the
binary case (2.2). Although we can reduce the number of multiplications to one, as in (2.12) for
a single relation, e.g. by the substitution z = µ3[x2, x3, x4], this cannot be done in all the cycled
relations (4.8)–(4.11): the third relations (4.10) cannot be presented in terms of z as well as the right
hand sides.
Remark 4. If we take the definition of regularity for a single relation, reduce it to one (k-ary) multi-
plication and then “artificially” cycle it (see [17] and references citing it), we find a conflict with the
arity invariance principle. Indeed, in the ternary case we obtain µ3[x, y, x] = x, µ3[y, x, y] = y.
Despite the similarity to the standard binary regularity (3.1)-(3.2), it contains only one multipli-
cation, and therefore it could be better treated as a symmetry property of the ternary product µ3,
rather than as a relation between variables, e.g., as a regularity which needs at least two products.
Also, the length of the sequence is two, as in the binary case, but it should be n(k − 1) = 4, as in
the ternary 2-regular sequence (4.8)–(4.11) according to the arity invariance principle.
Example 7. For a 3-regular ternary semigroup S3 = hS | µ3i with n = 3 and k = 3 we have a
3-regular sequence of 6 (mutual) inverses X(6) = (x1, x2, x3, x4, x5, x6) ∈ S×6, which satisfy the
following six 3-regularity conditions
µ
[3]
3 [x1, x2, x3, x4, x5, x6, x1] = x1, (4.16)
µ
[3]
3 [x2, x3, x4, x5, x6, x1, x2] = x2, (4.17)
µ
[3]
3 [x3, x4, x5, x6, x1, x2, x3] = x3, (4.18)
µ
[3]
3 [x4, x5, x6, x1, x2, x3, x4] = x4, (4.19)
µ
[3]
3 [x5, x6, x1, x2, x3, x4, x5] = x5, (4.20)
µ
[3]
3 [x6, x1, x2, x3, x4, x5, x6] = x6, xi ∈ S. (4.21)
This is the first nontrivial case in both arity k 6= 2 and regularity n 6= 2 (see Remark 2).
Remark 5. In this example we can reduce the number of multiplications, as in Example 4, to
one single relation, but not to all of them. For instance, we can put in the first relations (4.16)
t = µ3[x2, x3, x4] and further z = µ3[t, x5, x6], but, for instance, the third relation (4.18) cannot
be presented in terms of t, z, as in the first one in (4.16)–(4.21), because of the splitting of variables
in t, z, as well as in the right hand sides.
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The polyadic analog of the Thierrin theorem [25] in n-regular setting is given by:
Lemma 5. Every n-regular element in a polyadic (k-ary) semigroup has a polyadic n-inverse tuple
with (n(k − 1) − 1) elements.
Its proof literally repeats that of Lemma 1, but with different lengths of sequences.
The same is true for the Lemma 2 and Theorem 1 by exchanging S2 → Sk.
Definition 15. A higher n-regular polyadic semigroup Sk is called a higher n-inverse polyadic
semigroup, if each element xi ∈ S has a unique n-inverse (n(k − 1) − 1)-tuple xi ∈ S×(n(k−1)−1)
(4.4)–(4.6).
In searching for polyadic idempotents we observe that the binary regularity (3.13) and
polyadic regularity (4.7) differ considerably in lengths of tuples. In the binary case any
length is allowed, and one can define idempotents ei by (3.31), as a left neutral element
for xi, such that (no summation) eixi = xi , xi ∈ S2 (see (3.33)–(3.35)). At first sight, for
polyadic n-regularity (4.7), we could proceed in a similar way. However we have
Proposition 1. In the n-regular polyadic (k-ary) semigroup the length of the tuple (xi, xi) is
allowed to give an idempotent, only if k = 2, i.e. the semigroup is binary.
Proof. The allowed length of the tuple (xi, xi) (to give one element, an idempotent, see (
2.5)), where xi are in (4.4)–(4.6), is `(k − 1) + 1. While the tuple (xi, xi, xi) in (4.7) has the
given length n(k − 1) + 1, and we obtain the equation for ` as
`(k − 1) + 1 = n(k − 1) =⇒ ` = n −
1
k − 1
(4.22)
The equation (4.22) has only one solution over N namely ` = n − 1, iff k = 2.
This means that to go beyond the binary semigroups k 2 and have idempotents,
one needs to introduce a different regularity condition to (2.10) and (4.7), which we will do
below.
4.2. Idempotents and sandwich polyadic regularity
Here we go in the opposite direction to that above: we will define the idempotents
and then construct the needed regularity conditions using them, which in the limit k = 2
and n = 2 will give the ordinary binary regularity (3.1)–(3.2).
Let us formulate the binary higher n-regularity (3.13) in terms of the local polyadic
identities (2.9). We write the idempotents (3.31) in the form
ei = µ2[xi, xi], (4.23)
where xi are n-inverses being (n − 1)-tuples (3.9)–(3.11).
In terms of the idempotents ei (4.23) the higher n-regularity conditions (3.13) become
µ2[ei, xi] = xi. (4.24)
Assertion 3. The binary n-regularity conditions coincide with the definition of the local left
identities.
Proof. Compare (4.24) and the definition (2.9) with k = 2 (see [21]).
Thus, the main idea is to generalize (using the arity invariance principle) to the
polyadic case the binary n-regularity in the form (4.24) and also idempotents (4.23), but not
(3.12)–(3.13), as it was done in (4.7).
By analogy with (4.23) let us introduce
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13. 13 of 16
e0
i = µk
xi,
n−1
z }| {
x0
i, . . . , x0
i
, xi ∈ S, i = 1, . . . , k, (4.25)
where x0
i is the polyadic (k − 1)-tuple for xi (which differs from (4.4)–(4.6))
x0
1 = (x2, x3, . . . , xk), (4.26)
x0
2 = (x3, x4, . . . , xk, x1), (4.27)
.
.
.
x0
k = (x1, x2, . . . , xk−2, xk−1). (4.28)
Definition 16. Sandwich polyadic n-regularity conditions are defined as
µk
k−1
z }| {
e0
i . . . e0
i, xi
= xi. (4.29)
It follows form (4.29), that e0
i are polyadic idempotents (see (2.8) and (2.9)). It is seen
that in the limit k = 2 and n = 2 (4.25)–(4.29) give the ordinary regularity (3.1).
Now, by analogy with the binary case Assertion 3, we have
Assertion 4. The sandwich polyadic n-regularity conditions (4.29) coincide with the definition of
the local left polyadic identities.
Proof. Compare (4.29) and the definition (2.9).
By analogy with Definition 13 we have
Definition 17. A k-ary semigroup Sk is called a sandwich polyadic n-regular semigroup, if each
element has a (not necessary unique) k-tuple of inverses satisfying (4.25)–(4.29), or an element
xi ∈ S has its local left polyadic identity e0
i (4.25).
Now instead of (4.4)–(4.6) we have for the inverses
Definition 18. Each element of a sandwich polyadic (k-ary) n-regular semigroup xi ∈ S, i =
1, . . . , k − 1 has its (sandwich) polyadic k-inverse, as the (k − 1)-tuple x
0[k]
i ∈ S×(k−1) (see (4.26)–(
4.28)).
Example 8. In the ternary case k = 3 and n = 2 we have the following 3 ternary idempotents
e0
1 = µ3[x1, x2, x3], (4.30)
e0
2 = µ3[x2, x3, x1], (4.31)
e0
3 = µ3[x3, x1, x2], xi ∈ S, (4.32)
such that
µ3
e0
i, e0
i, e0
i
= e0
i, i = 1, 2, 3. (4.33)
The sandwich ternary regularity conditions become (cf. (4.8)–(4.11))
µ
[3]
3 [x1, x2, x3, x1, x2, x3, x1] = x1, (4.34)
µ
[3]
3 [x2, x3, x1, x2, x3, x1, x2] = x2, (4.35)
µ
[3]
3 [x3, x1, x2, x3, x1, x2, x3] = x3. (4.36)
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14. 14 of 16
It can be seen from (4.34)–(4.36) why we call such regularity “sandwich”: each xi appears in
the l.h.s. not only 2 times, on the first and the last places, as in the previous definitions, but also in
the middle, which gives the possibility for us to define idempotents. In general, the ith condition of
sandwich regularity (4.29) will contain k − 2 middle elements xi.
Each of the elements x1, x2, x3 ∈ S has a 2-tuple of the ternary inverses (cf. the inverses for
the binary 3-regularity (3.17)–(3.19) and the ternary regularity (4.12)–(4.15)) as
x0
1 = (x2, x3), (4.37)
x0
2 = (x3, x1), (4.38)
x0
3 = (x1, x2). (4.39)
Remark 6. In (4.34) we can also reduce the number of multiplications to one (see [17]), but only
in a single relation (see Example 4 and Remark 5). However, in this case the whole system (4.34)–(
4.36) will lose its self-consistency, since we are using the multi-relation definition of the sandwich
regular semigroup (see (3.1) for the ordinary regularity).
Example 9. The non-trivial case in both higher arity k 6= 2 and higher sandwich regularity n 6= 2
is the sandwich 3-regular 4-ary semigroup S4 in which there exist 4 elements satisfying the higher
sandwich 3-regularity relations
µ
[4]
4 [x1, x2, x3, x4, x1, x2, x3, x4, x1, x2, x3, x4, x1] = x1, (4.40)
µ
[4]
4 [x2, x3, x4, x1, x2, x3, x4, x1, x2, x3, x4, x1, x2] = x2, (4.41)
µ
[4]
4 [x3, x4, x1, x2, x3, x4, x1, x2, x3, x4, x1, x2, x3] = x3, (4.42)
µ
[4]
4 [x4, x1, x2, x3, x4, x1, x2, x3, x4, x1, x2, x3, x4] = x4, xi ∈ S. (4.43)
The 4-ary idempotents are
e0
1 = µ4[x1, x2, x3, x4], (4.44)
e0
2 = µ4[x2, x3, x4, x1],
e0
3 = µ4[x3, x4, x1, x2], (4.45)
e0
3 = µ4[x4, x1, x2, x3], (4.46)
and they obey the following commutation relations with elements (cf. (3.33)–(3.35))
µ4
e0
1, e0
1, e0
1, x1
= µ4
x1, e0
2, e0
2, e0
2
= x1, (4.47)
µ4
e0
2, e0
2, e0
2, x2
= µ4
x2, e0
3, e0
3, e0
3
= x2, (4.48)
µ4
e0
3, e0
3, e0
3, x3
= µ4
x3, e0
4, e0
4, e0
4
= x3, (4.49)
µ4
e0
4, e0
4, e0
4, x4
= µ4
x4, e0
1, e0
1, e0
1
= x4. (4.50)
Each of the elements x1, x2, x3, x4 has its triple of 4-ary inverses
x0
1 = (x2, x3, x4), (4.51)
x0
2 = (x3, x4, x1), (4.52)
x0
3 = (x4, x1, x2), (4.53)
x0
4 = (x1, x2, x3). (4.54)
By analogy with Corollary 3 we now have
Corollary 4. Each polyadic idempotent e0
i of the sandwich higher n-regular k-ary semigroup Sk is
a local left polyadic identity (neutral element) for xi and a local right polyadic identity for xi−1.
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15. 15 of 16
Definition 19. A sandwich higher n-regular polyadic semigroup Sk is called a sandwich n-inverse
polyadic semigroup, if each element xi ∈ S has a unique n-inverse (k − 1)-tuple x0
i ∈ S×(k−1) (
4.26)–(4.28).
In search of a polyadic analog of the Lemma 4, we have found that the commutation
of polyadic idempotents does not lead to uniqueness of the n-inverse elements (4.26)–(
4.28). The problem appears because of the presence of the middle elements in (4.25) and,
e.g., in (4.34) (see the discussion after (4.36)), while the middle elements do not exist in the
previous formulations of regularity.
Acknowledgement. The author is deeply grateful to Vladimir Akulov, Mike He-
witt, Mikhail Krivoruchenko, Thomas Nordahl, Vladimir Tkach and Raimund Vogl for
numerous fruitful and useful discussions, valuable help and support.
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