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.
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.
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.
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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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.
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.
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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
Plan: 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hosszú-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 9. Polyadic rings and fields. 10. Polyadic analogs of the integer number ring Z and the Galois field GF(p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers.
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed.
MSC classes: 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Fixed Point Theorems for Weak K-Quasi Contractions on a Generalized Metric Sp...IJERA Editor
In this paper we obtain conditions for a k- quasi contraction on a generalized metric space with a partial order to have a fixed point. Using this, we derive certain known results as corollaries.
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 Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
Plan: 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hosszú-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 9. Polyadic rings and fields. 10. Polyadic analogs of the integer number ring Z and the Galois field GF(p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers.
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed.
MSC classes: 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Fixed Point Theorems for Weak K-Quasi Contractions on a Generalized Metric Sp...IJERA Editor
In this paper we obtain conditions for a k- quasi contraction on a generalized metric space with a partial order to have a fixed point. Using this, we derive certain known results as corollaries.
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.
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.
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.
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.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
EVEN GRACEFUL LABELLING OF A CLASS OF TREESFransiskeran
A labelling or numbering of a graph G with q edges is an assignment of labels to the vertices of G that
induces for each edge uv a labelling depending on the vertex labels f(u) and f(v). A labelling is called a
graceful labelling if there exists an injective function f: V (G) → {0, 1,2,......q} such that for each edge xy,
the labelling │f(x)-f(y)│is distinct. In this paper, we prove that a class of Tn trees are even graceful.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Similar to Steven Duplij, "Higher braid groups and regular semigroups from polyadic binary correspondence" (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
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
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
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, 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
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.
https://arxiv.org/abs/2011.04370
A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
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.
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.
Richard's entangled aventures in wonderlandRichard 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.
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.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
2. Mathematics 2021, 9, 972 2 of 17
M(k − 1) ≡ M((k−1)×(k−1))
=
0 m(i1×i2) 0 . . . 0
0 0 m(i2×i3) . . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... m(ik−2×ik−1)
m(ik−1×i1) 0 0 . . . 0
. (1)
In other words, it is given by the cyclic shift (k − 1) × (k − 1) matrix, in which identi-
ties are replaced by blocks of suitable sizes and with arbitrary entries.
The set {M(k − 1)} is closed with respect to the product of k matrices, while the
product of two (or less than k of such matrices) has not the same form as (1). In this
sense, the k-ary multiplication is not reducible (or derived) to the binary multiplication;
therefore, we will call them k-ary matrices. This “non-reducibility” is the key property of
k-ary matrices, which will be used in the below constructions. The matrices of the shape (1)
form a k-ary semigroup (which cannot be reduced to binary semigroups), and, when the
blocks are over an associative binary ring, then the total k-ary associativity follows from
the ordinary associativity of the binary matrix multiplication of the blocks.
Our proposal is to use single arbitrary elements (from rings with associative multipli-
cation) in place of the blocks m(i×j), supposing that the elements of the multiplicative part
G of the rings form binary (semi)groups having some special properties. Then, we inves-
tigate the similar correspondence between the (multiplicative) properties of the matrices
M(k−1), related to idempotence and order, and the appearance of the relations in G leading
to regular semigroups and braid groups, respectively. We call this connection a polyadic
matrix-binary (semi)group correspondence (or in short the polyadic-binary correspondence).
In the lowest arity case k = 3, the ternary case, the 2 × 2 matrices M(2) are anti-triangle.
From (M(2))3
= M(2) and (M(2))3
∼ E(2) (where E(2) is the ternary identity; see below),
we obtain the correspondences of the above conditions on M(2) with the ordinary regular
semigroups and braid groups, respectively. In this way, we extend the polyadic-binary
correspondence on -arities k ≥ 4 to get the higher relations
(M(k − 1))k
=
= M(k − 1) corresponds to higher k-degree regular semigroups,
= qE(k − 1) corresponds to higher k-degree braid groups,
(2)
where E(k − 1) is the k-ary identity (see below), and q is a fixed element of the braid group.
3. Ternary Matrix Group Corresponding to the Regular Semigroup
Let Gfree =
n
G | µ
g
2
o
be a free semigroup with the underlying set G =
n
g(i)
o
and the
binary multiplication. The anti-diagonal matrices over Gfree
Mg
(2) = M(2×2)
g(1)
, g(2)
=
0 g(1)
g(2) 0
, g(1)
, g(2)
∈ Gfree (3)
form a ternary semigroup M
g
3 ≡ M
g
k=3 =
n
Mg(2) | µ
g
3
o
, where Mg(2) = {Mg(2)} is the
set of ternary matrices (3) closed under the ternary multiplication
µ
g
3
h
M
g
1 (2), M
g
2 (2), M
g
3 (2)
i
= M
g
1 (2)M
g
2 (2)M
g
3 (2), ∀M
g
1 (2), M
g
2 (2), M
g
3 (2) ∈ M
g
3, (4)
being the ordinary matrix product. Recall that an element Mg(2) ∈ M
g
3 is idempotent, if
µ
g
3[Mg
(2), Mg
(2), Mg
(2)] = Mg
(2), (5)
3. Mathematics 2021, 9, 972 3 of 17
which, in the matrix form (4), leads to
(Mg
(2))3
= Mg
(2). (6)
We denote the set of idempotent ternary matrices by M
g
id(2) =
n
M
g
id(2)
o
.
Definition 1. A ternary matrix semigroup in which every element is idempotent (6) is called an
idempotent ternary semigroup.
Using (3) and (6), the idempotence expressed in components gives the regularity conditions
g(1)
g(2)
g(1)
= g(1)
, (7)
g(2)
g(1)
g(2)
= g(2)
, ∀g(1)
, g(2)
∈ Gfree. (8)
Definition 2. A binary semigroup Gfree in which any two elements are mutually regular (7) and (8)
is called a regular semigroup Greg.
Proposition 1. The set of idempotent ternary matrices (6) form a ternary semigroup M
g
3,id =
{Mid(2) | µ3}, if Greg is abelian.
Proof. It follows from (7) and (8) that idempotence (and following from it regularity) is
preserved with respect to the ternary multiplication (4), only when any g(1), g(2) ∈ Gfree
commute.
Definition 3. We say that the set of idempotent ternary matrices M
g
id(2) (6) is in ternary-binary
correspondence with the regular (binary) semigroup Greg and write this as
M
g
id(2) m Greg. (9)
This means that such property of the ternary matrices as their idempotence (6) leads
to the regularity conditions (7) and (8) in the correspondent binary group Gfree.
Remark 1. The correspondence (9) is not a homomorphism and not a bi-element mapping [17],
and also not a heteromorphism in the sense of Reference [18], because we do not demand that the set
of idempotent matrices M
g
id(2) form a ternary semigroup (which is possible in commutative case of
Gfree only; see Proposition 1).
4. Polyadic Matrix Semigroup Corresponding to the Higher Regular Semigroup
Next we extend the ternary-binary correspondence (9) to the k-ary matrix case (1) and
thereby obtain higher k-regular binary semigroups. We use the following notation:
• Round brackets: (k) is size of matrix k × k, as well as the sequential number of a
matrix element.
• Square brackets: [k] is number of multipliers in the regularity and braid conditions.
• Angle brackets: h`ik is the polyadic power (number of k-ary multiplications).
Let us introduce the (k − 1) × (k − 1) matrix over a binary group Gfree of the form (1)
Mg
(k − 1) ≡ M((k−1)×(k−1))
g(1)
, g(2)
, . . . , g(k−1)
=
0 g(1) 0 . . . 0
0 0 g(2) . . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... g(k−2)
g(k−1) 0 0 . . . 0
, (10)
4. Mathematics 2021, 9, 972 4 of 17
where g(i) ∈ Gfree.
Definition 4. The set of k-ary matrices Mg(k − 1) (10) over Gfree is a k-ary matrix semigroup
M
g
k =
n
Mg(k − 1) | µ
g
k
o
, where the multiplication
µ
g
k
h
M
g
1 (k − 1), M
g
2 (k − 1), . . . , M
g
k (k − 1)
i
= M
g
1 (k − 1)M
g
2 (k − 1) . . . M
g
k (k − 1), M
g
i (k − 1) ∈ M
g
k (11)
is the ordinary product of k matrices M
g
i (k − 1) ≡ M((k−1)×(k−1))
g
(1)
i , g
(2)
i , . . . , g
(k−1)
i
; see (10).
Recall that the polyadic power ` of an element M from a k-ary semigroup Mk is
defined by (e.g., Reference [19])
Mh`ik = (µk)`
`(k−1)+1
z }| {
M, . . . , M
, (12)
such that ` coincides with the number of k-ary multiplications. In the binary case k = 2 the
polyadic power is connected with the ordinary power p (number of elements in the product)
as p = ` + 1, i.e., Mh`i2 = M`+1 = Mp. In the ternary case k = 3, we have h`i3 = 2` + 1,
and so the l.h.s. of (6) is of polyadic power ` = 1.
Definition 5. An element of a k-ary semigroup M ∈ M3 is called idempotent, if its first polyadic
power coincides with itself
Mh1ik = M, (13)
and h`i-idempotent, if
Mh`ik = M, Mh`−1ik 6= M. (14)
Definition 6. A k-ary semigroup Mk is called idempotent (`-idempotent), if each of its elements
M ∈ Mk is idempotent (h`i-idempotent).
Assertion 1. From Mh1ik = M it follows that Mh`ik = M, but not vice-versa; therefore, all h1i-
idempotent elements are h`i-idempotent, but an h`i-idempotent element need not be h1i-idempotent.
Therefore, the definition given in(14) makes sense.
Proposition 2. If a k-ary matrix Mg(k − 1) ∈ M
g
k is idempotent (13), then its elements satisfy
the (k − 1) relations
g(1)
g(2)
, . . . g(k−2)
g(k−1)
g(1)
= g(1)
, (15)
g(2)
g(3)
, . . . g(k−1)
g(1)
g(2)
= g(2)
, (16)
.
.
.
g(k−1)
g(1)
g(2)
, . . . g(k−2)
g(k−1)
= g(k−1)
, ∀g(1)
, . . . , g(k−1)
∈ Gfree. (17)
Proof. This follows from (10), (11) and (13).
Definition 7. The relations (15)–(17) are called (higher) [k]-regularity (or higher k-degree regular-
ity). The case k = 3 is the standard regularity ([3]-regularity in our notation) (7) and (8).
5. Mathematics 2021, 9, 972 5 of 17
Proposition 3. If a k-ary matrix Mg(k − 1) ∈ M
g
k is h`i-idempotent (14), then its elements
satisfy the following (k − 1) relations
`
z }| {
g(1)
g(2)
. . . g(k−2)
g(k−1)
. . .
g(1)
g(2)
, . . . g(k−2)
g(k−1)
g(1)
= g(1)
, (18)
`
z }| {
g(2)
g(3)
, . . . g(k−2)
g(k−1)
g(1)
. . .
g(2)
g(3)
. . . g(k−2)
g(k−1)
g(1)
g(2)
= g(2)
, (19)
.
.
.
`
z }| {
g(k−1)
g(1)
g(2)
. . . g(k−3)
g(k−2)
. . .
g(k−1)
g(1)
g(2)
. . . g(k−3)
g(k−2)
g(k−1)
= g(k−1)
, (20)
∀g(1)
, . . . , g(k−1)
∈ Gfree.
Proof. This also follows from (10), (11), and (14).
Definition 8. The relations (15)–(17) are called (higher) [k]-h`i-regularity. The case k = 3 (7)
and (8) is the standard regularity ([3]-h1i-regularity in this notation).
Definition 9. A binary semigroup Gfree, in which any k − 1 elements are [k]-regular ([k]-h`i-
regular), is called a higher [k]-regular ([k]-h`i-regular) semigroup Greg[k] (Gh`i-reg[k]).
Similarly to Assertion 1, it is seen that [k]-h`i-regularity (18)–(20) follows from [k]-
regularity (15)–(17), but not the other way around; therefore, we have:
Assertion 2. If a binary semigroup Greg[k] is [k]-regular, then it is [k]-h`i-regular as well, but not
vice-versa.
Proposition 4. The set of idempotent (h`i-idempotent) k-ary matrices M
g
id(k − 1) form a k-ary
semigroup M
g
3,id =
n
M
g
id(k − 1) | µ
g
k
o
, if and only if Greg[k] (G`-reg[k]) is abelian.
Proof. It follows from (15)–(20) that the idempotence (h`i-idempotence) and the following
[k]-regularity ([k]-h`i-regularity) are preserved with respect the k-ary multiplication (11)
only in the case, when all g(1), . . . , g(k−1) ∈ Gfree mutually commute.
By analogy with (9), we have:
Definition 10. We will say that the set of k-ary (k − 1) × (k − 1) matrices M
g
id(k − 1) (10) over
the underlying set G is in polyadic-binary correspondence with the binary [k]-regular semigroup
Greg[k] and write this as
M
g
id(k − 1) m Greg[k]. (21)
Thus, using the idempotence condition for k-ary matrices in components (being simulta-
neously elements of a binary semigroup Gfree) and the polyadic-binary correspondence (21)
we obtain the higher regularity conditions (15)–(20) generalizing the ordinary regularity
(7) and (8), which allows us to define the higher [k]-regular binary semigroups Greg[k]
(Gh`i-reg[k]).
Example 1. The lowest nontrivial (k ≥ 3) case is k = 4, where the 3 × 3 matrices over Gfree are
of the shape
M(3) = M(3×3)
=
0 a 0
0 0 b
c 0 0
, a, b, c ∈ Gfree, (22)
6. Mathematics 2021, 9, 972 6 of 17
and they form the 4-ary matrix semigroup M
g
4. The idempotence (M(3))h1i4 = (M(3))3
= M(3)
gives three [4]-regularity conditions
abca = a, (23)
bcab = b, (24)
cabc = c. (25)
According to the polyadic-binary correspondence (21), the conditions (23)–(25) are [4]-
regularity relations for the binary semigroup Gfree, which defines to the higher [4]-regular binary
semigroup Greg[4].
In the case ` = 2, we have (M(3))h2i4 = (M(3))7
= M(3), which gives three [4]-h2i-
regularity conditions (they are different from [7]-regularity)
abcabca = a, (26)
bcabcab = b, (27)
cabcabc = c, (28)
and these define the higher [4]-h2i-regular binary semigroup Gh2i-reg[4]. Obviously, (26)–(28)
follow from (23)–(25), but not vice-versa.
The higher regularity conditions (23)–(25) obtained above from the idempotence
of polyadic matrices using the polyadic-binary correspondence, appeared first in Refer-
ence [20] and were then used for transition functions in the investigation of semisuperman-
ifolds [6] and higher regular categories in TQFT [7,21].
Now, we turn to the second line of (2), and in the same way as above introduce higher
degree braid groups.
5. Ternary Matrix Group Corresponding to the Braid Group
Recall the definition of the Artin braid group [22] in terms of generators and rela-
tions [4] (we follow the algebraic approach; see, e.g., Reference [23]).
The Artin braid group Bn (with n strands and the identity e ∈ Bn) has the presentation
by n − 1 generators σ1, . . . , σn−1 satisfying n(n − 1)/2 relations
σiσi+1σi = σi+1σiσi+1, 1 ≤ i ≤ n − 2, (29)
σiσj = σjσi, |i − j| ≥ 2, (30)
where (29) are called the braid relations, and (30) are called far commutativity. A general
element of Bn is a word of the form
w = σ
p1
i1
. . . σ
pr
ir
. . . σ
pm
im
, im = 1, . . . , n, (31)
where pr ∈ Z are (positive or negative) powers of the generators σir
, r = 1, . . . , m and
m ∈ N.
For instance, B3 is generated by σ1 and σ2 satisfying one relation σ1σ2σ1 = σ2σ1σ2, and
is isomorphic to the trefoil knot group. The group B4 has 3 generators σ1, σ2, σ3 satisfying
σ1σ2σ1 = σ2σ1σ2, (32)
σ2σ3σ2 = σ3σ2σ3, (33)
σ1σ3 = σ3σ1. (34)
The representation theory of Bn is well known and well established [4,5]. The connec-
tions with the Yang-Baxter equation were investigated, e.g., in Reference [9].
Now, we build a ternary group of matrices over Bn having generators satisfying
relations which are connected with the braid relations (29) and (30). We then generalize
7. Mathematics 2021, 9, 972 7 of 17
our construction to a k-ary matrix group, which gives us the possibility to “go back” and
define some special higher analogs of the Artin braid group.
Let us consider the set of anti-diagonal 2 × 2 matrices over Bn
M(2) = M(2×2)
b(1)
, b(2)
=
0 b(1)
b(2) 0
, b(1)
, b(2)
∈ Bn. (35)
Definition 11. The set of matrices M(2) = {M(2)} (35) over Bn form a ternary matrix semigroup
Mk=3 = M3 = {M(2) | µ3}, where k = 3 is the arity of the following multiplication
µ3[M1(2), M2(2), M3(2)] ≡ M1(2), M2(2), M3(2) = M(2), (36)
b
(1)
1 b
(2)
2 b
(1)
3 = b(1)
, (37)
b
(2)
1 b
(1)
2 b
(2)
3 = b(2)
, b
(1)
i , b
(2)
i ∈ Bn, Mi(2) =
0 b
(1)
i
b
(2)
i 0
!
(38)
and the associativity is governed by the associativity of both the ordinary matrix product in the r.h.s.
of (36) and Bn.
Proposition 5. M(3) is a ternary matrix group.
Proof. Each element of the ternary matrix semigroup M(2) ∈ M3 is invertible (in the
ternary sense) and has a querelement M̄(2) (a polyadic analog of the group inverse [24])
defined by
µ3[M(2), M(2), M̄(2)] = µ3[M(2), M̄(2), M(2)] = µ3[M̄(2), M(2), M(2)] = M(2). (39)
It follows from (36)–(38) that
M̄(2) = (M(2))−1
=
0
b(1)
−1
b(2)
−1
0
, b(1)
, b(2)
∈ Bn, (40)
where
M(2)
−1
denotes the ordinary matrix inverse (but not the binary group inverse
which does not exist in the k-ary case, k ≥ 3). Non-commutativity of µ3 is provided by
(37) and (38).
The ternary matrix group M3 has the ternary identity
E(2) =
0 e
e 0
, e ∈ Bn, (41)
where e is the identity of the binary group Bn, and
µ3[M(2), E(2), E(2)] = µ3[E(2), M(2), E(2)] = µ3[E(2), E(2), M(2)] = M(2). (42)
We observe that the ternary product µ3 in components is “naturally braided”
(37) and (38). This allows us to ask the question: which generators of the ternary group
M3 can be constructed using the Artin braid group generators σi ∈ Bn and the relations
(29) and (30)?
6. Ternary Matrix Generators
Let us introduce (n − 1)2
ternary 2 × 2 matrix generators
Σij(2) = Σ
(2×2)
ij σi, σj
=
0 σi
σj 0
, (43)
8. Mathematics 2021, 9, 972 8 of 17
where σi ∈ Bn, i = 1 . . . , n − 1 are generators of the Artin braid group. The querelement of
Σij(2) is defined by analogy with (40) as
Σ̄ij(2) = Σij(2)
−1
=
0 σ−1
j
σ−1
i 0
!
. (44)
Now, we are in a position to present a ternary matrix group with multiplication µ3 in
terms of generators and relations in such a way that the braid group relations (29) and (30)
will be reproduced.
Proposition 6. The relations for the matrix generators Σij(2) corresponding to the braid group
relations for σi (29) and (30) have the form
µ3[Σi,i+1(2), Σi,i+1(2), Σi,i+1(2)] = µ3[Σi+1,i(2), Σi+1,i(2), Σi+1,i(2)]
= q
[3]
i E(2), 1 ≤ i ≤ n − 2, (45)
µ3
Σij(2), Σij(2), E(2)
= µ3
Σji(2), Σji(2), E(2)
, |i − j| ≥ 2, (46)
where q
[3]
i = σiσi+1σi = σi+1σiσi+1, and E(2) is the ternary identity (41).
Proof. Use µ3 as the triple matrix product (36)–(38) and the braid relations (29) and (30).
Definition 12. We say that the ternary matrix group M
gen-Σ
3 generated by the matrix generators
Σij(2) satisfying the relations (45) and (46) is in ternary-binary correspondence with the braid
(binary) group Bn, which is denoted as (cf. (9))
M
gen-Σ
3 m Bn. (47)
Indeed, in components the relations (45) give (29), and (46) leads to (30).
Remark 2. Note that the above construction is totally different from the bi-element representations
of ternary groups considered in Reference [17] (for k-ary groups see [18]).
Definition 13. An element M(2) ∈ M3 is of finite polyadic (ternary) order, if there exists a finite
` such that
M(2)h`i3 = M(2)2`+1
= E(2), (48)
where E(2) is the ternary matrix identity (41).
Definition 14. An element M(2) ∈ M3 is of finite q-polyadic (q-ternary) order, if there exists a
finite ` such that
M(2)h`i3 = M(2)2`+1
= qE(2), q ∈ Bn. (49)
The relations (45), therefore, say that the ternary matrix generators Σi,j+1(2) are of
finite q-ternary order. Each element of M
gen-Σ
3 is a ternary matrix word (analogous to the
binary word (31)), being the ternary product of the polyadic powers (12) of the 2 × 2 matrix
generators Σ
(2)
ij and their querelements Σ̄
(2)
ij (on choosing the first or second row)
W =
Σi1j1
(2)
Σ̄i1j1
(2)
h`1i3
, . . . ,
Σir jr
(2)
Σ̄ir jr
(2)
h`ri3
, . . . ,
Σim jm
(2)
Σ̄im jm
(2)
h`mi3
=
Σi1j1
(2)
Σ̄i1j1
(2)
2`1+1
, . . . ,
Σir jr
(2)
Σ̄ir jr
(2)
2`r+1
, . . . ,
Σim jm
(2)
Σ̄im jm
(2)
2`m+1
, (50)
9. Mathematics 2021, 9, 972 9 of 17
where r = 1, . . . , m, ir, jr = 1, . . . , n (from Bn), `r, m ∈ N. In the ternary case, the total num-
ber of multipliers in (50) should be compatible with (12), i.e., (2`1 + 1) + . . . + (2`r + 1) +
. . . + (2`m + 1) = 2`W + 1, `W ∈ N, and m is, therefore, odd. Thus, we have:
Remark 3. The ternary words (50) in components give only a subset of the binary words (31),
and so M
gen-Σ
3 corresponds to Bn, but does not present it.
Example 2. For B3, we have only two ternary 2 × 2 matrix generators
Σ12(2) =
0 σ1
σ2 0
, Σ21(2) =
0 σ2
σ1 0
, (51)
satisfying
(Σ12(2))h1i3 = (Σ12(2))3
= q
[3]
1 E(2), (52)
(Σ21(2))h1i3 = (Σ21(2))3
= q
[3]
1 E(2), (53)
where q
[3]
1 = σ1σ2σ1 = σ2σ1σ2, and both matrix relations (52) and (53) coincide in components.
Example 3. For B4, the ternary matrix group M
gen-Σ
3 is generated by more generators satisfying
the relations
(Σ12(2))3
= q
(3)
1 E(2), (54)
(Σ21(2))3
= q
(3)
1 E(2), (55)
(Σ23(2))3
= q
(3)
2 E(2), (56)
(Σ32(2))3
= q
(3)
2 E(2), (57)
Σ13(2)Σ13(2)E(2) = Σ31(2)Σ31(2)E(2), (58)
where q
[3]
1 = σ1σ2σ1 = σ2σ1σ2 and q
[3]
2 = σ2σ3σ2 = σ3σ2σ3. The first two relations give the braid
relations (32) and (33), while the last relation corresponds to far commutativity (34).
7. Generated k-Ary Matrix Group Corresponding the Higher Braid Group
The above construction of the ternary matrix group M
gen-Σ
3 corresponding to the
braid group Bn can be naturally extended to the k-ary case, which will allow us to “go
in the opposite way” and build so called higher degree analogs of Bn (in our sense: the
number of factors in braid relations more than 3). We denote such a braid-like group
with n generators by Bn[k], where k is the number of generator multipliers in the braid
relations (as in the regularity relations (15)–(17)). Simultaneously, k is the -arity of the
matrices (35); therefore, we call Bn[k] a higher k-degree analog of the braid group Bn. In this
notation, the Artin braid group Bn is Bn[3]. Now, we build Bn[k] for any degree k exploiting
the “reverse” procedure, as for k = 3 and Bn in SECTION 5. For that, we need a k-ary
generalization of the matrices over Bn, which, in the ternary case, are the anti-diagonal
matrices M(2) (35), and the generator matrices Σij(2) (43). Then, using the k-ary analog
of multiplication (37) and (38) we will obtain the higher degree (than (29)) braid relations
which generate the so called higher k-degree braid group. In distinction to the higher degree
regular semigroup construction from Section 4, where the k-ary matrices form a semigroup
for the Abelian group Gfree, using the generator matrices, we construct a k-ary matrix
semigroup (presented by generators and relations) for any (even non-commutative) matrix
entries. In this way, the polyadic-binary correspondence will connect k-ary matrix groups
of finite order with higher binary braid groups (cf. idempotent k-ary matrices and higher
regular semigroups (21)).
10. Mathematics 2021, 9, 972 10 of 17
Let us consider a free binary group Bfree and construct over it a k-ary matrix group
along the lines of Reference [16], similarly to the ternary matrix group M3 in (35)–(38).
Definition 15. A set M(k − 1) = {M(k − 1)} of k-ary (k − 1) × (k − 1) matrices
M(k − 1) = M((k−1)×(k−1))
b(1), b(2), . . . , b(k−1)
=
0 b(1) 0 . . . 0
0 0 b(2) . . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... b(k−2)
b(k−1) 0 0 . . . 0
, (59)
b(j) ∈ Bfree, j = 1, . . . , k − 1,
form a k-ary matrix semigroup Mk = {M(k − 1) | µk}, where µk is the k-ary multiplication
µk[M1(k − 1), M2(k − 1), . . . , Mk(k − 1)] = M1(k − 1)M2(k − 1) . . . Mk(k − 1) = M(k − 1), (60)
b
(1)
1 b
(2)
2 , . . . b
(k−1)
k−1 b
(1)
k = b(1)
, (61)
b
(2)
1 b
(3)
2 , . . . b
(1)
k−1b
(2)
k = b(2)
, (62)
.
.
.
b
(k−1)
1 b
(1)
2 , . . . b
(k−2)
k−1 b
(k−1)
k = b(k−1)
, (63)
where the r.h.s. of (60) is the ordinary matrix multiplication of k-ary matrices (59) Mi(k − 1) =
M((k−1)×(k−1))
b
(1)
i , b
(2)
i , . . . , b
(k−1)
i
, i = 1, . . . , k.
Proposition 7. Mk is a k-ary matrix group.
Proof. Because Bfree is a (binary) group with the identity e ∈ Bfree, each element of the k-
ary matrix semigroup M(k − 1) ∈ Mk is invertible (in the k-ary sense) and has a querelement
M̄(k − 1) (see Reference [24]) defined by (cf. (42))
µk
k−1
z }| {
M(k − 1), . . . , M(k − 1), M̄(k − 1)
= . . . = M(k − 1), (64)
where M̄(k − 1) can be on any place, and so we have k conditions (cf. (39) for k = 3).
The k-ary matrix group has the polyadic identity
E(k − 1) = E((k−1)×(k−1))
=
0 e 0 . . . 0
0 0 e . . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... e
e 0 0 . . . 0
, e ∈ Bfree, (65)
satisfying
µk[M(k − 1), E(k − 1), . . . , E(k − 1)] = . . . = M(k − 1), (66)
where M(k − 1) can be on any place, and so we have k conditions (cf. (42)).
Definition 16. An element of a k-ary group M(k − 1) ∈ Mk has the polyadic order `, if
(M(k − 1))h`ik ≡ (M(k − 1))`(k−1)+1
= E(k − 1), (67)
11. Mathematics 2021, 9, 972 11 of 17
where E(k − 1) ∈ Mk is the polyadic identity (65), for k = 3; see (41).
Definition 17. An element of the (k − 1) × (k − 1)-matrix over Bfree that is M(k − 1) ∈ M{k}
is of finite q-polyadic order, if there exists a finite ` such that
(M(k − 1))h`ik ≡ (M(k − 1))`(k−1)+1
= qE(k − 1), q ∈ Bfree. (68)
Let us assume that the binary group Bfree is presented by generators and relations
(cf. the Artin braid group (29) and (30)), i.e., it is generated by n − 1 generators σi,
i = 1, . . . , n − 1. An element of B
gen-σ
n ≡ Bfree(e, σi) is the word of the form (31). To find
the relations between σi we construct the corresponding k-ary matrix generators analogous
to the ternary ones (43). Then, using a k-ary version of the relations (45) and (46) for the
matrix generators, as the finite order conditions (68), we will obtain the corresponding
higher degree braid relations for the binary generators σi and can, therefore, present a
higher degree braid group Bn[k] in the form of generators and relations.
Using n − 1 generators σi of B
gen-σ
n , we build (n − 1)k
polyadic (or k-ary) (k − 1) ×
(k − 1)-matrix generators having k − 1 indices i1, . . . , ik−1 = 1, . . . , n − 1, as follows
Σi1,...,ik−1
(k − 1) ≡ Σ
((k−1)×(k−1))
i1,...,ik−1
σi1
, . . . , σik−1
=
0 σi1
0 . . . 0
0 0 σi2
. . . 0
0 0
.
.
.
...
.
.
.
.
.
.
.
.
. 0
... σik−2
σik−1
0 0 . . . 0
. (69)
For the matrix generator Σi1,...,ik−1
(k − 1) (69), its querelement Σ̄i1,...,ik−1
(k − 1) is de-
fined by (64).
We now build a k-ary matrix analog of the braid relations (29), (45) and of far commu-
tativity (30), (46). Using (69), we obtain (k − 1) conditions that the matrix generators are of
finite polyadic order (analog of (45))
µk[Σi,i+1,...,i+k−2(k − 1), Σi,i+1,...,i+k−2(k − 1), . . . , Σi,i+1,...,i+k−2(k − 1)] (70)
= µk[Σi+1,i+2,...,i+k−2,i(k − 1), Σi+1,i+2,...,i+k−2,i(k − 1), . . . , Σi+1,i+2,...,i+k−2,i(k − 1)] (71)
.
.
.
µk[Σi+k−2,i,i+1,...,i+k−3(k − 1), Σi+k−2,i,i+1,...,i+k−3(k − 1), . . . , Σi+k−2,i,i+1,...,i+k−3(k − 1)] (72)
= q
[k]
i E(k − 1), 1 ≤ i ≤ n − k + 1, (73)
where E(k − 1) are polyadic identities (65) and q
[k]
i ∈ B
gen-σ
n .
We propose a k-ary version of the far commutativity relation (46) in the following form:
µk
k−1
z }| {
Σi1,...,ik−1
(k − 1), . . . , Σi1,...,ik−1
(k − 1),E(k − 1)
= . . . (74)
= µk
k−1
z }| {
Στ(i1),τ(i2),...,τ(ik−1)(k − 1), . . . , Στ(i1),τ(i2),...,τ(ik−1)(k − 1),E(k − 1)
, (75)
if all
20. ≥ k − 1, p, s = 1, . . . , k − 1,
where τ ∈ Sk−1 and q
[k]
i ∈ B
gen-σ
n .
Each element of M
gen-Σ
k is a k-ary matrix word (analogous to the binary word (31))
being the k-ary product of the polyadic powers (12) of the matrix generators Σi1,...,ik−1
(k − 1)
and their querelements Σ̄i1,...,ik−1
(k − 1) as in (50).
Similarly to the ternary case k = 3 (Section 5), we now develop the k-ary “reverse”
procedure and build from B
gen-σ
n the higher k-degree braid group Bn[k] using (69). Be-
cause the presentation of M
gen-Σ
k by generators and relations has already been given in
(77) and (80), we need to expand them into components and postulate that these new
relations between the (binary) generators σi present a new higher degree analog of the
braid group. This gives:
Definition 19. A higher k-degree braid (binary) group Bn[k] is presented by (n − 1) generators
σi ≡ σ
[k]
i (and the identity e) satisfying the following relations:
• (k − 1) higher braid relations
k
z }| {
σiσi+1 . . . σi+k−3σi+k−2σi (81)
= σi+1σi+2 . . . σi+k−2σiσi+1 (82)
.
.
.
= σi+k−2σiσi+1σi+2 . . . σiσi+1σi+k−2 ≡ q
[k]
i q
[k]
i ∈ Bn[k], (83)
i ∈ Ibraid = {1, . . . , n − k + 1}, (84)
• (k − 1)-ary far commutativity
k−1
z }| {
σi1
σi2
. . . σik−3
σik−2
σik−1
(85)
.
.
.
= στ(i1)στ(i2) . . . στ(ik−3)στ(ik−2)στ(ik−1), (86)
if all
24. ≥ k − 1, p, s = 1, . . . , k − 1, (87)
If ar = {n − k, . . . , n − 1}, (88)
where τ is an element of the permutation symmetry group τ ∈ Sk−1.
25. Mathematics 2021, 9, 972 13 of 17
A general element of the higher k-degree braid group Bn[k] is a word of the form
w = σ
p1
i1
. . . σ
pr
ir
. . . σ
pm
im
, im = 1, . . . , n, (89)
where pr ∈ Z are (positive or negative) powers of the generators σir
, r = 1, . . . , m and
m ∈ N.
Remark 4. The ternary case k = 3 coincides with the Artin braid group B
[3]
n = Bn (29) and (30).
Remark 5. The representation of the higher k-degree braid relations in Bn[k] in the tensor product
of vector spaces (similarly to Bn and the Yang-Baxter equation [9]) can be obtained using the n0-ary
braid equation introduced in Reference [8] (Proposition 7.2 and next there).
Definition 20. We say that the k-ary matrix group M
gen-Σ
k generated by the matrix generators
Σi1,i2,...,ik−1
(k − 1) satisfying the relations (77)–(80) is in polyadic-binary correspondence with the
higher k-degree braid group Bn[k], which is denoted as (cf. (47))
M
gen-Σ
k m Bn[k]. (90)
Example 4. Let k = 4; then, the 4-ary matrix group M
gen-Σ
4 is generated by the matrix generators
Σi1,i2,i3
(3) satisfying (77)–(80)
• 4-ary relations of q-polyadic order (48)
(Σi,i+1,i+2(3))4
= (Σi+2,i,i+1(3))4
= (Σi+1,i+2,i(3))4
= q
[3]
i E(3), 1 ≤ i ≤ n − 3, (91)
• far commutativity
Σi1,i2,i3
(3)
3
E(3) = Σi3,i1,i2
(3)
3
E(3) = Σi2,i3,i1
(3)
3
E(3)
= Σi1,i3,i2
(3)
3
E(3) = Σi3,i2,i1
(3)
3
E(3) = Σi2,i1,i3
(3)
3
E(3), (92)
|i1 − i2| ≥ 3, |i1 − i3| ≥ 3, |i2 − i3| ≥ 3.
Let σi ≡ σ
[4]
i ∈ Bn[4], i = 1, . . . , n − 1; then, we use the 4-ary 3 × 3 matrix presentation for
the generators (cf. Example 1):
Σi1,i2,i3
(3) ≡ Σ(3×3)
σi1
, σi2
, σi3
=
0 σi1
0
0 0 σi2
σi3
0 0
, i1, i2, i3 = 1, . . . , n − 1. (93)
The querelement Σ̄i1,i2,i3
(3) satisfying
Σi1,i2,i3
(3)
3
Σ̄i1,i2,i3
(3) = Σi1,i2,i3
(3) (94)
has the form
Σ̄i1,i2,i3
(3) =
0 σ−1
i3
σ−1
i2
0
0 0 σ−1
i1
σ−1
i3
σ−1
i2
σ−1
i1
0 0
. (95)
Expanding (91)–(92) in components, we obtain the relations for the higher 4-degree braid
group Bn[4] as follows.
• higher 4-degree braid relations
σiσi+1σi+2σi = σi+1σi+2σiσi+1 = σi+2σiσi+1σi+2 ≡ q
[4]
i , 1 ≤ i ≤ n − 3, (96)
26. Mathematics 2021, 9, 972 14 of 17
• ternary far (total) commutativity
σi1
σi2
σi3
= σi2
σi3
σi1
= σi3
σi1
σi2
= σi1
σi3
σi2
= σi2
σi1
σi3
= σi3
σi2
σi1
, (97)
|i1 − i2| ≥ 3, |i1 − i3| ≥ 3, |i2 − i3| ≥ 3. (98)
In the higher 4-degree braid group, the minimum number of generators is 4, which fol-
lows from (96). In this case, we have a braid relation for i = 1 only and no far commutativity
relations because of (98). Then:
Example 5. The higher 4-degree braid group B4[4] is generated by 3 generators σ1, σ2, σ3, which
satisfy only the braid relation
σ1σ2σ3σ1 = σ2σ3σ1σ2 = σ3σ1σ2σ3. (99)
If n ≤ 7, then there will be no far commutativity relations at all, which follows from
(98), and so the first higher 4-degree braid group containing far commutativity should have
n = 8 elements.
Example 6. The higher 4-degree braid group B8[4] is generated by 7 generators σ1, . . . , σ7, which
satisfy the braid relations with i = 1, . . . , 5
σ1σ2σ3σ1 = σ2σ3σ1σ2 = σ3σ1σ2σ3, (100)
σ2σ3σ4σ2 = σ3σ4σ2σ3 = σ4σ2σ3σ4, (101)
σ3σ4σ5σ3 = σ4σ5σ3σ4 = σ5σ3σ4σ5, (102)
σ4σ5σ6σ4 = σ5σ6σ4σ5 = σ6σ4σ5σ6, (103)
σ5σ6σ7σ5 = σ6σ7σ5σ6 = σ7σ5σ6σ7, (104)
together with the ternary far commutativity relation
σ1σ4σ7 = σ4σ7σ1 = σ7σ1σ4 = σ1σ7σ4 = σ4σ1σ7 = σ7σ4σ1. (105)
Remark 6. In polyadic group theory, there are several possible modifications of the commutativity
property; but, nevertheless, we assume here the total commutativity relations in the k-ary matrix
generators and the corresponding far commutativity relations in the higher degree braid groups.
If Bn[k] → Z is the abelianization defined by σ±
i → ±1, then σ
p
i = e, if and only
if p = 0, and σi are of infinite order. Moreover, we can prove (as in the ordinary case
k = 3 [25]).
Theorem 1. The higher k-degree braid group Bn[k] is torsion-free.
Recall (see, e.g., Reference [4]) that there exists a surjective homomorphism of the braid
group onto the finite symmetry group Bn → Sn by σi → si = (i, i + 1) ∈ Sn. The generators
si satisfy (29) and (30), together with the finite order demand
sisi+1si = si+1sisi+1, 1 ≤ i ≤ n − 2, (106)
sisj = sjsi, |i − j| ≥ 2, (107)
s2
i = e, i = 1, . . . , n − 1, (108)
which is called the Coxeter presentation of the symmetry group Sn. Indeed, multiplying
both sides of (106) from the right successively by si+1, si, and si+1, using (108), we obtain
27. Mathematics 2021, 9, 972 15 of 17
(sisi+1)3
= 1, and (107) on si and sj, we get sisj
2
= 1. Therefore, a Coxeter group [26]
corresponding to (106)–(108) is presented by the same generators si and the relations
(sisi+1)3
= 1, 1 ≤ i ≤ n − 2, (109)
sisj
2
= 1, |i − j| ≥ 2, (110)
s2
i = e, i = 1, . . . , n − 1. (111)
A general Coxeter group Wn = Wn(e, ri) is presented by n generators ri and the
relations [27]
rirj
mij
= e, mij =
1, i = j,
≥ 2, i 6= j.
(112)
By analogy with (106)–(108), we make the following.
Definition 21. A higher analog of Sn, the k-degree symmetry group Sn[k] = S
[k]
n (e, si), is
presented by generators si, i = 1, . . . , n − 1 satisfying (81)–(86) together with the additional
condition of finite (k − 1)-order s
(k−1)
i = e, i = 1, . . . , n.
Example 7. The lowest higher degree case is S4[4] which is presented by three generators s1, s2,
s3 satisfying (see (99))
s1s2s3s1 = s2s3s1s2 = s3s1s2s3, (113)
s3
1 = s3
2 = s3
3 = e. (114)
In a similar way, we define a higher degree analog of the Coxeter group (112).
Definition 22. A higher k-degree Coxeter group Wn[k] = W
[k]
n (e, ri) is presented by n generators
ri obeying the relations
ri1
ri2
. . . rik−1
mi1i2
,...,ik−1 = e, (115)
mi1i2
,...,ik−1
=
1, i1 = i2 = . . . = ik−1,
≥ k − 1,
31. ≥ k − 1, p, s = 1, . . . , k − 1.
(116)
It follows from (116) that all generators are of (k − 1) order rk−1
i = e. A higher k-degree
Coxeter matrix is a hypermatrix M
[k−1]
n,Cox
k−1
z }| {
n × n × . . . × n
having 1 on the main diagonal
and other entries mi1i2
,...,ik−1
.
Example 8. In the lowest higher degree case, k = 4 and all mi1i2
,...,ik−1
= 3, we have (instead of
commutativity in the ordinary case k = 3)
rirj
2
= r2
j r2
i , (117)
rirjri = r2
j r2
i r2
j . (118)
Example 9. A higher 4-degree analog of (109)–(111) is given by
(riri+1ri+2)4
= 1, 1 ≤ i ≤ n − 3, (119)
ri1
ri2
ri3
3
= 1, |i1 − i2| ≥ 3, |i1 − i3| ≥ 3, |i2 − i3| ≥ 3, (120)
r3
i = e, i = 1, . . . , n − 1. (121)
32. Mathematics 2021, 9, 972 16 of 17
It follows from (120) that
ri1
ri2
ri3
2
= r2
i3
r2
i2
r2
i1
, (122)
which cannot be reduced to total commutativity (97). From the first relation (119), we obtain
riri+1ri+2ri = r2
i+2r2
i+1, (123)
which differs from the higher 4-degree braid relations (96).
Example 10. In the simplest case, the higher 4-degree Coxeter group W4[4] has 3 generator r1, r2,
r3 satisfying
(r1r2r3)4
= r3
1 = r3
2 = r3
3 = e. (124)
Example 11. The minimal case, when the conditions (120) appear is W8[4]
r1r2r3r1 = r2
3r2
2, (125)
r2r3r4r2 = r2
4r2
3, (126)
r3r4r5r3 = r2
5r2
4, (127)
r4r5r6r4 = r2
6r2
5, (128)
r5r6r7r5 = r2
7r2
6, (129)
and an analog of commutativity
(r1r4r7)2
= r2
7r2
4r2
1. (130)
Thus, we arrive at:
Theorem 2. The higher k-degree Coxeter group can present the k-degree symmetry group in the
lowest case only, if and only if k = 3.
As a further development, it would be interesting to consider the higher degree (in
our sense) groups constructed here from a geometric viewpoint (e.g., Reference [5,28]).
Funding: This research received no external funding.
Acknowledgments: The author is grateful to Mike Hewitt, Thomas Nordahl, Vladimir Tkach and
Raimund Vogl for the numerous fruitful discussions and valuable support.
Conflicts of Interest: The authors declare no conflicts of interest.
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