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.
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.
We consider a general approach to describing the interaction in multigravity models in a D-dimensional space–time. We present various possibilities for generalizing the invariant volume. We derive the most general form of the interaction potential, which becomes a Pauli–Fierz-type model in the bigravity case. Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts leads to this bigravity model being completely equivalent to the Pauli–Fierz model. We thus in a concrete example show that introducing an interaction between metrics is equivalent to introducing the graviton mass.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
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.
We consider a general approach to describing the interaction in multigravity models in a D-dimensional space–time. We present various possibilities for generalizing the invariant volume. We derive the most general form of the interaction potential, which becomes a Pauli–Fierz-type model in the bigravity case. Analyzing this model in detail in the (3+1)-expansion formalism and also requiring the absence of ghosts leads to this bigravity model being completely equivalent to the Pauli–Fierz model. We thus in a concrete example show that introducing an interaction between metrics is equivalent to introducing the graviton mass.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
GREY LEVEL CO-OCCURRENCE MATRICES: GENERALISATION AND SOME NEW FEATURESijcseit
Grey Level Co-occurrence Matrices (GLCM) are one of the earliest techniques used for image texture
analysis. In this paper we defined a new feature called trace extracted from the GLCM and its implications
in texture analysis are discussed in the context of Content Based Image Retrieval (CBIR). The theoretical
extension of GLCM to n-dimensional gray scale images are also discussed. The results indicate that trace
features outperform Haralick features when applied to CBIR.
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
Global Domination Set in Intuitionistic Fuzzy Graphijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
In this paper we discussed about the pseudo integral for a measurable function based on a strict pseudo addition and pseudo multiplication. Further more we got several important properties of the pseudo integral of a measurable function based on a strict pseudo addition decomposable measure.
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.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
On the Family of Concept Forming Operators in Polyadic FCADmitrii Ignatov
Triadic Formal Concept Analysis (3FCA) was introduced by Lehman and Wille almost two decades ago. And many researchers work in Data Mining and Formal Concept Analysis using the notions of closed sets, Galois and closure operators, closure systems. However, up-to-date even though that different researchers actively work on mining triadic and n-ary relations, a proper closure operator for enumeration of triconcepts, i.e. maximal triadic cliques of tripartite hypergaphs, was not introduced. In this talk we show that the previously introduced operators for obtaining triconcepts are not always consistent, describe their family and study their properties. We also introduce the notion of maximal switching generator to explain why such concept-forming operators are not closure operators due to violation of monotonicity property.
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.
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 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.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
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
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
GREY LEVEL CO-OCCURRENCE MATRICES: GENERALISATION AND SOME NEW FEATURESijcseit
Grey Level Co-occurrence Matrices (GLCM) are one of the earliest techniques used for image texture
analysis. In this paper we defined a new feature called trace extracted from the GLCM and its implications
in texture analysis are discussed in the context of Content Based Image Retrieval (CBIR). The theoretical
extension of GLCM to n-dimensional gray scale images are also discussed. The results indicate that trace
features outperform Haralick features when applied to CBIR.
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
Global Domination Set in Intuitionistic Fuzzy Graphijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
In this paper we discussed about the pseudo integral for a measurable function based on a strict pseudo addition and pseudo multiplication. Further more we got several important properties of the pseudo integral of a measurable function based on a strict pseudo addition decomposable measure.
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.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
Existence results for fractional q-differential equations with integral and m...journal ijrtem
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
On the Family of Concept Forming Operators in Polyadic FCADmitrii Ignatov
Triadic Formal Concept Analysis (3FCA) was introduced by Lehman and Wille almost two decades ago. And many researchers work in Data Mining and Formal Concept Analysis using the notions of closed sets, Galois and closure operators, closure systems. However, up-to-date even though that different researchers actively work on mining triadic and n-ary relations, a proper closure operator for enumeration of triconcepts, i.e. maximal triadic cliques of tripartite hypergaphs, was not introduced. In this talk we show that the previously introduced operators for obtaining triconcepts are not always consistent, describe their family and study their properties. We also introduce the notion of maximal switching generator to explain why such concept-forming operators are not closure operators due to violation of monotonicity property.
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.
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 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.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
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
A new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized (“deformed”) using a parameter q which takes special integer values. A version of the
“q-deformed” analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The “q-deformed” homomorphism theorem is also 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
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.
This article continues the study of concrete algebra-like structures in our polyadic approach, when 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. In this way, the 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 algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but "quantized"; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as 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, 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.
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.
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.
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.
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.
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and leads to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, oper- ator norms, isometries and projections, as well as polyadic C* -algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with num- ber theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the derived polyadic ring operations neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented.
https://arxiv.org/abs/1703.10132
https://arxiv.org/abs/2006.07865
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 e-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" e-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and e-commutativity are studied.
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.
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with number theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the nonderivedpolyadic ring operations (polyadic numbers) neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented.
In a polytope P the faces F (from vertex to facets) define the combinatorics of P. In particular a flag F0, F1, ...., F(n-1) with Fi being a face of dimension i and Fi a subset of F(i+1).
From such a flag system and a subset S of {1,....,n} we can define a new flag system named the Wythoff construction. We consider the l1-embedding of the obtained graphs. We also expose an application of the Wythoff construction to the computation of homology of the Mathieu group M24.
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.
Similar to Steven Duplij, "Polyadic analogs of direct product" (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
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
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 .
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
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.
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.
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.
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.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
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.
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/
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.
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.
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.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
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.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
2. Universe 2022, 8, 230 2 of 21
binary fields that are a direct product of fields does not hold in the polyadic case. We
construct polyadic fields which are products of zeroless fields, which can lead to a new
category (which does not exist for binary fields): the category of polyadic fields.
The proposed constructions are accompanied by concrete illustrative examples.
2. Preliminaries
In this section we briefly introduce the usual notation; for details see [15]. For a non-
empty (underlying) set G the n-tuple (or polyad [16]) of elements is denoted by (g1, . . . , gn),
gi ∈ G, i = 1, . . . , n, and the Cartesian product is denoted by G×n ≡
n
z }| {
G × . . . × G and
consists of all such n-tuples. For all elements equal to g ∈ G, we denote n-tuple (polyad) by
a power (gn). To avoid unneeded indices we denote with one bold letter (g) a polyad for
which the number of elements in the n-tuple is clear from the context, and sometimes we
will write
g(n)
. On the Cartesian product G×n we define a polyadic (or n-ary) operation
µ(n) : G×n → G such that µ(n)[g] 7→ h, where h ∈ G. The operations with n = 1, 2, 3 are
called unary, binary and ternary.
Recall the definitions of some algebraic structures and their special elements (in the
notation of [15]). A (one-set) polyadic algebraic structure G is a set that is G-closedwith
respect to polyadic operations. In the case of one n-ary operation µ(n) : G×n → G, it
is called polyadic multiplication (or n-ary multiplication). A one-set n-ary algebraic structure
M(n) =
D
G | µ(n)
E
or polyadic magma (n-ary magma) is a set that is G-closed with respect
to one n-ary operation µ(n) and without any other additional structure. In the binary case
M(2) was also called a groupoid by Hausmann and Ore [17] (and [18]). Since the term
“groupoid” was widely used in category theory for a different construction, the so-called
Brandt groupoid [19,20], Bourbaki [21] later introduced the term “magma”.
Denote the number of iterating multiplications by `µ, and call the resulting composi-
tion an iterated product
µ(n)
◦`µ
, such that
µ0(n0)
=
µ(n)
◦`µ de f
=
`µ
z }| {
µ(n)
◦
µ(n)
◦ . . .
µ(n)
× id×(n−1)
. . . × id×(n−1)
, (1)
where the arities are connected by
n0
= niter = `µ(n − 1) + 1, (2)
which gives the length of an iterated polyad (g) in our notation
µ(n)
◦`µ
[g].
A polyadic zero of a polyadic algebraic structure G(n)
D
G | µ(n)
E
is a distinguished
element z ∈ G (and the corresponding 0-ary operation µ
(0)
z ) such that for any (n − 1)-tuple
(polyad) g(n−1)∈G×(n−1) we have
µ(n)
h
g(n−1)
, z
i
= z, (3)
where z can be in any place on the l.h.s. of (3). If its place is not fixed it can be a single zero.
As in the binary case, an analog of positive powers of an element [16] should coincide with
the number of multiplications `µ in the iteration (1).
A (positive) polyadic power of an element is
gh`µi =
µ(n)
◦`µ
h
g`µ(n−1)+1
i
. (4)
3. Universe 2022, 8, 230 3 of 21
We define associativity as the invariance of the composition of two n-ary multipli-
cations. An element of a polyadic algebraic structure g is called `µ-nilpotent (or simply
nilpotent for `µ = 1), if there exist `µ such that
gh`µi = z. (5)
A polyadic (n-ary) identity (or neutral element) of a polyadic algebraic structure is a
distinguished element e (and the corresponding 0-ary operation µ
(0)
e ) such that for any
element g ∈ G we have
µ(n)
h
g, en−1
i
= g, (6)
where g can be in any place on the l.h.s. of (6).
In polyadic algebraic structures, there exist neutral polyads n ∈ G×(n−1) satisfying
µ(n)
[g, n] = g, (7)
where g can be in any of n places on the l.h.s. of (7). Obviously, the sequence of polyadic
identities en−1 is a neutral polyad (6).
A one-set polyadic algebraic structure
D
G | µ(n)
E
is called totally associative if
µ(n)
◦2
[g, h, u] = µ(n)
h
g, µ(n)
[h], u
i
= invariant, (8)
with respect to the placement of the internal multiplication µ(n)[h] on the r.h.s. on any
of n places, with a fixed order of elements in the any fixed polyad of (2n − 1) elements
t(2n−1) = (g, h, u) ∈ G×(2n−1).
A polyadic semigroup S(n) is a one-set S one-operation µ(n) algebraic structure in which
the n-ary multiplication is associative, S(n) =
D
S | µ(n) | associativity (8)
E
. A polyadic
algebraic structure G(n) =
D
G | µ(n)
E
is σ-commutative, if µ(n) = µ(n) ◦ σ, or
µ(n)
[g] = µ(n)
[σ ◦ g], g ∈ G×n
, (9)
where σ ◦ g =
gσ(1), . . . , gσ(n)
is a permutated polyad and σ is a fixed element of Sn, the
permutation group on n elements. If (9) holds for all σ ∈ Sn, then a polyadic algebraic
structure is commutative. A special type of the σ-commutativity
µ(n)
h
g, t(n−2)
, h
i
= µ(n)
h
h, t(n−2)
, g
i
, (10)
where t(n−2) ∈ G×(n−2) is any fixed (n − 2)-polyad, is referred to as semicommutativity. If
an n-ary semigroup S(n) is iterated from a commutative binary semigroup with identity,
then S(n) is semicommutative. A polyadic algebraic structure is called (uniquely) i-solvable,
if for all polyads t, u and element h, one can (uniquely) resolve the equation (with respect
to h) for the fundamental operation
µ(n)
[u, h, t] = g (11)
where h can be on any place, and u, t are polyads of the needed length.
A polyadic algebraic structure which is uniquely i-solvable for all places i = 1, . . . , n
is called a n-ary (or polyadic) quasigroup Q(n) =
D
Q | µ(n) | solvability
E
. An associative
polyadic quasigroup is called an n-ary (or polyadic) group. In an n-ary group G(n) =
D
G | µ(n)
E
the only solution of (11) is called a querelement of g and is denoted by ḡ [22],
such that
µ(n)
[h, ḡ] = g, g, ḡ ∈ G, (12)
4. Universe 2022, 8, 230 4 of 21
where ḡ can be on any place. Any idempotent g coincides with its querelement ḡ = g. The
unique solvability relation (12) in an n-ary group can be treated as a definition of the unary
(multiplicative) queroperation
µ̄(1)
[g] = ḡ. (13)
We observe from (12) and (7) that the polyad
ng =
gn−2
ḡ
(14)
is neutral for any element of a polyadic group, where ḡ can be on any place. If this i-th
place is important, then we write ng;i. In a polyadic group the Dörnte relations [22]
µ(n)
[g, nh;i] = µ(n)
h
nh;j, g
i
= g (15)
hold true for any allowable i, j. In the case of a binary group, the relations (15) become
g · h · h−1 = h · h−1 · g = g.
Using the queroperation (13) one can give a diagrammatic definition of a polyadic
group [23]: an n-ary group is a one-set algebraic structure (universal algebra)
G(n)
=
D
G | µ(n)
, µ̄(1)
| associativity (8), Dörnte relations (15)
E
, (16)
where µ(n) is an n-ary associative multiplication and µ̄(1) is the queroperation (13).
3. Polyadic Products of Semigroups and Groups
We start from the standard external direct product construction for semigroups. Then
we show that consistent “polyadization” of the semigroup direct product, which preserves
associativity, can lead to additional properties:
(1) The arities of the polyadic direct product and power can differ from that of the initial
semigroups.
(2) The components of the polyadic power can contain elements from different multipliers.
We use here a vector-like notation for clarity and convenience in passing to higher
arity generalizations. Begin from the direct product of two (binary) semigroups G1,2 ≡
G
(2)
1,2 =
D
G1,2 | µ
(2)
1,2 ≡ (·1,2) | assoc
E
, where G1,2 are underlying sets, whereas µ
(2)
1,2 are mul-
tiplications in G1,2. On the Cartesian product of the underlying sets G0 = G1 × G2 we
define a direct product G1 × G2 = G0 =
D
G0 | µ0(2) ≡ (•0)
E
of the semigroups G1,2 via the
componentwise multiplication of the doubles G =
g1
g2
∈ G1 × G2 (being the Kronecker
product of doubles in our notation) , as
G(1)
•0
G(2)
=
g1
g2
(1)
•0
g1
g2
(2)
=
g
(1)
1 ·1 g
(2)
1
g
(1)
2 ·2 g
(2)
2
!
, (17)
and in the “polyadic” notation
µ0(2)
h
G(1)
, G(2)
i
=
µ
(2)
1
h
g
(1)
1 , g
(2)
1
i
µ
(2)
2
h
g
(1)
2 , g
(2)
2
i
. (18)
Obviously, the associativity of µ0(2) follows immediately from that of µ
(2)
1,2 , because of
the componentwise multiplication in (18). If G1,2 are groups with the identities e1,2 ∈ G1,2,
then the identity of the direct product is the double E =
e1
e2
, such that µ0(2)[E, G] =
µ0(2)[G, E] = G ∈ G.
5. Universe 2022, 8, 230 5 of 21
3.1. Full Polyadic External Product
The “polyadization” of (18) is straightforward
Definition 1. An n0-ary full direct product semigroup G0(n0) = G
(n)
1 × G
(n)
2 consists of (two or k)
n-ary semigroups (of the same arity n0 = n)
µ0(n)
h
G(1)
, G(2)
, . . . , G(n)
i
=
µ
(n)
1
h
g
(1)
1 , g
(2)
1 , . . . , g
(n)
1
i
µ
(n)
2
h
g
(1)
2 , g
(2)
2 , . . . , g
(n)
2
i
, (19)
where the (total) polyadic associativity (8) of µ0(n0) is governed by those of the constituent semigroups
G
(n)
1 and G
(n)
2 (or G
(n)
1 . . . G
(n)
k ) and the componentwise construction (19).
If G
(n)
1,2 =
D
G1,2 | µ
(n)
1,2 , µ̄
(1)
1,2
E
are n-ary groups (where µ̄
(1)
1,2 are the unary multiplicative
queroperations (13)), then the queroperation µ̄0(1) of the full direct product group G0(n0) =
D
G0 ≡ G1 × G2 | µ0(n0), µ̄0(1)
E
(n0 = n) is defined componentwise as follows:
Ḡ ≡ µ̄0(1)
[G] =
µ̄
(1)
1 [g1]
µ̄
(1)
2 [g2]
!
, or Ḡ =
ḡ1
ḡ2
, (20)
which satisfies µ0(n)[G, G, . . . , Ḡ] = G with Ḡ on any place (cf. (12)).
Definition 2. A full polyadic direct product G0(n) = G
(n)
1 × G
(n)
2 is called derived if its constituents
G
(n)
1 and G
(n)
2 are derived, such that the operations µ
(n)
1,2 are compositions of the binary operations
µ
(2)
1,2 , correspondingly.
In the derived case, all the operations in (19) have the form (see (1) and (2))
µ
(n)
1,2 =
µ
(2)
1,2
◦(n−1)
, µ(n)
=
µ(2)
◦(n−1)
. (21)
The operations of the derived polyadic semigroup can be written as (cf., the binary
direct product (17) and (18))
µ0(n)
h
G(1)
, G(2)
, . . . , G(n)
i
= G(1)
•0
G(2)
•0
. . . •0
G(n)
=
g
(1)
1 ·1 g
(2)
1 ·1 . . . ·1 g
(n)
1
g
(1)
2 ·2 g
(2)
2 ·2 . . . ·2 g
(n)
2
. (22)
We will be more interested in nonderived polyadic analogs of the direct product.
Example 1. Let us have two ternary groups: the unitless nonderived group G
(3)
1 =
D
iR | µ
(3)
1
E
,
where i2 = −1, µ
(3)
1
h
g
(1)
1 , g
(2)
1 , g
(3)
1
i
= g
(1)
1 g
(2)
1 g
(3)
1 is a triple product in C, the querelement
is µ̄
(1)
1 [g1] = 1/g1, and G
(3)
2 =
D
R | µ
(3)
2
E
with µ
(3)
2
h
g
(1)
2 , g
(2)
2 , g
(3)
2
i
= g
(1)
2
g
(2)
2
−1
g
(3)
2 , the
querelement µ̄
(1)
2 [g2] = g2. Then, the ternary nonderived full direct product group becomes
G0(3) =
D
iR × R | µ0(3), µ̄0(1)
E
, where
µ0(3)
h
G(1)
, G(2)
, G(3)
i
=
g
(1)
1 g
(2)
1 g
(3)
1
g
(1)
2
g
(2)
2
−1
g
(3)
2
, Ḡ ≡ µ̄0(1)
[G] =
1/g1
g2
, (23)
which contains no identity, because G
(3)
1 is unitless and nonderived.
6. Universe 2022, 8, 230 6 of 21
3.2. Mixed-Arity Iterated Product
In the polyadic case, the following question arises, which cannot even be stated in
the binary case: is it possible to build a version of the associative direct product such
that it can be nonderived and have different arity than the constituent semigroup arities?
The answer is yes, which leads to two arity-changing constructions: componentwise and
noncomponentwise.
(1) Iterated direct product (~). In each of the constituent polyadic semigroups we use the
iterating (1) componentwise, but with different numbers of compositions, because the
same number of compositions evidently leads to the iterated polyadic direct product.
In this case the arity of the direct product is greater than or equal to the arities of the
constituents n0 ≥ n1, n2.
(2) Hetero product (). The polyadic product of k copies of the same n-ary semigroup
is constructed using the associativity quiver technique, which mixes (“entangles”)
elements from different multipliers, it is noncomponentwise (by analogy with het-
eromorphisms in [15]), and so it can be called a hetero product or hetero power (for
coinciding multipliers, i.e., constituent polyadic semigroups or groups). This gives
the arity of the hetero product which is less than or equal to the arities of the equal
multipliers n0 ≤ n.
In the first componentwise case 1), the constituent multiplications (19) are composed
from the lower-arity ones in the componentwise manner, but the initial arities of up and
down components can be different (as opposed to the binary derived case (21))
µ
(n)
1 =
µ
(n1)
1
◦`µ1
, µ
(n)
2 =
µ
(n2)
2
◦`µ2
, 3 ≤ n1,2 ≤ n − 1, (24)
where we exclude the limits: the derived case n1,2 = 2 (21) and the undecomposed case
n1,2 = n (19). Since the total size of the up and down polyads is the same and coincides
with the arity of the double G multiplication n0, using (2) we obtain the arity compatibility
relations
n0
= `µ1(n1 − 1) + 1 = `µ2(n2 − 1) + 1. (25)
Definition 3. A mixed-arity polyadic iterated direct product semigroup G0(n0) = G
(n1)
1 ~ G
(n2)
2
consists of (two) polyadic semigroups G
(n1)
1 and G
(n2)
2 of the different arity shapes n1 and n2
µ0(n0)
h
G(1)
, G(2)
, . . . , G(n0)
i
=
µ
(n1)
1
◦`µ1
h
g
(1)
1 , g
(2)
1 , . . . , g
(n)
1
i
µ
(n2)
2
◦`µ2
h
g
(1)
2 , g
(2)
2 , . . . , g
(n)
2
i
, (26)
and the arity compatibility relations (25) hold.
Observe that it is not the case that any two polyadic semigroups can be composed in
the mixed-arity polyadic direct product.
Assertion 1. If the arity shapes of two polyadic semigroups G
(n1)
1 and G
(n2)
2 satisfy the compatibil-
ity condition
a(n1 − 1) = b(n2 − 1) = c, a, b, c ∈ N, (27)
then they can form a mixed-arity direct product G0(n0) = G
(n1)
1 ~ G
(n2)
2 , where n0 = c + 1 (25).
7. Universe 2022, 8, 230 7 of 21
Example 2. In the case of 4-ary and 5-ary semigroups G
(4)
1 and G
(5)
2 the direct product arity of
G0(n0) is “quantized” n0 = 3`µ1 + 1 = 4`µ2 + 1, such that
n0
= 12k + 1 = 13, 25, 37, . . . , (28)
`µ1 = 4k = 4, 8, 12, . . . , (29)
`µ2 = 3k = 3, 6, 9, . . . , k ∈ N, (30)
and only the first mixed-arity 13-ary direct product semigroup G0(13) is nonderived. If G
(4)
1 and
G
(5)
2 are polyadic groups with the queroperations µ̄
(1)
1 and µ̄
(1)
2 correspondingly, then the iterated
direct G0(n0) is a polyadic group with the queroperation µ̄0(1) given in (20).
In the same way one can consider the iterated direct product of any number of
polyadic semigroups.
3.3. Polyadic Hetero Product
In the second noncomponentwise case 2) we allow multiplying elements from different
components, and therefore we should consider the Cartesian k-power of sets G0 = G×k
and endow the corresponding k-tuple with a polyadic operation in such a way that the
associativity of G(n) will govern the associativity of the product G0(n). In other words we
construct a k-power of the polyadic semigroup G(n) such that the result G0(n0) is an n0-ary
semigroup.
The general structure of the hetero product formally coincides “reversely” with the
main heteromorphism equation [15]. The additional parameter which determines the arity
n0 of the hetero power of the initial n-ary semigroup is the number of intact elements `id.
Thus, we arrive at
Definition 4. The hetero (“entangled”) k-power of the n-ary semigroup G(n) =
D
G | µ(n)
E
is the
n0-ary semigroup defined on the k-th Cartesian power G0 = G×k, such that G0(n0) =
D
G0 | µ0(n0)
E
,
G0(n0)
=
G(n)
k
≡
k
z }| {
G(n)
. . . G(n)
, (31)
and the n0-ary multiplication of k-tuples GT = (g1, g2, . . . , gk) ∈ G×k is given (informally) by
µ0(n0)
g1
.
.
.
gk
, . . . ,
gk(n0−1)
.
.
.
gkn0
=
µ(n)[g1, . . . , gn],
.
.
.
µ(n)
h
gn(`µ−1), . . . , gn`µ
i
`µ
gn`µ+1,
.
.
.
gn`µ+`id
`id
, gi ∈ G, (32)
where `id is the number of intact elements on the r.h.s., and `µ = k − `id is the number of
multiplications in the resulting k-tuple of the direct product. The hetero power parameters are
connected by the arity-changing formula [15]
n0
= n −
n − 1
k
`id, (33)
with the integer
n − 1
k
`id ≥ 1.
8. Universe 2022, 8, 230 8 of 21
The concrete placement of elements and multiplications in (32) to obtain the associative
µ0(n0) is governed by the associativity quiver technique [15].
There exist important general numerical relations between the parameters of the
twisted direct power n0, n, k, `id, which follow from (32) and (33). First, there are non-strict
inequalities for them
0 ≤ `id ≤ k − 1, (34)
`µ ≤ k ≤ (n − 1)`µ, (35)
2 ≤ n0
≤ n. (36)
Second, the initial and final arities n and n0 are not arbitrary, but “quantized” such that
the fraction in (33) has to be an integer (see Table 1).
Table 1. Hetero power “quantization”.
k `µ `id n/n0
2 1 1
n = 3, 5, 7, . . .
n0 = 2, 3, 4, . . .
3 1 2
n = 4, 7, 10, . . .
n0 = 2, 3, 4, . . .
3 2 1
n = 4, 7, 10, . . .
n0 = 3, 5, 7, . . .
4 1 3
n = 5, 9, 13, . . .
n0 = 2, 3, 4, . . .
4 2 2
n = 3, 5, 7, . . .
n0 = 2, 3, 4, . . .
4 3 1
n = 5, 9, 13, . . .
n0 = 4, 7, 10, . . .
Assertion 2. The hetero power is not unique in both directions, if we do not fix the initial n and
final n0 arities of G(n) and G0(n0).
Proof. This follows from (32) and the hetero power “quantization” shown in Table 1.
The classification of the hetero powers consists of two limiting cases.
(1) Intactless power: there are no intact elements `id = 0. The arity of the hetero power
reaches its maximum and coincides with the arity of the initial semigroup n0 = n (see
Example 5).
(2) Binary power: the final semigroup is of lowest arity, i.e., binary n0 = 2. The number of
intact elements is (see Example 4)
`id = k
n − 2
n − 1
. (37)
Example 3. Consider the cubic power of a 4-ary semigroup G0(3) =
G(4)
3
with the identity e,
then the ternary identity triple in G0(3) is ET = (e, e, e), and therefore this cubic power is a ternary
semigroup with identity.
Proposition 1. If the initial n-ary semigroup G(n) contains an identity, then the hetero power
G0(n0) =
G(n)
k
can contain an identity in the intactless case and the Post-like quiver [15]. For
the binary power k = 2 only the one-sided identity is possible.
9. Universe 2022, 8, 230 9 of 21
Let us consider some concrete examples.
Example 4. Let G(3) =
D
G | µ(3)
E
be a ternary semigroup, then we can construct its power
k = 2 (square) of the doubles G =
g1
g2
∈ G × G = G0 in two ways to obtain the associative
hetero power
µ0(2)
h
G(1)
, G(2)
i
=
µ(3)
h
g
(1)
1 , g
(1)
2 , g
(2)
1
i
g
(2)
2
!
,
µ(3)
h
g
(1)
1 , g
(2)
2 , g
(2)
1
i
g
(1)
2
!
,
g
(j)
i ∈ G. (38)
This means that the Cartesian square can be endowed with the associative multiplication
µ0(2), and therefore G0(2) =
D
G0 | µ0(2)
E
is a binary semigroup, being the hetero product G0(2) =
G(3) G(3). If G(3) has a ternary identity e ∈ G, then G0(2) has only the left (right) identity
E =
e
e
∈ G0, since µ0(2)[E, G] = G (µ0(2)[G, E] = G), but not the right (left) identity. Thus,
G0(2) can be a semigroup only, even if G(3) is a ternary group.
Example 5. Take G(3) =
D
G | µ(3)
E
a ternary semigroup, then the multiplication on the double
G =
g1
g2
∈ G × G = G0 is ternary and noncomponentwise
µ0(3)
h
G(1)
, G(2)
, G(3)
i
=
µ(3)
h
g
(1)
1 , g
(2)
2 , g
(3)
1
i
µ(3)
h
g
(1)
2 , g
(2)
1 , g
(3)
2
i
, g
(j)
i ∈ G, (39)
and µ0(3) is associative (and described by the Post-like associative quiver [15]), and therefore the cubic
hetero power is the ternary semigroup G0(3) =
D
G × G | µ0(3)
E
, such that G0(3) = G(3) G(3). In
this case, as opposed to the previous example, the existence of a ternary identity in G(3) implies the
ternary identity in the direct cube G0(3) by E =
e
e
. If G(3) is a ternary group with the unary
queroperation µ̄(1), then the cubic hetero power G0(3) is also a ternary group of the special class [24]:
all querelements coincide (cf., (20)), such that ḠT = gquer, gquer
, where µ̄(1)[g] = gquer, ∀g ∈ G.
This is because in (12) the querelement can be foundon any place.
Theorem 1. If G(n) is an n-ary group, then the hetero k-power G0(n0) =
G(n)
k
can contain
queroperations in the intactless case only.
Corollary 1. If the power multiplication (32) contains no intact elements `id = 0 and does not
change arity n0 = n, a hetero power can be a polyadic group which has only one querelement.
Next we consider more complicated hetero power (“entangled”) constructions with
and without intact elements, as well as Post-like and non-Post associative quivers [15].
Example 6. Let G(4) =
D
G | µ(4)
E
be a 4-ary semigroup, then we can construct its 4-ary associa-
tive cubic hetero power G0(4) =
D
G0 | µ0(4)
E
using the Post-like and non-Post-associative quivers
without intact elements. Taking in (32) n0 = n, k = 3, `id = 0, we obtain two possibilities for the
multiplication of the triples GT = (g1, g2, g3) ∈ G × G × G = G0
10. Universe 2022, 8, 230 10 of 21
(1) Post-like associative quiver. The multiplication of the hetero cubic power case takes the form
µ0(4)
h
G(1)
, G(2)
, G(3)
, G(4)
i
=
µ(4)
h
g
(1)
1 , g
(2)
2 , g
(3)
3 , g
(4)
1
i
µ(4)
h
g
(1)
2 , g
(2)
3 , g
(3)
1 , g
(4)
2
i
µ(4)
h
g
(1)
3 , g
(2)
1 , g
(3)
2 , g
(4)
3
i
, g
(j)
i ∈ G, (40)
and it can be shown that µ0(4) is totally associative; therefore, G0(4) =
D
G0 | µ0(4)
E
is a
4-ary semigroup.
(2) Non-Post associative quiver. The multiplication of the hetero cubic power differs from (40)
µ0(4)
h
G(1)
, G(2)
, G(3)
, G(4)
i
=
µ(4)
h
g
(1)
1 , g
(2)
3 , g
(3)
2 , g
(4)
1
i
µ(4)
h
g
(1)
2 , g
(2)
1 , g
(3)
3 , g
(4)
2
i
µ(4)
h
g
(1)
3 , g
(2)
2 , g
(3)
1 , g
(4)
3
i
, g
(j)
i ∈ G, (41)
and it can be shown that µ0(4) is totally associative; therefore, G0(4) =
D
G0 | µ0(4)
E
is a 4-ary
semigroup.
The following is valid for both the above cases. If G(4) has the 4-ary identity satisfying
µ(4)
[e, e, e, g] = µ(4)
[e, e, g, e] = µ(4)
[e, g, e, e] = µ(4)
[g, e, e, e] = g, ∀g ∈ G, (42)
then the hetero power G0(4) has the 4-ary identity
E =
e
e
e
, e ∈ G. (43)
In the case where G(3) is a ternary group with the unary queroperation µ̄(1), then the cubic hetero
power G0(4) is also a ternary group with one querelement (cf., Example 5)
Ḡ =
g1
g2
g3
=
gquer
gquer
gquer
, gquer ∈ G, gi ∈ G, (44)
where gquer = µ̄(1)[g], ∀g ∈ G.
A more nontrivial example is a cubic hetero power which has different arity to the
initial semigroup.
Example 7. Let G(4) =
D
G | µ(4)
E
be a 4-ary semigroup, then we can construct its ternary
associative cubic hetero power G0(3) =
D
G0 | µ0(3)
E
using the associative quivers with one intact
element and two multiplications [15]. Taking in (32) the parameters n0 = 3, n = 4, k = 3,
`id = 1 (see third line of Table 1), we obtain for the ternary multiplication µ0(3) for the triples
GT = (g1, g2, g3) ∈ G × G × G = G0 of the hetero cubic power case the form
µ0(3)
h
G(1)
, G(2)
, G(3)
i
=
µ(4)
h
g
(1)
1 , g
(2)
2 , g
(3)
3 , g
(3)
1
i
µ(4)
h
g
(1)
2 , g
(2)
3 , g
(2)
1 , g
(3)
2
i
g
(1)
3
, g
(j)
i ∈ G, (45)
11. Universe 2022, 8, 230 11 of 21
which is totally associative, and therefore the hetero cubic power of 4-ary semigroup
G(4) =
D
G | µ(4)
E
is a ternary semigroup G0(3) =
D
G0 | µ0(3)
E
, such that G0(3) =
G(4)
3
. If
the initial 4-ary semigroup G(4) has the identity satisfying (42), then the ternary hetero power G0(3)
has only the right ternary identity (43) satisfying one relation
µ0(3)
[G, E, E] = G, ∀G ∈ G×3
, (46)
and therefore G0(3) is a ternary semigroup with a right identity. If G(4) is a 4-ary group with
the queroperation µ̄(1), then the hetero power G0(3) can only be a ternary semigroup , because in
D
G0 | µ0(3)
E
we cannot define the standard queroperation [16].
4. Polyadic Products of Rings and Fields
Now we show that the thorough “polyadization” of operations can lead to some
unexpected new properties of ring and field external direct products. Recall that in the
binary case the external direct product of fields does not exist at all (see, e.g., [2]). The main
new peculiarities of the polyadic case are:
(1) The arity shape of the external product ring and its constituent rings can be different.
(2) The external product of polyadic fields can be a polyadic field.
4.1. External Direct Product of Binary Rings
First, we recall the ordinary (binary) direct product of rings in notation which would be
convenient to generalize to higher-arity structures [14]. Let us have two binary rings R1,2 ≡
R
(2,2)
1,2 =
D
R1,2 | ν
(2)
1,2 ≡ (+1,2), µ
(2)
1,2 ≡ (·1,2)
E
, where R1,2 are underlying sets, whereas ν
(2)
1,2
and µ
(2)
1,2 are additions and multiplications (satisfying distributivity) in R1,2, correspond-
ingly. On the Cartesian product of the underlying sets R0 = R1 × R2 one defines the external
direct product ring R1 × R2 = R0 =
D
R0 | ν0(2) ≡ (+0), µ0(2) ≡ (•0)
E
by the componentwise
operations (addition and multiplication) on the doubles X =
x1
x2
∈ R1 × R2 as follows:
X(1)
+0
X(2)
=
x1
x2
(1)
+0
x1
x2
(2)
≡
x
(1)
1
x
(1)
2
!
+0 x
(2)
1
x
(2)
2
!
=
x
(1)
1 +1 x
(2)
1
x
(1)
2 +2 x
(2)
2
!
, (47)
X(1)
•0
X(2)
=
x1
x2
(1)
•0
x1
x2
(2)
=
x
(1)
1 ·1 x
(2)
1
x
(1)
2 ·2 x
(2)
2
!
, (48)
or in the polyadic notation (with manifest operations)
ν0(2)
h
X(1)
, X(2)
i
=
ν
(2)
1
h
x
(1)
1 , x
(2)
1
i
ν
(2)
2
h
x
(1)
2 , x
(2)
2
i
, (49)
µ0(2)
h
X(1)
, X(2)
i
=
µ
(2)
1
h
x
(1)
1 , x
(2)
1
i
µ
(2)
2
h
x
(1)
2 , x
(2)
2
i
. (50)
The associativity and distributivity of the binary direct product operations ν0(2) and
µ0(2) are obviously governed by those of the constituent binary rings R1 and R2, because
of the componentwise construction on the r.h.s. of (49) and (50). In the polyadic case,
the construction of the direct product is not so straightforward and can have additional
unusual peculiarities.
12. Universe 2022, 8, 230 12 of 21
4.2. Polyadic Rings
Here we recall definitions of polyadic rings [25–27] in our notation [14,15]. Consider
a polyadic structure
D
R | µ(n), ν(m)
E
with two operations on the same set R: the m-ary
addition ν(m) : R×m → R and the n-ary multiplication µ(n) : R×n → R. The “interaction”
between operations can be defined using the polyadic analog of distributivity.
Definition 5. The polyadic distributivity for µ(n) and ν(m) consists of n relations
µ(n)
h
ν(m)
[x1, . . . xm], y2, y3, . . . yn
i
= ν(m)
h
µ(n)
[x1, y2, y3, . . . yn], µ(n)
[x2, y2, y3, . . . yn], . . . µ(n)
[xm, y2, y3, . . . yn]
i
(51)
µ(n)
h
y1, ν(m)
[x1, . . . xm], y3, . . . yn
i
= ν(m)
h
µ(n)
[y1, x1, y3, . . . yn], µ(n)
[y1, x2, y3, . . . yn], . . . µ(n)
[y1, xm, y3, . . . yn]
i
(52)
.
.
.
µ(n)
h
y1, y2, . . . yn−1, ν(m)
[x1, . . . xm]
i
= ν(m)
h
µ(n)
[y1, y2, . . . yn−1, x1], µ(n)
[y1, y2, . . . yn−1, x2], . . . µ(n)
[y1, y2, . . . yn−1, xm]
i
, (53)
where xi, yj ∈ R.
The operations µ(n) and ν(m) are totally associative, if (in the invariance definition
[14,15])
ν(m)
h
u, ν(m)
[v], w
i
= invariant, (54)
µ(n)
h
x, µ(n)
[y], t
i
= invariant, (55)
where the internal products can be on any place, and y ∈ R×n, v ∈ R×m, and the polyads x,
t, u, w are of the needed lengths. In this way both algebraic structures
D
R | µ(n) | assoc
E
and
D
R | ν(m) | assoc
E
are polyadic semigroups S(n) and S(m).
Definition 6. A polyadic (m, n)-ring R(m,n) is a set R with two operations µ(n) : R×n → R and
ν(m) : R×m → R, such that:
(1) they are distributive (51)–(53);
(2)
D
R | µ(n) | assoc
E
is a polyadic semigroup;
(3)
D
R | ν(m) | assoc, comm, solv
E
is a commutative polyadic group.
In case the multiplicative semigroup
D
R | µ(n) | assoc
E
of R(m,n) is commutative,
µ(n)[x] = µ(n)[σ ◦ x], for all σ ∈ Sn, then R(m,n) is called a commutative polyadic ring,
and if it contains the identity, then R(m,n) is a (m, n)-semiring. A polyadic ring R(m,n) is
called derived, if (m) and µ(n) are repetitions of the binary addition (+) and multiplica-
tion (·), whereas hR | (+)i and hR | (·)i are commutative (binary) group and semigroup,
respectively.
13. Universe 2022, 8, 230 13 of 21
4.3. Full Polyadic External Direct Product of (m, n)-Rings
Let us consider the following task: for a given polyadic (m, n)-ring R0(m,n) =
D
R0 | ν0(m), µ0(n)
E
to construct a product of all possible (in arity shape) constituent rings
R
(m1,n1)
1 and R
(m2,n2)
2 . The first-hand “polyadization” of (49) and (50) leads to
Definition 7. A full polyadic direct product ring R0(m,n) = R
(m,n)
1 × R
(m,n)
2 consists of (two)
polyadic rings of the same arity shape, such that
ν0(m)
h
X(1)
, X(2)
, . . . , X(m)
i
=
ν
(m)
1
h
x
(1)
1 , x
(2)
1 , . . . , x
(m)
1
i
ν
(m)
2
h
x
(1)
2 , x
(2)
2 , . . . , x
(m)
2
i
, (56)
µ0(n)
h
X(1)
, X(2)
, . . . , X(n)
i
=
µ
(n)
1
h
x
(1)
1 , x
(2)
1 , . . . , x
(n)
1
i
µ
(n)
2
h
x
(1)
2 , x
(2)
2 , . . . , x
(n)
2
i
, (57)
where the polyadic associativity (8) and polyadic distributivity (51)–(53) of the direct product
operations ν(m) and µ(n) follow from those of the constituent rings and the componentwise operations
in (56) and (57).
Example 8. Consider two (2, 3)-rings R
(2,3)
1 =
D
{ix} | ν
(2)
1 = (+), µ
(3)
1 = (·), x ∈ Z, i2 = −1
E
and R
(2,3)
2 =
0 a
b 0
| ν
(2)
2 = (+), µ
(3)
2 = (·), a, b ∈ Z
, where (+) and (·) are op-
erations in Z, then their polyadic direct product on the doubles XT =
ix,
0 a
b 0
∈
iZ, GLadiag(2, Z)
is defined by
ν0(2)
h
X(1)
, X(2)
i
=
ix(1) + ix(2)
0 a(1) + a(2)
b(1) + b(2) 0
, (58)
µ0(3)
h
X(1)
, X(2)
, X(3)
i
=
ix(1)x(2)x(3)
0 a(1)b(2)a(3)
b(1)a(2)b(3) 0
. (59)
The polyadic associativity and distributivity of the direct product operations ν0(2) and µ0(3) are
evident, and therefore R(2,3) =
ix,
0 a
b 0
| ν0(2), µ0(3)
is a (2, 3)-ring R(2,3) =
R
(2,3)
1 × R
(2,3)
2 .
Definition 8. A polyadic direct product R(m,n) is called derived if both constituent rings R
(m,n)
1
and R
(m,n)
2 are derived, such that the operations ν
(m)
1,2 and µ
(n)
1,2 are compositions of the binary
operations ν
(2)
1,2 and µ
(2)
1,2 , correspondingly.
So, in the derived case (see (1) all the operations in (56) and (57) have the form (cf., (21))
ν
(m)
1,2 =
ν
(2)
1,2
◦(m−1)
, µ
(n)
1,2 =
ν
(2)
1,2
◦(n−1)
, (60)
ν(m)
=
ν(2)
◦(m−1)
, µ(n)
=
ν(2)
◦(n−1)
. (61)
14. Universe 2022, 8, 230 14 of 21
Thus, the operations of the derived polyadic ring can be written as (cf., the binary
direct product (47) and (48))
ν0(m)
h
X(1)
, X(2)
, . . . , X(m)
i
=
x
(1)
1 +1 x
(2)
1 +1 . . . +1 x
(m)
1
x
(1)
2 +2 x
(2)
2 +2 . . . +2 x
(m)
2
, (62)
µ0(n)
h
X(1)
, X(2)
, . . . , X(n)
i
=
x
(1)
1 ·1 x
(2)
1 ·1 . . . ·1 x
(n)
1
x
(1)
2 ·2 x
(2)
2 ·2 . . . ·2 x
(n)
2
, (63)
The external direct product (2, 3)-ring R(2,3) from Example 8 is not derived, because
both multiplications µ
(3)
1 and µ
(3)
2 there are nonderived.
4.4. Mixed-Arity Iterated Product of (m, n)-Rings
Recall that some polyadic multiplications can be iterated, i.e., composed (1) from those
of lower arity (2), as well as those larger than 2, and so being nonderived, in general. The
nontrivial “polyadization” of (49) and (50) can arise, when the composition of the separate
(up and down) components on the r.h.s. of (56) and (57) will be different, and therefore
the external product operations on the doubles X ∈ R1 × R2 cannot be presented in the
iterated form (1).
Let the constituent operations in (56) and (57) be composed from lower-arity cor-
responding operations, but in different ways for the up and down components, such
that
ν
(m)
1 =
ν
(m1)
1
◦`ν1
, ν
(m)
2 =
ν
(m2)
2
◦`ν2
, 3 ≤ m1,2 ≤ m − 1, (64)
µ
(n)
1 =
µ
(n1)
1
◦`µ1
, µ
(n)
2 =
µ
(n2)
2
◦`µ2
, 3 ≤ n1,2 ≤ n − 1, (65)
where we exclude the limits: the derived case m1,2 = n1,2 = 2 (60) and (61) and the
uncomposed case m1,2 = m, n1,2 = n (56) and (57). Since the total size of the up and
down polyads is the same and coincides with the arities of the double addition m and
multiplication n, using (2) we obtain the arity compatibility relations
m = `ν1(m1 − 1) + 1 = `ν2(m2 − 1) + 1, (66)
n = `µ1(n1 − 1) + 1 = `µ2(n2 − 1) + 1. (67)
Definition 9. A mixed-arity polyadic direct product ring R(m,n) = R
(m1,n1)
1 ~ R
(m2,n2)
2 consists
of two polyadic rings of the different arity shape, such that
ν0(m)
h
X(1)
, X(2)
, . . . , X(m)
i
=
ν
(m1)
1
◦`ν1
h
x
(1)
1 , x
(2)
1 , . . . , x
(m)
1
i
ν
(m2)
2
◦`ν2
h
x
(1)
2 , x
(2)
2 , . . . , x
(m)
2
i
, (68)
µ0(n)
h
X(1)
, X(2)
, . . . , X(n)
i
=
µ
(n1)
1
◦`µ1
h
x
(1)
1 , x
(2)
1 , . . . , x
(n)
1
i
µ
(n2)
2
◦`µ2
h
x
(1)
2 , x
(2)
2 , . . . , x
(n)
2
i
, (69)
and the arity compatibility relations (66) and (67) hold valid.
Thus, two polyadic rings cannot always be composed in the mixed-arity polyadic
direct product.
15. Universe 2022, 8, 230 15 of 21
Assertion 3. If the arity shapes of two polyadic rings R
(m1,n1)
1 and R
(m2,n2)
2 satisfy the compatibil-
ity conditions
a(m1 − 1) = b(m2 − 1), (70)
c(n1 − 1) = d(n2 − 1), a, b, c, d ∈ N, (71)
then they can form a mixed-arity direct product.
The limiting cases, undecomposed (56) and (57) and derived (62) and (63), satisfy the
compatibility conditions (70) and (71) as well.
Example 9. Let us consider two (nonderived) polyadic rings
R
(9,3)
1 =
D
{8l + 7} | ν
(9)
1 , µ
(3)
1 , l ∈ Z
E
, (72)
R
(5,5)
2 =
D
{M} | ν
(5)
2 , µ
(5)
2
E
, (73)
where
M =
0 4k1 + 3 0 0
0 0 4k2 + 3 0
0 0 0 4k3 + 3
4k4 + 3 0 0 0
, ki ∈ Z, (74)
and ν
(5)
2 and µ
(5)
2 are the ordinary sum and product of 5 matrices. Using (66) and (67) we obtain
m = 9, n = 5, if we choose the smallest “numbers of multiplications” `ν1 = 1, `ν2 = 2, `µ1 = 2,
`µ2 = 1, and therefore the mixed-arity direct product nonderived (9, 5)-ring becomes
R(9,5)
=
D
{X} | ν0(9)
, µ0(5)
E
, (75)
where the doubles are X =
8l + 7
M
and the nonderived direct product operations are
ν0(9)
h
X(1)
, X(2)
, . . ., X(9)
i
=
8
l(1) + l(2) + l(3) + l(4) + l(5) + l(6) + l(7) + l(8) + l(9) + 7
+ 7
0 4K1 + 3 0 0
0 0 4K2 + 3 0
0 0 0 4K3 + 3
4K4 + 3 0 0 0
, (76)
µ0(5)
h
X(1)
, X(2)
, X(3)
, X(4)
, X(5)
i
=
8lµ + 7
0 4Kµ,1 + 3 0 0
0 0 4Kµ,2 + 3 0
0 0 0 4Kµ,3 + 3
4Kµ,4 + 3 0 0 0
, (77)
where, in the first line, Ki = k
(1)
i + k
(2)
i + k
(3)
i + k
(4)
i + k
(5)
i + k
(6)
i + k
(7)
i + k
(8)
i + k
(9)
i + 6 ∈ Z,
lµ ∈ Z is a cumbersome integer function of l(j) ∈ Z, j = 1, . . . , 9, and in the second line Kµ,i ∈ Z
are cumbersome integer functions of k
(s)
i , i = 1, . . . , 4, s = 1, . . . , 5. Therefore, the polyadic
ring (75) is the nonderived mixed arity polyadic external product R(9,5) = R
(9,3)
1 ~ R
(5,5)
2 (see
Definition 9).
16. Universe 2022, 8, 230 16 of 21
Theorem 2. The category of polyadic rings PolRing can exist (having the class of all polyadic
rings for objects and ring homomorphisms for morphisms) and can be well-defined, because it has a
product as the polyadic external product of rings.
In the same way one can construct the iterated full and mixed-arity products of any
number k of polyadic rings, merely by passing from the doubles X to k-tuples XT
k =
(x1, . . . , xk).
4.5. Polyadic Hetero Product of (m, n)-Fields
The most crucial difference between the binary direct products and the polyadic ones
arises for fields, because a direct product’s two binary fields are not a field [2].The reason
for this lies in the fact that each binary field F(2,2) necessarily contains 0 and 1, by definition.
As follows from (48), a binary direct product contains nonzero idempotent doubles
1
0
and
0
1
which are noninvertible, and therefore the external direct product of fields
F
(2,2)
1 × F
(2,2)
2 can never be a field. In the opposite case,polyadic fields (see Definition 10)
can be zeroless (we denote them by b
F),and the above arguments do not hold for them.
Recall the definitions of (m, n)-fields (see [27,28]). Denote R∗ = R {z}, if the zero z
exists (3). Observe that (in distinction to binary rings)
D
R∗ | µ(n) | assoc
E
is not a polyadic
group, in general. If
D
R∗ | µ(n)
E
is the n-ary group, then R(m,n) is called a (m, n)-division
ring D(m,n).
Definition 10. A (totally) commutative (m, n)-division ring R(m,n) is called a (m, n)-field F(m,n).
In n-ary groups there exists an “intermediate” commutativity, known as semicommu-
tativity (10).
Definition 11. A semicommutative (m, n)-division ring R(m,n) is called a semicommutative
(m, n)-field F(m,n).
The definition of a polyadic field can be expressed in a diagrammatic form, analogous
to (16). We introduce the double Dörnte relations: for n-ary multiplication µ(n) (15) and for
m-ary addition ν(m), as follows
ν(m)
my, x
= x, (78)
where the (additive) neutral sequence is my = ym−2, ỹ
, and ỹ is the additive querelement
for y ∈ R (see (14)). In distinction with (15), we have only one (additive) Dörnte relation
(78) and one diagram from (16) only, because of the commutativity of ν(m).
By analogy with the multiplicative queroperation µ̄(1) (13), introduce the additive unary
queroperation by
ν̃(1)
(x) = x̃, ∀x ∈ R, (79)
where x̃ is the additive querelement (13). Thus, we have
Definition 12 (Diagrammatic definition of (m, n)-field). A (polyadic) (m, n)-field is a one-set
algebraic structure with 4 operations and 3 relations
D
R | ν(m)
, ν̃(1)
, µ(n)
, µ̄(1)
| associativity, distributivity, double Dörnte relations
E
, (80)
where ν(m) and µ(n) are commutative associative m-ary addition and n-ary associative multiplication
connected by polyadic distributivity (51)–(53), ν̃(1) and µ̄(1) are unary additive queroperation (79)
and multiplicative queroperation (13).
17. Universe 2022, 8, 230 17 of 21
There is no initial relation between ν̃(1) and µ̄(1); nevertheless the possibility of their
“interaction” can lead to further thorough classification of polyadic fields.
Definition 13. A polyadic field F(m,n) is called quer-symmetric if its unary queroperations com-
mute
ν̃(1)
◦ µ̄(1)
= µ̄(1)
◦ ν̃(1)
, (81)
e
x = e
x, ∀x ∈ R, (82)
in the other case F(m,n) is called quer-nonsymmetric.
Example 10. Consider the nonunital zeroless (denoted by b
F)polyadic field b
F(3,3) =
D
{ia/b} | ν(3), µ(3)
E
, i2 = −1, a, b ∈ Zodd. The ternary addition ν(3)[x, y, t] = x + y + t
and the ternary multiplication µ(3)[x, y, t] = xyt are nonderived, ternary associative and distribu-
tive (operations are in C). For each x = ia/b (a, b ∈ Zodd) the additive querelement is x̃ = −ia/b,
and the multiplicative querelement is x̄ = −ib/a (see (12)). Therefore, both
D
{ia/b} | µ(3)
E
and
D
{ia/b} | ν(3)
E
are ternary groups, but they both contain no neutral elements (no unit, no zero).
The nonunital zeroless (3, 3)-field b
F(3,3) is quer-symmetric, because (see (82))
e
x = e
x = i
b
a
. (83)
Finding quer-nonsymmetric polyadic fields is not a simple task.
Example 11. Consider the set of real 4 × 4 matrices over the fractions 4k+3
4l+3 , k, l ∈ Z, of the form
M =
0
4k1 + 3
4l1 + 3
0 0
0 0
4k2 + 3
4l2 + 3
0
0 0 0
4k3 + 3
4l3 + 3
4k4 + 3
4l4 + 3
0 0 0
, ki, li ∈ Z. (84)
The set {M} is closed with respect to the ordinary addition of m ≥ 5 matrices, because the sum
of fewer of the fractions 4k+3
4l+3 does not give a fraction of the same form [14], and with respect to the
ordinary multiplication of n ≥ 5 matrices, since the product of fewer matrices (84) does not have the
same shape [29]. The polyadic associativity and polyadic distributivity follow from the binary ones
of the ordinary matrices over R, and the product of 5 matrices is semicommutative (see 10). Taking
the minimal values m = 5, n = 5, we define the semicommutative zeroless (5, 5)-field (see (11))
F
(5,5)
M =
D
{M} | ν(5)
, µ(5)
, ν̃(1)
, µ̄(1)
E
, (85)
where ν(5) and µ(5) are the ordinary sum and product of 5 matrices, whereas ν̃(1) and µ̄(1) are
additive and multiplicative queroperations
ν̃(1)
[M] ≡ M̃ = −3M, µ̄(1)
[M] ≡ M̄ =
4l1 + 3
4k1 + 3
4l2 + 3
4k2 + 3
4l3 + 3
4k3 + 3
4l4 + 3
4k4 + 3
M. (86)
18. Universe 2022, 8, 230 18 of 21
The division ring D
(5,5)
M is zeroless, because the fraction 4k+3
4l+3 , is never zero for k, l ∈ Z, and it
is unital with the unit
Me =
0 1 0 0
0 0 1 0
0 0 0 1
1 0 0 0
. (87)
Using (84) and (86), we obtain
ν̃(1)
h
µ̄(1)
[M]
i
= −3
4l1 + 3
4k1 + 3
4l2 + 3
4k2 + 3
4l3 + 3
4k3 + 3
4l4 + 3
4k4 + 3
M, (88)
µ̄(1)
h
ν̃(1)
[M]
i
= −
1
27
4l1 + 3
4k1 + 3
4l2 + 3
4k2 + 3
4l3 + 3
4k3 + 3
4l4 + 3
4k4 + 3
M, (89)
or
e
M = 81 e
M, (90)
and therefore the additive and multiplicative queroperations do not commute independently of
the field parameters. Thus, the matrix (5, 5)-division ring D
(5,5)
M (85) is a quer-nonsymmetric
division ring.
Definition 14. The polyadic zeroless direct product field b
F0(m,n) =
D
R0 | ν0(m), µ0(n)
E
consists of
(two) zeroless polyadic fields b
F
(m,n)
1 =
D
R1 | ν
(m)
1 , µ
(n)
1
E
and b
F
(m,n)
2 =
D
R2 | ν
(m)
2 , µ
(n)
2
E
of the
same arity shape, whereas the componentwise operations on the doubles X ∈ R1 × R2 in (56) and
(57) still remain valid, and
D
R1 | µ
(n)
1
E
,
D
R2 | µ
(n)
2
E
,
D
R0 = {X} | µ0(n)
E
are n-ary groups.
Following Definition 11, we have
Corollary 2. If at least one of the constituent fields is semicommutative, and another one is totally
commutative, then the polyadic product will be a semicommutative (m, n)-field.
The additive and multiplicative unary queroperations (13) and (79) for the direct
product field b
F(m,n) are defined componentwise on the doubles X as follows
ν̃0(1)
[X] =
ν̃
(1)
1 [x1]
ν̃
(1)
2 [x2]
!
, (91)
µ̄0(1)
[X] =
µ̄
(1)
1 [x1]
µ̄
(1)
2 [x2]
!
, x1 ∈ R1, x2 ∈ R2. (92)
Definition 15. A polyadic direct product field b
F0(m,n) =
D
R0 | ν0(m), ν̃0(1), µ0(n), µ̄0(1)
E
is called
quer-symmetric if its unary queroperations (91) and (92) commute
ν̃0(1)
◦ µ̄0(1)
= µ̄0(1)
◦ ν̃0(1)
, (93)
e
X = e
X, ∀X ∈ R0
, (94)
in the other case, b
F0(m,n) is called a quer-nonsymmetric direct product (m, n)-field.
Example 12. Consider two nonunital zeroless (3, 3)-fields
b
F
(3,3)
1,2 =
i
a1,2
b1,2
| ν
(3)
1,2 , µ
(3)
1,2 , ν̃
(1)
1,2 , µ̄
(1)
1,2
, i2
= −1, a1,2, b1,2 ∈ Zodd
, (95)
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where ternary additions ν
(3)
1,2 and ternary multiplications µ
(3)
1,2 are the sum and product in Zodd,
correspondingly, and the unary additive and multiplicative queroperations are ν̃
(1)
1,2 [ia1,2/b1,2] =
−ia1,2/b1,2 and µ̄
(1)
1,2 [ia1,2/b1,2] = −ib1,2/a1,2 (see Example 10). Using (56) and (57) we build the
operations of the polyadic nonderived nonunital zeroless product (3, 3)-field b
F0(3,3) = b
F
(3,3)
1 ×
b
F
(3,3)
2 on the doubles XT = (ia1/b1, ia2/b2) as follows
ν0(3)
h
X(1)
, X(2)
, X(3)
i
=
i
a
(1)
1 b
(2)
1 b
(3)
1 + b
(1)
1 a
(2)
1 b
(3)
1 + b
(1)
1 b
(2)
1 a
(3)
1
b
(1)
1 b
(2)
1 b
(3)
1
i
a
(1)
2 b
(2)
2 b
(3)
2 + b
(1)
2 a
(2)
2 b
(3)
2 + b
(1)
2 b
(2)
2 a
(3)
2
b
(1)
2 b
(2)
2 b
(3)
2
, (96)
µ0(3)
h
X(1)
, X(2)
, X(3)
i
=
−i
a
(1)
1 a
(2)
1 a
(3)
1
b
(1)
1 b
(2)
1 b
(3)
1
−i
a
(1)
2 a
(2)
2 a
(3)
2
b
(1)
2 b
(2)
2 b
(3)
2
, a
(j)
i , b
(j)
i ∈ Zodd
, (97)
and the unary additive and multiplicative queroperations (91) and (92) of the direct product b
F0(3,3)
are
ν̃0(1)
[X] =
−i
a1
b1
−i
a2
b2
, (98)
µ̄0(1)
[X] =
−i
b1
a1
−i
b2
a2
, ai, bi ∈ Zodd
. (99)
Therefore, both
D
{X} | ν0(3), ν̃0(1)
E
and
D
{X} | µ0(3), µ̄0(1)
E
are commutative ternary groups,
which means that the polyadic direct product b
F0(3,3) = b
F
(3,3)
1 × b
F
(3,3)
2 is the nonunital zeroless
polyadic field. Moreover, b
F0(3,3) is quer-symmetric, because (93) and (94) remain valid
µ̄0(1)
◦ ν̃0(1)
[X] = ν̃0(1)
◦ µ̄0(1)
[X] =
i
b1
a1
i
b2
a2
, ai, bi ∈ Zodd
. (100)
Example 13. Let us consider the polyadic direct product of two zeroless fields, one of them being
the semicommutative (5, 5)-field b
F
(5,5)
1 = F
(5,5)
M from (85), and the other one being the nonderived
nonunital zeroless (5, 5)-field of fractions b
F
(5,5)
2 =
Dn√
i4r+1
4s+1
o
| ν
(5)
2 , µ
(5)
2
E
, r, s ∈ Z, i2 = −1.
The double is XT =
√
i4r+1
4s+1 , M
, where M is in (84). The polyadic nonunital zeroless direct
20. Universe 2022, 8, 230 20 of 21
product field b
F0(5,5) = b
F
(5,5)
1 × b
F
(5,5)
2 is nonderived and semicommutative, and is defined by
b
F(5,5) =
D
X | ν0(5), µ0(5), ν̃0(1), µ̄(1)
E
, where its addition and multiplication are
ν0(5)
h
X(1)
, X(2)
, X(3)
, X(4)
, X(5)
i
=
√
i
4Rν + 1
4Sν + 1
0
4Kν,1 + 3
4Lν,1 + 3
0 0
0 0
4Kν,2 + 3
4Lν,2 + 3
0
0 0 0
4Kν,3 + 3
4Lν,3 + 3
4Kν,4 + 3
4Lν,4 + 3
0 0 0
, (101)
µ0(5)
h
X(1)
, X(2)
, X(3)
, X(4)
, X(5)
i
=
√
i
4Rµ + 1
4Sµ + 1
0
4Kµ,1 + 3
4Lµ,1 + 3
0 0
0 0
4Kµ,2 + 3
4Lµ,2 + 3
0
0 0 0
4Kµ,3 + 3
4Lµ,3 + 3
4Kµ,4 + 3
4Lµ,4 + 3
0 0 0
, (102)
where Rν,µ, Sν,µ ∈ Z are cumbersome integer functions of r(i), s(i) ∈ Z, i = 1, . . . , 5, and
Kν,i, Kµ,i, Lν,i, Lµ,i ∈ Z are cumbersome integer functions of k
(i)
j , l
(i)
j ∈ Z, j = 1, . . . , 4, i =
1, . . . , 5 (see (84)). The unary queroperations (91) and (92) of the direct product b
F(5,5) are
ν̃0(1)
[X] =
−3
√
i
4r + 1
4s + 1
−3M
, (103)
µ̄0(1)
[X] =
−
√
i
4s + 1
4r + 1
3
4l1 + 3
4k1 + 3
4l2 + 3
4k2 + 3
4l3 + 3
4k3 + 3
4l4 + 3
4k4 + 3
M
, r, s, ki, li ∈ Z, (104)
where M is in (84). Therefore,
D
{X} | ν0(5), ν̃0(1)
E
is a commutative 5-ary group, and
D
{X} | µ0(5), µ̄0(1)
E
is a semicommutative 5-ary group, which means that the polyadic direct
product b
F0(5,5) = b
F
(5,5)
1 × b
F
(5,5)
2 is the nonunital zeroless polyadic semicommutative (5, 5)-field.
Using (90) we obtain
ν̃0(1)
µ̄0(1)
[X] = 81µ̄0(1)
ν̃0(1)
[X], (105)
and therefore the direct product (5, 5)-field b
F0(5,5) is quer-nonsymmetric (see (81)).
Thus, we arrive at
21. Universe 2022, 8, 230 21 of 21
Theorem 3. The category of zeroless polyadic fields zlessPolField can exist (having the class
of all zeroless polyadic fields for objects and field homomorphisms for morphisms) and can be
well-defined, because it has a product as the polyadic field product.
5. Conclusions
For physical applications, for instance, the particle content of any elementary particle
model is connected with the direct decomposition of its gauge symmetry group. Thus, the
proposed generalization of the direct product can lead to principally new physical models
having unusual mathematical properties.
For mathematical applications, further analysis of the direct product constructions
introduced here and their examples for polyadic rings and fields would be interesting, and
could lead to new kinds of categories.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Acknowledgments: The author is deeply grateful to Vladimir Akulov, Mike Hewitt, Vladimir
Tkach, Raimund Vogl and Wend Werner for numerous fruitful discussions, important help and
valuable support.
Conflicts of Interest: The authors declare no conflict of interest.
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