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Search Results (6)

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Keywords = polyadic semigroup

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8 pages, 303 KiB  
Article
Polyadic Rings of p-Adic Integers
by Steven Duplij
Symmetry 2022, 14(12), 2591; https://doi.org/10.3390/sym14122591 - 7 Dec 2022
Viewed by 1596
Abstract
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 [...] Read more.
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 very short spacetime scales, such rings could lead to new symmetries of modern particle models. Full article
(This article belongs to the Special Issue Symmetry in Cosmology and Gravity: Topic and Advance)
20 pages, 373 KiB  
Article
Polyadization of Algebraic Structures
by Steven Duplij
Symmetry 2022, 14(9), 1782; https://doi.org/10.3390/sym14091782 - 26 Aug 2022
Viewed by 1746
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry in Strong-Field Physics)
21 pages, 432 KiB  
Article
Polyadic Analogs of Direct Product
by Steven Duplij
Universe 2022, 8(4), 230; https://doi.org/10.3390/universe8040230 - 8 Apr 2022
Cited by 3 | Viewed by 1989
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Selected Topics in Gravity, Field Theory and Quantum Mechanics)
20 pages, 379 KiB  
Article
Higher Regularity, Inverse and Polyadic Semigroups
by Steven Duplij
Universe 2021, 7(10), 379; https://doi.org/10.3390/universe7100379 - 13 Oct 2021
Viewed by 1873
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Gauge Theory, Strings and Supergravity)
48 pages, 597 KiB  
Article
Polyadic Braid Operators and Higher Braiding Gates
by Steven Duplij and Raimund Vogl
Universe 2021, 7(8), 301; https://doi.org/10.3390/universe7080301 - 15 Aug 2021
Cited by 3 | Viewed by 2338
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Foundations of Quantum Mechanics and Quantum Gravity)
17 pages, 356 KiB  
Article
Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence
by Steven Duplij
Mathematics 2021, 9(9), 972; https://doi.org/10.3390/math9090972 - 26 Apr 2021
Cited by 5 | Viewed by 1894
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 [...] Read more.
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. Full article
(This article belongs to the Section A: Algebra and Logic)
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