Polyadic Algebraic Structures

Duplij, Steven

doi:10.1088/978-0-7503-2648-3

Cited by 7 publications

(33 citation statements)

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“…Here we recall that representatives of special residue (congruence) classes can form polyadic rings, as was found in [11,12] (see also notation from [13]).…”

Section: (M N)-rings Of Integer Numbers From Residue Classesmentioning

confidence: 90%

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Polyadic rings of p-adic integers

Duplij¹

2022

Preprint

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“…Here we recall that representatives of special residue (congruence) classes can form polyadic rings, as was found in [11,12] (see also notation from [13]).…”

Section: (M N)-rings Of Integer Numbers From Residue Classesmentioning

confidence: 90%

“…We have found that in the study of p-adic integers some polyadic structures, that is (m, n)-rings, can appear naturally, if we introduce informally a p-adic analog of the residue classes for ordinary integers and investigate the set of its representatives along the lines of [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Polyadic rings of p-adic integers

Duplij¹

2022

Preprint

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“…The relation (2.1) can be considered as a definition of the unary queroperation μp1q ras " ā GLEICHGEWICHT AND GŁAZEK [1967]. For further details and definitions, see DUPLIJ [2022a].…”

Section: Preliminariesmentioning

confidence: 99%

“…Because the binary addition in R transfers to the matrix addition without restrictions (as opposed to the polyadic case, see below), we will consider only the multiplicative part of the resulting polyadic matrix ring. In this way, we propose a special block-shift matrix method to obtain n-ary semigroups (n-ary groups) from the binary ones, but the former are not derived from the latter GAL'MAK [2003], DUPLIJ [2022a]. In general, this can lead to new algebraic structures that were not known before.…”

Section: Polyadization Conceptmentioning

confidence: 99%

Polyadization of algebraic structures

Duplij¹

2022

Preprint

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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. CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SEMISIMPLICITY 3.1. Simple polyadic structures 3.2. Semisimple polyadic structures 4. POLYADIZATION CONCEPT 4.1. Polyadization of binary algebraic structures 4.2. Concrete examples of the polyadization procedure 4.2.1. Polyadization of GL p2, Cq 4.2.2. Polyadization of SO p2, Rq 4.3. "Deformation" of binary operations by shifts 4.4. Polyadization of binary supergroups 4.4.1.

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“…We found that the set of representatives becomes a polyadic or (m, n)-ring, if the parameters of a class satisfy special "quantization" conditions. We have found that similar polyadic structures appear naturally for p-adic integers, if we introduce informally a p-adic analog of the residue classes, and investigate here the set of its representatives along the lines of [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Polyadic Rings of p-adic Integers

Duplij¹

2022

Preprint

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In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-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 $\left( m,n\right) $-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.

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Polyadic Algebraic Structures

Cited by 7 publications

References 0 publications

Polyadic rings of p-adic integers

Polyadic rings of p-adic integers

Polyadization of algebraic structures

Polyadic Rings of p-adic Integers

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