Polyadization of algebraic structures

TL;DR

This paper introduces the concept of polyadization, a method to construct polyadic algebraic structures from binary ones, generalizing semisimplicity and providing new forms and decompositions for these structures.

Contribution

It proposes the polyadization concept, enabling the construction of nonderived polyadic structures from binary structures, and extends semisimplicity to polyadic algebraic frameworks.

Findings

Polyadization allows constructing polyadic structures from binary ones.

A general form of semisimple polyadic structures is characterized by block-shift matrices.

Concrete examples illustrate the new algebraic constructions.

Abstract

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic structures is 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.