Polyadic analogs of direct product

TL;DR

This paper generalizes the concept of direct product to polyadic algebraic structures, allowing for different arities and entangled elements, while preserving associativity through a quiver technique, and explores conditions under which these products form new polyadic structures.

Contribution

It introduces novel external product constructions for polyadic semigroups, groups, rings, and fields, expanding the algebraic framework beyond traditional binary operations.

Findings

External products can have different arities from their factors.

Conditions identified for products to form new polyadic groups and fields.

Examples demonstrate the diversity and properties of these generalized products.

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 \textquotedblleft entangled\textquotedblright\ 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. Many illustrative concrete examples are presented.