$\sigma$-matching and interchangeable structures on certain associative algebras

TL;DR

This paper introduces $\sigma$-matching and interchangeable structures on various associative algebras, providing a unified framework for compatible product operations across multiple algebra classes.

Contribution

It defines $\sigma$-matching and interchangeable products, extending the understanding of compatible algebraic structures in several classes of associative algebras.

Findings

$\sigma$-matching structures are characterized for specific algebra classes

Interchangeable and totally compatible products are constructed on these algebras

The framework applies to unital, semigroup, and free associative algebras

Abstract

We describe $\sigma$-matching, interchangeable and, as a consequence, totally compatible products on some classes of associative algebras, including unital algebras, the semigroup algebras of rectangular bands, algebras with enough idempotents, free non-unital associative algebras and free non-unital commutative associative algebras.