Stable varieties of semigroups and groupoids

TL;DR

This paper characterizes stable and s-stable varieties of semigroups and groupoids using algebraic and reduction system methods, providing comprehensive classifications and new insights into term replacement techniques.

Contribution

It offers a complete description of stable varieties of semigroups and certain groupoids, introducing a novel approach with an abstract reduction system for analyzing term replacements.

Findings

Full classification of stable varieties of semigroups

Description of stable and s-stable varieties of groupoids

Introduction of an abstract reduction system for term analysis

Abstract

The paper deals with $\Sigma-$composition and $\Sigma$-essential composition of terms, which lead to stable and s-stable varieties of algebras. A full description of all stable varieties of semigroups, commutative and idempotent groupoids is obtained. We use an abstract reduction system which simplifies the presentations of terms of type $\tau=(2)$ to study the varietiy of idempotent groupoids and s-stable varieties of groupoids. They are used as an alternating of the stable varieties, aiming to highlight replacing the subterms of a term in a deductive systems instead of the usual replacing the variables with terms.