Equationally defined classes of semigroups

TL;DR

This paper explores equationally defined classes of semigroups, demonstrating that certain classes can be characterized by quantifier-free equations and providing examples requiring multiple quantifiers.

Contribution

It extends the main theorem from algebraic logic to semigroups, showing conditions under which classes are defined by quantifier-free equations and presenting examples with complex quantifier requirements.

Findings

Classes closed under products and homomorphic images are defined by systems of equations.

Some semigroup classes require equations with more than two quantifiers.

A dual to Birkhoff's theorem is established for certain semigroup classes.

Abstract

We apply, in the context of semigroups, the main theorem from~\cite{higjac} that an elementary class $\mathcal{C}$ of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of $\mathcal{C}$ is free of the $\forall$ quantifier. Examples are given of EHP-classes that require more than two quantifiers in some equation of any equational basis.