Algebras defined by equations
Group Theory
2022-06-23 v4 Logic
Abstract
We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular semigroups in particular this allows an interpretation of a universal algebraic nature that is formulated entirely in terms of the associative binary operation of the semigroup, which serves as an alternative to the approach via so called e-varieties. In particular we prove that classes of Inverse semigroups, Orthodox semigroups, and -solid semigroups are equational in our sense.
Cite
@article{arxiv.1810.13012,
title = {Algebras defined by equations},
author = {Peter M Higgins and Marcel Jackson},
journal= {arXiv preprint arXiv:1810.13012},
year = {2022}
}