Equationally defined classes of semigroups

Peter M. Higgins; Marcel Jackson

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arXiv · 2212.02699

Equationally defined classes of semigroups

Peter M. Higgins·Marcel Jackson

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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.

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BibTeX RIS

Peter M. Higgins, Marcel Jackson. 2022-12-06. Equationally defined classes of semigroups. https://arxiv.org/abs/2212.02699

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