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Equationally defined classes of semigroups

journal contribution
posted on 2025-04-01, 23:20 authored by PM Higgins, Marcel JacksonMarcel Jackson
We apply, in the context of semigroups, the main theorem from the authors’ paper “Algebras defined by equations” (Higgins and Jackson in J Algebra 555:131–156, 2020) that an elementary class 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 C is free of the ∀ quantifier. We also observe the decidability of the class of equation systems satisfied by semigroups, via a link to systems of rationally constrained equations on free semigroups. Examples are given of EHP-classes for which neither (∀ ⋯) (∃ ⋯) equation systems nor (∃ ⋯) (∀ ⋯) systems suffice.

History

Publication Date

2023-10-01

Journal

Semigroup Forum

Volume

107

Issue

2

Pagination

19p. (p. 459-477)

Publisher

Springer

ISSN

0037-1912

Rights Statement

© The Author(s) 2023. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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