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arXiv · 2302.08461

Regular semigroups weakly generated by one element

Luís Oliveira
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Abstract

In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup $F_1$ weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of $F_1$. We define $F_1$ using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of $F_1$. In particular, we show that the `free regular semigroup $FI_2$ weakly generated by two idempotents "is isomorphic to a regular subsemigroup of $F_1$ weakly generated by {xx',x'x}.

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Luís Oliveira. 2023-02-16. Regular semigroups weakly generated by one element. https://arxiv.org/abs/2302.08461

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