Regular semigroups weakly generated by one element

TL;DR

This paper investigates the structure of regular semigroups generated by a single element, establishing a universal semigroup with a decidable word problem and exploring its properties and subsemigroups.

Contribution

It introduces a universal regular semigroup weakly generated by one element, with a decidable word problem and a canonical form for its congruence classes.

Findings

Existence of a universal regular semigroup weakly generated by one element

Decidability of the word problem for this semigroup

Structural relationship between free regular semigroups and this universal semigroup

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}.