Regular semigroups weakly generated by one element
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 F1 weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of F1. We define F1 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 F1. In particular, we show that the `free regular semigroup FI2 weakly generated by two idempotents "is isomorphic to a regular subsemigroup of F1 weakly generated by xx',x'x.