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 {textrm{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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