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 $$F_{1}$$ F 1 weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of $$F_{1}$$ F 1 . We define $$F_{1}$$ 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 $$F_{1}$$ F 1 . In particular, we show that the ‘free regular semigroup $${\textrm{FI}}_2$$ FI 2 weakly generated by two idempotents’ is isomorphic to a regular subsemigroup of $$F_{1}$$ F 1 weakly generated by $$\{xx',x'x\}$$ { x x ′ , x ′ x } .