Abstract
AbstractIn 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
}
.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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