Embedding Regular Semigroups into Idempotent Generated Ones

Abstract

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties [Formula: see text], we prove that S and ΨS simultaneously satisfy [Formula: see text] or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice [Formula: see text] of varieties of completely regular semigroups [Formula: see text] regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on [Formula: see text] and evaluate it on a small sublattice of [Formula: see text].