Bifree locally inverse semigroups as expansions.
In a preceding paper [Stud.~Sci.~Math.~Hung.~41, No. 1, 39--58 (2004; Zbl 1070.20074)], the author assigns to each completely simple semigroup \(S\) an expansion \({\mathcal C}(S)\) which is a locally inverse semigroup and which coincides with the Birget-Rhodes expansion in case \(S\) is a group. In the present paper, the author further explores the properties of \({\mathcal C}(S)\). The main result states that if \(S\) is the bifree completely simple semigroup on a set \(X\) then \({\mathcal C}(S)\) contains a copy of the bifree locally inverse semigroup on \(X\). As a consequence, new proofs of two classical results by Pastijn are obtained: (i) each locally inverse semigroup divides a semidirect product of a semilattice by a completely simple semigroup; (ii) each locally inverse semigroup divides a perfect rectangular band of inverse monoids.
perfect rectangular bands of inverse monoids
scientific article; zbMATH DE number 2136594
scientific article