Left proper E-dense monoids

Abstract

By a left proper E-dense monoid M we mean a left type-A proper and E-dense monoid. More precisely, we say that M is left type-A if it can be embedded in an inverse monoid M̂ in such a way that the left congruence R ∗ is respected. The monoid M is said to be E-dense if E(M) is a semilattice and, for all a ∈ M, there exist x,y ∈ M such that ax and ya are idempotents. On a left type-A monoid M we define a relation σ by, for all a,b ∈ M, a σ b if and only if ea = eb, for some idempotent e ∈ M. The monoid M is called proper if R ∗∩σ=ι, the identity relation. In this paper, left proper E-dense monoids are characterised as: (I) monoids C / G where C is a connected proper left type-A category and G is a group acting on C transitively and fixpoint freely; (II) McAlister monoids M( G, X , Y ), where X and Y are defined in terms of a category C and a group G, as in (I); (III) submonoids M of a semidirect product S= J ∗ G, of a semilattice J by a group G, such that R ∗ M ⊆ R S. Also, we prove that every left type-A E-dense monoid has a left proper E-dense cover.