Inverse semigroup homomorphisms via partial group actions

Abstract

This papar constructs all homomorphisms of inverse semigroups which factor through anE-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domainIof βE-unitary; moreover, theP-representation ofIis explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism isE-unitary. Stronger results are obtained for the case ofF-inverse monoids.Special cases of our results include theP-theorem and the factorisation theorem for homomorphisms fromE-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence ofE-unitary covers over a group, as well as a generalisation toF-inverse covers, allowing a quick proof that every inverse monoid has anF-inverse cover.