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Inverse semigroup homomorphisms via partial group actions

2001; Cambridge University Press; Volume: 64; Issue: 1 Linguagem: Inglês

10.1017/s0004972700019778

1755-1633

Benjamin Steinberg,

Geometric and Algebraic Topology

This papar constructs all homomorphisms of inverse semigroups which factor through an E -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 domain I of β E -unitary; moreover, the P -representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E -unitary. Stronger results are obtained for the case of F -inverse monoids. Special cases of our results include the P -theorem and the factorisation theorem for homomorphisms from E -unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E -unitary covers over a group, as well as a generalisation to F -inverse covers, allowing a quick proof that every inverse monoid has an F -inverse cover.