Combinatorially factorizable inverse monoids

Abstract

An inverse semigroup S is factorizable if it can be written as a product of a semilattice and a group; more generally, S is combinatorially factorizable if it is the product of a combinatorial semigroup and a group. In this paper we explore combinatorially factorizable monoids S on which $\mathcal{H}$ is a congruence. Such semigroups will be characterized as certain idempotent-separating extensions of a factorizable Clifford semigroup by a combinatorial semigroup. Necessary and sufficient conditions will be given for S to be a direct product of a combinatorial semigroup and a group. We also give a means to construct all combinatorially factorizable submonoids of any inverse monoid on which $\mathcal{H}$ is a congruence.