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 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 H is a congruence.

Factorizable inverse monoids

“…Appropriate generalisations of the concept were developed: signi…cantly, Lawson [17] identi…ed the appropriate generalisation from monoids to semigroups as almost factorizable semigroups, which had been used in McAlister [25]; for an account, see section 7.1 of [18]. Tirasupa [35] examined the Cli¤ord by semilattice case and Mills [29] the group by aperiodic case. Consideration of these generalisations is beyond the scope of this article.…”

Combinatorially Factorizable Cryptic Inverse Semigroups

“…An inverse semigroup S is factorizable if S = TG where T is a semilattice and G is a group. There is a modest literature concerning the structure of these semigroups, see [1]. This concept was generalized to include the case when T is a combinatorial inverse monoid by Mills [1] under the label of combinatorially factorizable inverse semigroups.…”

“…There is a modest literature concerning the structure of these semigroups, see [1]. This concept was generalized to include the case when T is a combinatorial inverse monoid by Mills [1] under the label of combinatorially factorizable inverse semigroups. In [1], she successfully analyzed such inverse semigroups as subsemigroups of cryptic inverse semigroups and some related cases.…”

“…This concept was generalized to include the case when T is a combinatorial inverse monoid by Mills [1] under the label of combinatorially factorizable inverse semigroups. In [1], she successfully analyzed such inverse semigroups as subsemigroups of cryptic inverse semigroups and some related cases. In her study appear two types of semigroups as essential ingredients: combinatorial inverse monoids and factorizable inverse monoids.…”

“…Hence we can say that, within cryptic inverse monoids, factorizable ones are completely determined. Analyzing subsemigroups of cryptic inverse monoids S that are susceptible to being of the form TG, where T is a combinatorial inverse monoid and G is a group, in [1] Mills arrived at a factorizable Clifford submonoid A of S having its semilattice of idempotents in common with T. In terms of structure, she arrived at a submonoid of a semidirect product of A and T.…”