Factorizable inverse monoids of cosets of subgroups of a group

The study of coset monoids was initiated by Schein in 1966. The coset monoid of a group G, denoted ...(G), consists of all cosets of all subgroups of G. We show how to generalize ...(G) by constructing a monoid ...ℒ(G) of cosets of subgroups from a semilattice (ℒ, ∨ℒ) of subgroups of G which satisfies certain conditions. When (ℒ, ∨ℒ) = (...(G), ∨) is the join semilattice of all subgroups of G , we recover the coset monoid ...(G). The class of semigroups which are isomorphic to some ...ℒ(G) has a very simple description. Elements of the monoids in this class may be represented by cosets of subgroups of their group of units. We show that the factorizable part of the symmetric (resp. dual symmetric) inverse semigroup does not (resp. does) belong to this class.