Almost Factorizable Glrac Semigroups

Glrac semigroups are common generalizations of left GC-lpp semigroups and left inverse semigroups. And, such a semigroup is a left restriction semigroup if and only if the projection set is a semilattice. So, glrac semigroup is also a generalization of left restriction semigroup. Permissible subsets of a glrac semigroup are introduced in this paper. In terms of permissible subsets, we define (uniquely) factorizable glrac semigroups and (uniquely) almost factorizable glrac semigroups. Many characterizations of (uniquely) factorizable glrac semigroups and (uniquely) almost factorizable glrac semigroups are obtained. As their applications, we establish the structures of uniquely factorizable left GC-lpp semigroups (left inverse semigroups, inverse semigroups, ample semigroups, left restriction semigroup, restriction semigroups) and uniquely almost factorizable left GC-lpp semigroups (left inverse semigroups, inverse semigroups, ample semigroups, left restriction semigroup, restriction semigroups). Our results enrich and extend the related results of almost factorizable restriction semigroups.