Factorizable inverse monoids

Abstract

Factorizable inverse monoids constitute the algebraic theory of those partial symmetries which are restrictions of automorphisms; the formal de…nition is that each element is the product of an idempotent and an invertible. This class of monoids has theoretical signi…cance, and includes concrete instances which are important in various contexts. This survey is organised around the idea of group acts on semilattices and contains a large range of examples. Topics also include methods for construction of factorizable inverse monoids, and aspects of their inner structure, morphisms, and presentations.

Partial symmetry, reflection monoids and Coxeter groups

“…At several points in the paper we will want to identify a monoid of partial isomorphisms with some pre-existing monoid in the literature. This is possible if the group of units are the same, the idempotents are the same and the actions of the groups on the idempotents are the same: We mention that this result also occurs in [8]; as there, we leave the straightforward proof to the reader.…”

“…Let Φ n be a root system of type A n−1 , B n or D n as in Table 1 and B the Boolean system (8) for W (Φ n ). Then the Boolean reflection monoids have orders,…”

Primitive permutation groups and strongly factorizable transformation semigroups

“…Finally we note that in the context of inverse semigroups (where strongly factorizable semigroups are simply called factorizable) this topic is an entire body of knowledge on its own. (For details we suggest the lovely survey paper [12] and the references therein.) In the context of groups, it is usually assumed that at least one of A and B is a subgroup; many classifications are known, notably that of Liebeck, Praeger and Saxl [22,23], who found all factorizations of almost simple groups where both factors are subgroups.…”

Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups

“…Finally we note that in the context of inverse semigroups (where strongly factorizable semigroups are simply called factorizable) this topic is an entire body of knowledge on its own. (For details we suggest the lovely survey paper [12] and the references therein.) In the context of groups, it is usually assumed that at least one of A and B is a subgroup; many classifications are known, notably that of Liebeck, Praeger and Saxl [20,21], who found all factorizations of almost simple groups where both factors are subgroups.…”