Finite and semisimple Boolean inverse monoids

Abstract

We describe the structure of finite Boolean inverse monoids and apply our results to the representation theory of finite inverse semigroups. We then generalize to semisimple Boolean inverse semigroups.

Non-commutative Stone duality

“…If S is a semisimple Boolean inverse semigroup, then the Stone groupoid G(S) is isomorphic to the set of atoms of S equipped with the restricted product; see [29] for the structure of semisimple Boolean inverse semigroups. The significance of semisimple Boolean inverse semigroups is explained by the following result.…”

Characterizations of classes of countable Boolean inverse monoids

“…The theory of finite Boolean inverse monoids generalizes that of finite Boolean algebras by replacing the finite sets of the theory of finite Boolean algebras by the finite groupoids of the theory of finite Boolean inverse monoids. The theory of finite Boolean inverse monoids is summarized in [22]. The goal of our paper is to generalize that theory to the countably infinite case.…”

“…• By [22,Lemma 4.6], S is 0-simplifying if and only if at(S) is connected. We see that the groupoid structure of at(S) determines the structure of S. If G is a group, we denote by G 0 the group G with a zero adjoined, and we denote by R n (G 0 ) the Boolean inverse monoid of all n × n rook matrices over G 0 .…”