Tarski monoids: Matui’s spatial realization theorem

Abstract

Abstract. We introduce a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class ofétale topological groupoids under a non-commutative generalization of classical Stone duality and, significantly, they arise naturally in the theory of dynamical systems as developed by Matui. We are thereby able to reinterpret a theorem of Matui (à la Rubin) on a class ofétale groupoids as an equivalent theorem about a class of Tarski monoids: two simple Tarski monoids are isomorphic if and only if their groups of units are isomorphic. The inverse monoids in question may also be viewed as countably infinite generalizations of finite symmetric inverse monoids. Their groups of units therefore generalize the finite symmetric groups and include amongst their number the Thompson groups Vn.

Higher dimensional generalizations of the Thompson groups

“…Let C be a k-graph. We say that C is aperiodic if for all a, b ∈ C such that a The following is a souped up version of [15,Lemma 3.2]. In the proof of the lemma below, we use the result that in a Boolean algebra e ≤ f if and only if ef = 0.…”

“…A non-zero element a of an inverse semigroup with zero is called an infinitesimal if a 2 = 0. Observe that a is an infinitesimal if and only if d(a) ⊥ by [15,Lemma 2.5]. An inverse semigroup S is said to be fundamental if the only elements of S that commute with all the idempotents of S are themselves idempotents.…”

“…The following is a souped up version of [15,Lemma 3.2]. In the proof of the lemma below, we use the result that in a Boolean algebra e ≤ f if and only if e f = 0.…”

Non-commutative Stone duality

“…This theorem generalizes what you will find in [24] since we shall not assume that our topological groupoids are Hausdorff. Although this theorem can be gleaned from our papers [24,25,26,27,30], what I describe here has not been reported in one place before.…”

Finite and semisimple Boolean inverse monoids

“…A countable atomless Boolean inverse monoid is called a Tarski monoid. The theory of such monoids is discussed in [8,9]. In Theorem 5.…”