The tight groupoid of the inverse semigroups of left cancellative small categories

Abstract

We fix a path model for the space of filters of the inverse semigroup S Λ associated to a left cancellative small category Λ. Then, we compute its tight groupoid, thus giving a representation of its C * -algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses. 2010 Mathematics Subject Classification. Primary: 46L05; Secondary: 46L80, 46L55, 20L05. . α ∨ β := {the minimal extensions of α and β} .Notice that if α ∨ β = ∅ then α ⋓ β, but the converse fails in general.When Λ is a finitely aligned LCSC, we can always assume that α ∨ β = Γ where Γ is a finite set of minimal common extensions of α and β. 1.2. Inverse semigroups. Definition 1.9. A semigroup S is an inverse semigroup if for every s ∈ S there exists a unique s * ∈ S such that s = ss * s and s * = s * ss * . Equivalently, S is an inverse semigroup if and only if the subsemigroup E(S) := {e ∈ S : e 2 = e} of idempotents of S is commutative [12, Theorem 1.1.3].A monoid is a semigroup with unit. We say that a semigroup S has zero if there exists 0 ∈ S such that 0s = s0 = 0 for every s ∈ S.Definition 1.10. Given a set X, we define the (symmetric) inverse semigroup on X as

Zappa-Szép products for partial actions of groupoids on Left Cancellative Small Categories

“…In the present paper, we extend the scope of [2] to actions of groupoids, To this end, we define groupoid actions on a left cancellative small category and their Zappa-Szép products, and we show that Zappa-Szép products appear naturally in the context of left cancellative small categories with length functions. Finally, we will extend the results of [15,Sections 7 & 8] to determine the essential properties of the tight groupoid associated to Zappa-Szép products of groupoid actions on a left cancellative small category, including the amenability of its tight groupoid.…”

“…In this section, we will define the Zappa-Szép product of a left cancellative small category categories. This is inspired in the construction of the Zappa-Szép product of a groupoid on a finite graph [9, Section 3] and the Zappa-Szép product of a group acting on a left cancellative small category [3,15]. This has also been recently done in [12] where they construct Zappa-Szép products of groupoids acting on higher-rank graphs.…”

“…Observe that in particular ϕ(r(α), α) ∈ G (0) for every α ∈ Λ. Now, we state the properties that a cocycle will enjoy (in a similar list as that of [15,Section 7]). But instead of impose them, we will try to deduce from the definition, as in [9, Section 3].…”

“…In [15], the authors studied Spielberg's construction, using a groupoid approach based in the Exel's tight groupoid construction [6], showing that the tight groupoid for these inverse semigroups coincide with Spielberg's groupoid [17]. With this tool at hand, they were able to characterize simplicity for the algebras associated to finitely aligned left cancellative small categories, and in particular in the case of Exel-Pardo systems [8].…”

“…In this section we collect all the basic background material about small categories we need for the rest of the paper. For more details see [15].…”