Remarks on the Farthing--Muhly--Yeend inverse semigroup of a finitely aligned higher-rank graph

Abstract

Given a finitely aligned higher-rank $k$-graph $Λ$, Farthing, Muhly and Yeend (Semigroup Forum 2005) constructed an inverse semigroup $S_Λ$ to provide a groupoid model for the $C^*$-algebra of $Λ$. In this paper, we extend some of the existing results related to the semigroup structure of a graph inverse semigroup to the more general setting of the inverse semigroup $S_Λ$. We obtain geometric descriptions of the Green's relations on $S_Λ$, which effectively generalize the existing characterization of Green's relations on a graph inverse semigroup. We provide a simple and useful description of the Cuntz-Krieger ideal of the Booleanization of $S_Λ$ and, based on this, identify a certain $X$-to-join Booleanization of $S_Λ$, where the abstract and concrete Cuntz-Krieger relations agree. Besides the natural partial order, $S_Λ$ is also ordered by set-inclusion. This helps us to realize $S_Λ$ as a semigroup \emph{graded-ordered} by $\mathbb{Z}^k$ and, subsequently, gives rise to a full inverse subsemigroup $S_Λ^h$ of $S_Λ$, which is graded by $\mathbb{Z}^k$ in the usual sense. We show that the Kumjian-Pask algebra of $Λ$ is graded isomorphic to the quotient of the contracted semigroup algebra of $S_Λ^h$ by the tight ideal. In the final part of the paper, we show that a $k$-graph morphism $p:Λ_1\longrightarrow Λ_2$ that is injective and satisfies a certain path lifting property induces, in a natural way, an injective $0$-preserving semigroup homomorphism from $S_{Λ_1}$ to $S_{Λ_2}$. In view of this, we build a suitable categorical setup to conclude that the isomorphism established between the Kumjian-Pask algebra of $Λ$ and the tight ideal quotient of the semigroup algebra of $S_Λ^h$ is natural.