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 SΛ 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 SΛ. We obtain geometric descriptions of the Green's relations on SΛ, 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 SΛ and, based on this, identify a certain X-to-join Booleanization of SΛ, where the abstract and concrete Cuntz-Krieger relations agree. Besides the natural partial order, SΛ is also ordered by set-inclusion. This helps us to realize SΛ as a semigroup \emph{graded-ordered} by ℤk and, subsequently, gives rise to a full inverse subsemigroup SΛh of SΛ, which is graded by ℤk 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 SΛh by the tight ideal. In the final part of the paper, we show that a k-graph morphism p:Λ1⟶ Λ2 that is injective and satisfies a certain path lifting property induces, in a natural way, an injective $0$-preserving semigroup homomorphism from SΛ1 to SΛ2. 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 SΛh is natural.
Related
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BibTeX
@misc{mukherjee2026remarks,
title = {{Remarks on the Farthing--Muhly--Yeend inverse semigroup of a finitely aligned higher-rank graph}},
author = {Promit Mukherjee},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04398},
eprint = {2611.04398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04398},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Remarks on the Farthing--Muhly--Yeend inverse semigroup of a finitely aligned higher-rank graph
AU - Promit Mukherjee
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04398
UR - https://arxiv.org/abs/2611.04398
AB - 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 ℤ^k and, subsequently, gives rise to a full inverse subsemigroup S_Λ^h of S_Λ, which is graded by 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⟶ 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.
ER -