2024/03/08 by Pere Ara, Ara, Pere, Alcides Buss +3 · 2 citations
Computer Science · Decision Sciences · #Advanced Algebra and Logic #semigroups and automata theory #Fuzzy and Soft Set Theory
paper · pdf · doi:10.48550/arxiv.2403.05295
In this paper we introduce an inverse semigroup S(E,C) associated to a separated graph (E,C) and describe its internal structure. In particular we show that it is strongly E^*-unitary and can be realized as a partial semidirect product of the form Y\rtimes\mathbbF for a certain partial action of the free group \mathbbF=\mathbbF(E1) on the edges of E on a semilattice Y realizing the idempotents of S(E,C). In addition we also describe the spectrum as well as the tight spectrum of Y. We then use the inverse semigroup S(E,C) to describe several "tame" algebras associated to (E,C), including its Cohn algebra, its Leavitt-path algebra, and analogues in the realm of C^*-algebras, like the tame C^*-algebra O(E,C) and its Toeplitz extension T(E,C), proving that these algebras are canonically isomorphic to certain algebras attached to S(E,C). Our structural results on S(E,C) imply that these algebras can be realized as partial crossed products, revealing a great portion of their structure.