The Leavitt inverse semigroup of a separated graph
Pere Ara, Alcides Buss, Ado Dalla Costa
TL;DR
The paper develops a comprehensive inverse-semigroup framework for tame Leavitt path algebras of separated graphs by introducing the Leavitt inverse semigroup $\mathcal{LI}(E,C)$ as a natural quotient of the separated graph inverse semigroup. It provides a normal form via Leavitt–Munn trees, proves an embedding of $\mathcal{LI}(E,C)$ into the abelianized Leavitt path algebra, and establishes a canonical basis for $\mathcal{L}_K^{\mathrm{ab}}(E,C)$. It further connects Cohn and Leavitt theories through corner isomorphisms, computes a linear basis for the kernel $\mathcal{Q}$, analyzes the spectrum and tight groupoid (with a Leavitt model), and determines the socle of tame Leavitt algebras. The framework yields new structural insights, including models for the tight groupoid and explicit descriptions of socle blocks, illustrated via key examples such as the Cuntz separated graph and free inverse monoid algebras.
Abstract
We introduce and study a new inverse semigroup associated to a separated graph $(E,C)$, which we call the \emph{Leavitt inverse semigroup}. This semigroup is obtained as a quotient of the separated graph inverse semigroup $\mathcal{S}(E,C)$, introduced in our previous paper [9], and it provides a canonical inverse semigroup model for the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E,C)$ over a commutative unital ring $K$. Our first main result describes the Leavitt inverse semigroup $\mathcal{LI}(E,C)$ as a restricted semidirect product of the free group on the edges of $E$ acting partially on a certain semilattice, which is isomorphic to the semilattice of idempotents of $\mathcal{LI}(E,C)$. This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of $\mathcal{LI}(E,C)$. We obtain a normal form for elements of $\mathcal{L}_K^\mathrm{ab}(E,C)$, leading to explicit linear bases for $\mathcal{L}_K^\mathrm{ab}(E,C)$. Building on this and on the structural properties of $\mathcal{LI} (E,C)$, we prove that the natural homomorphism from $\mathcal{LI}(E,C)$ to $\mathcal{L}_K^\mathrm{ab}(E,C)$ is injective, so that $\mathcal{LI}(E,C)$ embeds as the inverse semigroup generated by the canonical partial isometries in $\mathcal{L}_K^\mathrm{ab}(E,C)$. Further applications include the determination of natural bases of the kernel $\mathcal Q$ of the natural map from the tame Cohn algebra $\mathcal{C}_K^\mathrm{ab} (E,C)$ to the tame Leavitt path algebtra $\mathcal{L}_K^\mathrm{ab} (E,C)$, the computation of the socle, and a characterization of the isolated points of the spectrum. Several examples, such as the Cuntz separated graph and free separations, are discussed to illustrate the theory.