The Leavitt inverse semigroup of a separated graph

TL;DR

This paper introduces the Leavitt inverse semigroup associated with separated graphs, providing a structural description, normal forms, and embeddings into Leavitt path algebras, with applications to bases, kernels, and spectra.

Contribution

It defines and analyzes the Leavitt inverse semigroup for separated graphs, offering a normal form, embedding results, and structural insights into associated algebras.

Findings

Normal form for elements of the Leavitt inverse semigroup

Explicit linear bases for the tame Leavitt path algebra

Embedding of the inverse semigroup into the algebra

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.