The Leavitt inverse semigroup of a separated graph

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 $\mathcal{L}\mathcal{I}(E,C)$ 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 $\mathcal{L}\mathcal{I}(E,C)$. This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of $\mathcal{L}\mathcal{I}(E,C)$. 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 $\mathcal{L}\mathcal{I}(E,C)$, we prove that the natural homomorphism from $\mathcal{L}\mathcal{I}(E,C)$ to ${\mathcal{L}}_K^{\mathrm{ab}}(E,C)$ is injective, so that $\mathcal{L}\mathcal{I}(E,C)$ 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.

40 pages