The Leavitt inverse semigroup of a separated graph

We introduce and study a new inverse semigroup associated to a separated graph (E, C), which we call the Leavitt inverse semigroup. This semigroup is obtained as a quotient of the separated graph inverse semigroup S (E, C), introduced in our previous paper 9, and it provides a canonical inverse semigroup model for the tame Leavitt path algebra LKᵃb (E, C) over a commutative unital ring K. Our first main result describes the Leavitt inverse semigroup 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 LI (E, C). This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of LI (E, C). We obtain a normal form for elements of LKᵃb (E, C), leading to explicit linear bases for LKᵃb (E, C). Building on this and on the structural properties of LI (E, C), we prove that the natural homomorphism from LI (E, C) to LKᵃb (E, C) is injective, so that LI (E, C) embeds as the inverse semigroup generated by the canonical partial isometries in LKᵃb (E, C). Further applications include the determination of natural bases of the kernel Q of the natural map from the tame Cohn algebra CKᵃb (E, C) to the tame Leavitt path algebtra LKᵃb (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.