The Leavitt inverse semigroup of a separated graph

We introduce and study a new inverse semigroup associated to a separated graph [Formula: see text], which we call the Leavitt inverse semigroup. This semigroup is obtained as a quotient of the separated graph inverse semigroup [Formula: see text], introduced in our previous paper [P. Ara, A. Buss and A. Dalla Costa, Inverse semigroups of separated graphs and associated algebras, Bull. Braz. Math. Soc. (N.S.) 56(3) (2025) 55, doi:10.1007/s00574-025-00462-7], and it provides a canonical inverse semigroup model for the tame Leavitt path algebra [Formula: see text] over a commutative unital ring [Formula: see text]. Our first main result describes the Leavitt inverse semigroup [Formula: see text] as a restricted semidirect product of the free group on the edges of [Formula: see text] acting partially on a certain semilattice, which is isomorphic to the semilattice of idempotents of [Formula: see text]. This description, given in terms of Leavitt–Munn trees, yields a normal form for the elements of [Formula: see text]. We obtain a normal form for elements of [Formula: see text], leading to explicit linear bases for [Formula: see text]. Building on this and on the structural properties of [Formula: see text], we prove that the natural homomorphism from [Formula: see text] to [Formula: see text] is injective, so that [Formula: see text] embeds as the inverse semigroup generated by the canonical partial isometries in [Formula: see text]. Further applications include the determination of natural bases of the kernel [Formula: see text] of the natural map from the tame Cohn algebra [Formula: see text] to the tame Leavitt path algebra [Formula: see text], 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.