Inverse semigroups of separated graphs and associated algebras

In this paper we introduce an inverse semigroup S (E, C) associated to a separated graph (E, C) and describe its internal structure. In particular we show that it is strongly E^*-unitary and can be realized as a partial semidirect product of the form Y for a certain partial action of the free group F=F (E¹) on the edges of E on a semilattice Y realizing the idempotents of S (E, C). In addition we also describe the spectrum as well as the tight spectrum of Y. We then use the inverse semigroup S (E, C) to describe several ''tame'' algebras associated to (E, C), including its Cohn algebra, its Leavitt-path algebra, and analogues in the realm of C^*-algebras, like the tame C^*-algebra O (E, C) and its Toeplitz extension T (E, C), proving that these algebras are canonically isomorphic to certain algebras attached to S (E, C). Our structural results on S (E, C) will then imply certain natural structure results for these algebras, like their description as partial crossed products.