The study of Leavitt path algebras developed from foundational work of W.G. Leavitt in the early 1960’s on the invariant basis problem for free modules over non-commutative rings, and also from work in the late 1990’s on Cuntz-Krieger graph $C^{\ast}$-algebras. In this talk I will discuss the construction of an inverse semigroup (referred to as a “Leavitt inverse semigroup”) associated with a directed graph. Leavitt inverse semigroups are homomorphic images of graph inverse semigroups: they provide some (limited) structural information and some information about the isomorphism problem for Leavitt path algebras.

This is joint work with David Milan and Zhengpan Wang.