The study of Leavitt path algebras is an outgrowth of work done by Bill Leavitt, long-time faculty member in mathematics at UNL, in the early 1960’s. Leavitt path algebras are F-algebras built essentially from a field F and paths in a directed graph. Reasonable necessary and sufficient graph-theoretic conditions for two directed graphs to have isomorphic Leavitt path algebras are not known. In this talk I will discuss a recent construction, due to Zhengpan Wang and myself, of a semigroup associated with a directed graph that we call the Leavitt inverse semigroup of the graph. Leavitt inverse semigroups are closely related to graph inverse semigroups and Leavitt path algebras. Leavitt inverse semigroups provide a certain amount of structural information about Leavitt path algebras. For example if two graphs have isomorphic Leavitt inverse semigroups then they have isomorphic Leavitt path algebras, but the converse is false. I will discuss some topological aspects of the structure of graph inverse semigroups and Leavitt inverse semigroups: in particular, I will provide necessary and sufficient conditions for two graphs to have isomorphic Leavitt inverse semigroups.

This is joint work with Zhengpan Wang, Southwest University, Chongqing, China.