Topological graph inverse semigroups

To every directed graph E one can associate a graph inverse semigroup  G(E)G(E), where elements roughly correspond to possible paths in E  . These semigroups generalize polycyclic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, graph C⁎C⁎-algebras, and Toeplitz C⁎C⁎-algebras. We investigate topologies that turn G(E)G(E) into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, G(E)∖{0}G(E)∖{0} must be discrete for any directed graph E  . On the other hand, G(E)G(E) need not be discrete in a Hausdorff semigroup topology, and for certain graphs E  , G(E)G(E) admits a T1T1 semigroup topology in which G(E)∖{0}G(E)∖{0} is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E)G(E) in larger topological semigroups.