Topological Graph Inverse Semigroups

Zak Mesyan, J. D. Mitchell, Michal Morayne, Y. H. Péresse

    Research output: Contribution to journalArticlepeer-review

    12 Citations (Scopus)
    205 Downloads (Pure)


    To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$. On the other hand, $G(E)$ need not be discrete in a Hausdorff semigroup topology, and for certain graphs $E$, $G(E)$ admits a $T_1$ semigroup topology in which $G(E)\setminus \{0\}$ is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of $G(E)$ in larger topological semigroups.
    Original languageEnglish
    Pages (from-to)106-126
    Number of pages21
    JournalTopology and its Applications
    Early online date24 May 2016
    Publication statusPublished - 1 Aug 2016


    • Topological Algebra
    • Abstract Algebra


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