Topological graph inverse semigroups

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• 作者：Z. Mesyan^a ; ^{zmesyan@uccs.edu" class="auth_mail" title="E-mail the corresponding author} ; J.D. Mitchell^b ; ^{jdm3@st-and.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; M. Morayne^c ; ^{michal.morayne@pwr.wroc.pl" class="auth_mail" title="E-mail the corresponding author} ; Y.H. Pé ; resse^d ; ^{y.peresse@herts.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 20M18 ; 22A15 ; secondary ; 54H99 ; 05C20
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：208
• 期：Complete
• 页码：106-126
• 全文大小：507 K

文摘

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}$G(E)\setminus \{0\}$ 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 T₁$T_1$ semigroup topology in which G(E)∖{0}$G(E)\setminus \{0\}$ is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of G(E)$G(E)$ in larger topological semigroups.