Topological graph inverse semigroups

Z. Mesyan; J. D. Mitchell; M. Morayne; Y. H. Péresse

doi:10.1016/j.topol.2016.05.012

Topological graph inverse semigroups

Z. Mesyan, J. D. Mitchell, M. Morayne, Y. H. Péresse

Research output: Contribution to journal › Article › peer-review

Abstract

To every directed graph E one can associate a graph inverse semigroup 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^⁎-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)∖{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 T₁ semigroup topology in which G(E)∖{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 language	English
Pages (from-to)	106-126
Number of pages	21
Journal	Topology and Its Applications
Volume	208
Early online date	24 May 2016
DOIs	https://doi.org/10.1016/j.topol.2016.05.012
Publication status	Published - 1 Aug 2016

Keywords

Graph inverse semigroup
Polycyclic monoid
Topological semigroup

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10.1016/j.topol.2016.05.012

Mitchell_2016_T&A_TopologicalGraph_AM
© 2016, Elsevier B.V. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://dx.doi.org/10.1016/j.topol.2016.05.012
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title = "Topological graph inverse semigroups",

abstract = "To every directed graph E one can associate a graph inverse semigroup 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⁎-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)∖{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 T1 semigroup topology in which G(E)∖{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.",

keywords = "Graph inverse semigroup, Polycyclic monoid, Topological semigroup",

author = "Z. Mesyan and Mitchell, {J. D.} and M. Morayne and P{\'e}resse, {Y. H.}",

note = "Micha{\l} Morayne was partially supported by NCN grant DEC-2011/01/B/ST1/01439 while this work was performed.",

year = "2016",

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doi = "10.1016/j.topol.2016.05.012",

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TY - JOUR

T1 - Topological graph inverse semigroups

AU - Mesyan, Z.

AU - Mitchell, J. D.

AU - Morayne, M.

AU - Péresse, Y. H.

N1 - Michał Morayne was partially supported by NCN grant DEC-2011/01/B/ST1/01439 while this work was performed.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - To every directed graph E one can associate a graph inverse semigroup 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⁎-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)∖{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 T1 semigroup topology in which G(E)∖{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.

AB - To every directed graph E one can associate a graph inverse semigroup 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⁎-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)∖{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 T1 semigroup topology in which G(E)∖{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.

KW - Graph inverse semigroup

KW - Polycyclic monoid

KW - Topological semigroup

UR - http://arxiv.org/abs/1306.5388

U2 - 10.1016/j.topol.2016.05.012

DO - 10.1016/j.topol.2016.05.012

M3 - Article

SN - 0166-8641

VL - 208

SP - 106

EP - 126

JO - Topology and Its Applications

JF - Topology and Its Applications

ER -

Topological graph inverse semigroups

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Keywords

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