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Alexandroff topologies and monoid actions

Alexandroff topologies and monoid actions AbstractGiven a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X.Conversely, we prove that any Alexandroff topology may be obtained through a monoid action.Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions.Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets.Then, in relation to such a dependence, we study the family 𝒜σ,X{\mathcal{A}_{\sigma,X}} of the closed subsets Y of X such that, for any y1,y2∈Y{y_{1},y_{2}\in Y}, there exists a third element y∈Y{y\in Y} whose closure contains both y1{y_{1}} and y2{y_{2}}.More in detail, relying on some specific properties of the maximal members of the family 𝒜σ,X{\mathcal{A}_{\sigma,X}}, we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence.Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Alexandroff topologies and monoid actions

Forum Mathematicum , Volume 32 (3): 32 – May 1, 2020

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References (32)

Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0283
Publisher site
See Article on Publisher Site

Abstract

AbstractGiven a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X.Conversely, we prove that any Alexandroff topology may be obtained through a monoid action.Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions.Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets.Then, in relation to such a dependence, we study the family 𝒜σ,X{\mathcal{A}_{\sigma,X}} of the closed subsets Y of X such that, for any y1,y2∈Y{y_{1},y_{2}\in Y}, there exists a third element y∈Y{y\in Y} whose closure contains both y1{y_{1}} and y2{y_{2}}.More in detail, relying on some specific properties of the maximal members of the family 𝒜σ,X{\mathcal{A}_{\sigma,X}}, we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence.Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2020

Keywords: Alexandroff spaces; closure operators; monoids; monoid actions; 54A05; 54A25; 20M30; 20M15

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