Principal topologies and transformation semigroups
type: Item
instance of: scholarly article
title: Principal topologies and transformation semigroups
zbMATH Open document ID: 1166.54017
published in: Topology and its Applications
publication date: 2008-09-23T00:00:00Z
review text: We quote the author's abstract: ``For a given set \(X\), the set \(F(X)\) of all maps from \(X\) to \(X\) forms a semigroup under composition. A subgroup \(S\) of \(F(X)\) is said to be saturated if for each \(x \in X\) there exists a set \(Q_X \subseteq X\) such that \(S= \{f \in F(X) |f(x)\in Q_X \forall x \in X\}\). It is shown that there exists a one-to-one correspondence between principal topologies on \(X\) and saturated subsemigroups of \(F(X)\). Some properties of principal topologies on \(X\) and the corresponding properties of their associated saturated subsemigroups of \(F(X)\) are discussed.'' The author undertakes to establish a one-to-one correspondence between principal topologies on a set \(X\) and appropriate transformation semigroups on \(X\), for which subspaces can be identified with subsemigroups in a natural way. Examples are discussed how topological properties of \(X\) translate to corresponding properties of the associated transformation semigroup on \(X\). Amongst the important results presented are: \begin{itemize} \item[1.] Characterization of a class to be a basis for some principal topology. \item[2.] If a semigroup \(S\) acts on a non empty set \(X\), then the collection of orbits is a basis consisting of all minimal open neighborhoods for some principal topology on \(X\). \item[3.] The one-to-one correspondence between the principal topologies on \(X\) and the saturated transformation semigroups on \(X\). \end{itemize} Examples are supplied to illustrate the bijective correspondence of the above theorem for principal topologies which happen to be infinite. A smallest saturated transformation semigroup corresponds to the principal topology on \(\mathbb R\) with a minimal open neighborhood basis. An example revealing a saturated transformation semigroup on \(\mathbb N\) and the corresponding topology having the minimal neighborhood of an element in \(\mathbb N\). \begin{itemize} \item[4.] The author gives a characterization of a disconn…
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 5346263
zbMATH Keywords: principal topology
zbMATH Keywords: Greens's left quasiorder
zbMATH Keywords: transformation semigroup
MaRDI profile type: Publication
full work available at URL: j.topol.2008.04.007
OpenAlex ID: W2041302060
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DOI: 10.1016/J.TOPOL.2008.04.007
author: M. Bettina Zoeller
description: scientific article; zbMATH DE number 5346263