On monoids of monotone injective partial selfmaps of integers with cofinite domains and images
type: Item
instance of: scholarly article
title: On monoids of monotone injective partial selfmaps of integers with cofinite domains and images
zbMATH Open document ID: 1256.22001
published in: Georgian Mathematical Journal
publication date: 2012-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.0587
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 54H15
zbMATH DE Number: 6092469
zbMATH Keywords: topological semigroup
zbMATH Keywords: semitopological semigroup
zbMATH Keywords: semigroup of bijective partial transformations
zbMATH Keywords: topological inverse semigroup
zbMATH Keywords: Baire space
zbMATH Keywords: bicyclic semigroup
zbMATH Keywords: embedding of of topological semigroup
Wikidata QID: Q114053032
author: O. Gutik
author: Dušan Repovš
MaRDI profile type: Publication
OpenAlex ID: W1994095589
arXiv ID: 1209.0587
DOI: 10.1515/GMJ-2012-0022
review text: To state the main results of this paper, first we need to recall some necessary definitions. Then we state the main results. For a topological space \(X\), a family \(\{A_s|s\in\mathcal{A}\}\) of subsets of \(X\) is called \textit{locally finite} if for every point \(x\in X\), there exists an open neighborhood \(U\) of \(x\) in \(X\) such that the set \(\{s\in \mathcal{A}|U\cap A_s\neq \emptyset\}\) is finite. A subset \(A\) of \(X\) is said to be \textit{codense} in \(X\) if \(X\setminus A\) is dense in \(X\). A space \(X\) is said to be a \textit{Baire space} if for each sequence \(A_1,\cdots,A_i,\cdots\) of nowhere dense subsets of \(X\), the union \(\bigcup_{i= 1}^\infty A_i\) is a codense subset of \(X\). A semigroup \(S\) is called \textit{bisimple} if \(S\) has only one \(\mathcal{D}\)-class, where \(\mathcal{D}=\mathcal{L}\circ \mathcal{R}\) (\(a\mathcal{R} b\) (\(a\mathcal{L} b\)) if and only if \(aS^1=bS^1\) (\(S^1a=S^1b\))). A \textit{semitopological} (resp. \textit{topological}) \textit{semigroup} is a Hausdorff topological space together with a separately (resp. jointly) continuous semigroup operation. An inverse topological semigroup with continuous inversion is called \textit{topological inverse semigroup}. A Hausdorff topology \(\tau\) on an (inverse) semigroup \(S\) such that \((S,\tau)\) is a topological inverse semigroup is called (\textit{inverse}) \textit{semigroup topology}. Let \(\mathbb{Z}\) be the set of integers with the usual order \(\leq\). A partial map \(\alpha:Z\to Z\) is said to be \textit{monotone} if \(n\leq m\) implies that \((n)\alpha\leq (m)\alpha\) for \(n,m\in\mathbb{Z}\). By \(\mathcal{F}^\nearrow_\infty(\mathbb{Z})\) we denote a subsemigroup of injective partial monotone selfmaps such that \(|\mathbb{Z}\setminus \mathrm{dom} \alpha|\) and \(|\mathbb{Z}\setminus \mathrm{ran} \alpha|\) are finite.NEWLINENEWLINEIn this paper, the authors study the semigroup \(\mathcal{F}^\nearrow_\infty(\mathbb{Z})\) and they show that \(\mathca…
reviewed by: Behnam Khosravi
description: scientific article; zbMATH DE number 6092469