Semigroups of order-decreasing transformations: The isomorphism theorem
type: Item
instance of: scholarly article
title: Semigroups of order-decreasing transformations: The isomorphism theorem
zbMATH Open document ID: 0858.20056
DOI: 10.1007/BF02574137
published in: Semigroup Forum
publication date: 1997-03-20T00:00:00Z
full work available at URL: https://eudml.org/doc/135486
review text: Let \(X\) be a totally ordered set. Let \(T(X)\) denote the full transformation semigroup on \(X\) and let \(S^-(X)=\{\alpha\in T(X):x\alpha\leq x\) for all \(x\in X\}\) and \(S^+(X)=\{\alpha\in T(X):x\alpha\geq x\) for all \(x\in X\}\). Then \(S^-(X)\) is a semigroup which consists of all order decreasing selfmaps of \(X\) and, similarly, \(S^+(X)\) is a semigroup which consists of all order increasing selfmaps of \(X\). In the main result of the paper, the author proves that the semigroups \(S^-(X)\) and \(S^-(Y)\) are isomorphic if and only if the totally ordered sets \(X\) and \(Y\) are order isomorphic. An immediate corollary is the fact that \(S^-(X)\) and \(S^+(Y)\) are isomorphic if and only if \(X\) and \(Y\) are order anti-isomorphic.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 936970
zbMATH Keywords: full transformation semigroups
zbMATH Keywords: order decreasing selfmaps
zbMATH Keywords: order increasing selfmaps
zbMATH Keywords: totally ordered sets
author: A. Umar
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
cites work: Q5721233
cites work: Q3848243
cites work: Abundant Semigroups
cites work: Q4126563
cites work: Q4059104
cites work: On the semigroups of order-decreasing finite full transformations
cites work: On certain infinite semigroups of order-decreasing transformations i
description: scientific article; zbMATH DE number 936970