The endomorphisms of a finite chain form a Rees congruence semigroup
type: Item
instance of: scholarly article
title: The endomorphisms of a finite chain form a Rees congruence semigroup
zbMATH Open document ID: 0931.20051
DOI: 10.1007/PL00006004
author: Timothy Lavers
author: Andrew Solomon
published in: Semigroup Forum
publication date: 1999-11-29T00:00:00Z
review text: Let \(O_n\) denote the semigroup of all order-preserving mappings on the finite chain \(1<2<\cdots<n\), under composition, that is to say the semigroup of all endomorphisms of the finite chain. It is proved that every non-identity congruence on \(O_n\) is a Rees congruence, meaning that the only non-singleton congruence class is an ideal. For this semigroup of transformations, the ideals themselves form a chain \(I_1\subseteq I_2\subseteq\cdots\subseteq I_n\), where \(I_k\) consists of all mappings of rank \(k\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 1330704
zbMATH Keywords: order-preserving mappings
zbMATH Keywords: finite chains
zbMATH Keywords: endomorphisms
zbMATH Keywords: Rees congruences
zbMATH Keywords: semigroups of transformations
zbMATH Keywords: ideals
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
full work available at URL: pl00006004
OpenAlex ID: W2069296875
description: scientific article; zbMATH DE number 1330704