Chain decomposition of ordered semigroups
type: Item
instance of: scholarly article
title: Chain decomposition of ordered semigroups
zbMATH Open document ID: 1007.06011
published in: Semigroup Forum
publication date: 2003-03-26T00:00:00Z
review text: A partially ordered (p.o.) semigroup \((S,\cdot ,\leq)\) is called a natural ordered semilattice, in particular, chain \(Y\) of subsemigroups \(S_\alpha \), \((\alpha\in Y)\) if \(a\leq b\), \(a\in S_\alpha\), \(b\in S_\beta\) imply that \(\alpha\leq \beta\) in \(Y\). If \(\rho\) is any binary relation on \(S\) then \(S\) is said to be \(\rho\)-simple if \(\rho\) is the universal relation. Using several particular relations \(\rho\) defined earlier by the author, conditions for \((S, \cdot ,\leq)\) are given to be a natural ordered chain of \(\rho\)-simple subsemigroups. The definitions of the \(\rho\)'s considered involve left, right, two-sided, or bi-ideals, where ``ideal'' is understood as semigroup and order-ideal. In particular, chain decompositions into \(t\)-simple subsemigroups \(S_\alpha\) are studied (as a generalization of semilattices of groups in the non-ordered case): \((S_\alpha,\cdot, \leq)\) is \(t\)-simple if \(S_\alpha\) is \({\mathcal L}\)- and \({\mathcal R}\)-simple, where \(a{\mathcal L}b\) if and only if the ``left-ideals'' of \(S_\alpha\) generated by \(a\) and \(b\), respectively, are equal \(({\mathcal R}\) is defined dually). Also, the case of 1-\((t\)-)archimedian components is considered: \((S_\alpha, \cdot,\leq)\) is 1-\((t\)-)archimedian if for all \(a,b\in S_\alpha\), \(b^m\leq xa\) \((b^m\leq axa)\) for some \(m\in\mathbb{N}\), \(x\in S\). Up to 15 equivalent conditions for a p.o. semigroup to be a natural ordered chain of subsemigroups of one of the above \(\rho\)-simple kinds are given.
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 1784750
zbMATH Keywords: partially ordered semigroup
zbMATH Keywords: natural ordered chain of \(\rho\)-simple subsemigroups
author: Yonglin Cao
reviewed by: H. Mitsch
MaRDI profile type: Publication
full work available at URL: s002330010120
OpenAlex ID: W2003560389
DOI: 10.1007/S002330010120
description: scientific article; zbMATH DE number 1784750