On a class of lattice ordered inverse semigroups
type: Item
instance of: scholarly article
title: On a class of lattice ordered inverse semigroups
zbMATH Open document ID: 0966.06021
author: Gracinda M. S. Gomes
author: Emilia Giraldes
author: Donald B. McAlister
published in: Journal of Algebra
publication date: 2001-07-26T00:00:00Z
review text: It was shown by \textit{T. Saito} [Proc. Japan Acad. 50, 837-838 (1974; Zbl 0338.06004)] that the free inverse semigroup \(S\) on a nonempty set cannot be totally ordered (i.e., there is no total order \(\leq\) on \(S\) such that \(a\leq b\) implies \(ac\leq bc\) and \(ca\leq cb\) for every \(c\in S\)). In the paper under review it is proved that every free inverse monoid can be lattice-ordered (i.e., there exists a lattice order such that multiplication distributes over join and meet from the left and the right). McAlister's representation theorem for \(E\)-unitary inverse semigroups is used to construct all lattice-ordered \(E\)-unitary inverse semigroups which are lexicographically ordered (i.e., there exists an order-preserving homomorphism \(\theta\) onto a partially ordered group such that \(a\theta< b\theta\) imlies \(a< b\)). The particular case where the lattice order \(\leq\) on \(S\) is natural (in the sense that \(e= ef= fe\) and \(e,f\in E(S)\) implies \(e\leq f\)) is also considered. Furthermore, the compatible lattice orderings on the free monogenic inverse monoid are specified: there are, up to conjugacy, precisely two such orderings. Also, it is shown that an inverse semigroup is an idempotent separating homomorphic image of a naturally lattice ordered inverse semigroup if and only if its idempotents form a distributive lattice under the usual ordering. Finally, it is proved that a lattice-ordered \(E\)-unitary inverse semigroup which is lexicographically ordered, is a distributive lattice under the imposed partial order if and only if its set of idempotents forms a distributive lattice.
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 20M18
zbMATH DE Number: 1512977
zbMATH Keywords: free inverse semigroup
zbMATH Keywords: free inverse monoid
zbMATH Keywords: lattice order
zbMATH Keywords: \(E\)-unitary inverse semigroups
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: Q5789040
cites work: Q4317872
cites work: Q5509706
cites work: Q5608798
cites work: Q4003180
cites work: Q4846425
cites work: Q4125028
cites work: Amenably ordered inverse semigroups
cites work: Compatible orders on the bicyclic semigroup
cites work: A Class of Irreducible matrix representations of an Arbitrary Inverse Semigroup
cites work: FUNDAMENTAL INVERSE SEMIGROUPS
cites work: Q3337662
cites work: Proper ordered inverse semigroups
cites work: Ordered Inverse Semigroups
cites work: No free inverse semigroup is orderable
cites work: Free Inverse Semigroups
description: scientific article; zbMATH DE number 1512977