Toward a description of monoids over which all polygons have \(\omega\)- stable theories
type: Item
instance of: scholarly article
title: Toward a description of monoids over which all polygons have \(\omega\)- stable theories
zbMATH Open document ID: 0820.20069
DOI: 10.1007/BF01978556
author: T. G. Mustafin
published in: Algebra and Logic
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/187545
review text: We recall some definitions and notation: A monoid is said to be a stabilizer (superstabilizer, \(\omega\)-stabilizer) if each left polygon over \(S\) has a stable (superstable, \(\omega\)-stable) theory. If \(\alpha, \beta \in S\), then \(\alpha \trianglelefteq \beta:=S\alpha \supseteq S\beta\), \(\alpha \sim \beta:=S_ \alpha = S_ \beta\). We denote \(| S/\sim|\) by \(I_ S\). A monoid \(S\) is said to be an LO-monoid (WO- monoid) if \(S/\sim\) is linearly ordered (well ordered) relative to the relation \(\trianglelefteq\). The following propositions were proved earlier: Proposition 1. A monoid \(S\) is a stabilizer (superstabilizer) if and only if \(S\) is an LO-monoid (WO-monoid). Proposition 2. If \(S\) is a countable group (i.e., \(I_ S = 1\)), then \(S\) is an \(\omega\)-stabilizer if and only if it has no more than a countable number of subgroups. In this paper we continue our investigation of \(\omega\)-stabilizers. Since the problem of \(\omega\)-stabilizers is solved by Proposition 2 for groups, we will consider monoids that are not groups. A countable monoid \(S\) is said to be admissible if it can be represented in the form \(G \cup J\), where \(J\) is a unique proper left ideal \(G = S\setminus J\), and \(G\) is the largest subgroup of \(S\) that contains the identity \(e \in S\). We call the group \(G\) the group part of \(S\). The fundamental results of the present paper are the following: 1. If \(S\) is an \(\omega\)-stabilizer that is not a group, then a) \(S\) is an admissible monoid (Lemmas 1.2 and 2.1); b) the group part of \(S\) has no more than a countable number of subgroups (Lemma 7.1); c) if the group part of \(S\) is finite, then \(S\) is finite (and regular) (\S4.5); d) if the group part of \(S\) is infinite it satisfies condition (*) (defined in \S7, Lemma 7.2). 2. A finite monoid \(S\) is an \(\omega\)-stabilizer if and only if \(S\) is a group or an admissible monoid (Lemma 6.5). 3. A complete algebraic description of the structure of adm…
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 59391
zbMATH Keywords: stable theory
zbMATH Keywords: superstable theory
zbMATH Keywords: \(\omega\)-stable theory
zbMATH Keywords: WO-monoids
zbMATH Keywords: regular monoids
zbMATH Keywords: left polygon
zbMATH Keywords: LO-monoids
zbMATH Keywords: \(\omega\)-stabilizers
zbMATH Keywords: monoids
zbMATH Keywords: groups
zbMATH Keywords: admissible monoids
zbMATH Keywords: finite monoids
MaRDI profile type: Publication
cites work: Classification theory and the number of non-isomorphic models
description: scientific article; zbMATH DE number 59391