Toward a description of monoids over which all polygons have \(\omega\)- stable theories (Q1191177)

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 admissible regular (and, in particular, finite) monoids (Proposition 3.1 and its corollary).