Model-theoretic properties of free, projective, and flat S-acts

Abstract

This is the second in a series of articles surveying the body of work on the model theory of S-acts over a monoid S. The first concentrated on the theory of regular S-acts. Here we review the material on model-theoretic properties of free, projective and (strongly, weakly) flat S-acts. We consider questions of axiomatisability, completeness, model completeness and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids S such that the class of free left S-acts is axiomatisable, is new.

Axiomatisability problems for S-posets

“…We can conclude that every ultrapower of S as a left S-act is projective. From [12,Theorem 8.6], S is left perfect, so from [13, Theorem 6.3], S is left poperfect. Hence SF = Pr.…”

“…(2) ⇒ (3) If Pr is axiomatisable, then every ultrapower of copies of S is projective as a left S-poset, and hence as a left S-act. From [12,Lemma 8.4], it follows that for any e ∈ E(S) and u ∈ S, there are only finitely many x ∈ S such that e = ux. This permits us to define the sentences ϕ e as in [12].…”

“…From [12,Lemma 8.4], it follows that for any e ∈ E(S) and u ∈ S, there are only finitely many x ∈ S such that e = ux. This permits us to define the sentences ϕ e as in [12]. Let Pr be the set of sentences axiomatising the projective left S-posets.…”

“…Let Pr be the set of sentences axiomatising the projective left S-posets. Then, as in [12,Theorem 9.1], Pr ∪ ϕ e : e ∈ E(S) \ {1} . axiomatises F r.…”

“…Corresponding questions for classes of M-acts over a monoid M have been answered in [10,19,1] and [11], see also the survey article [12]. The classes of projective (strongly flat, po-flat, weakly po-flat) left S-posets Pr(SF , PF , WPF ) have recently been considered in [15] (which uses slightly different terminology; the results also appearing in [16]) as has the class F r of free left S-posets in the case where S has only finitely many right ideals.…”