Axiomatizability and completeness of some classes of partially ordered polygons

Abstract

We study partially ordered monoids over which a class of free (over sets and over posets), projective, and (strongly, weakly) flat partially ordered polygons is axiomatizable, complete, or model complete. Similar issues for polygons were dealt with in papers by V. , classes of free, projective, and (strongly, weakly) flat polygons were explored in relation to their axiomatizability, completeness, and model completeness. In the present paper, we look into the same issues, but now for classes of partially ordered polygons. Here, the concept of a left perfect monoid is carried over the case of a partially ordered monoid, and certain conditions equivalent to this concept are formulated. We furnish a description of partially ordered monoids with an axiomatizable class of partially ordered flat, weakly flat, and strongly flat polygons. This result is similar to the appropriate result for polygons (see [2,3]). As in the unordered case, for a commutative partially ordered monoid, it is proved that completeness of a class of strongly flat partially ordered polygons over the monoid is equivalent to the class being model complete, categorical, and to the monoid being a partially ordered Abelian group. It is known that a partially ordered polygon is strongly flat iff it satisfies conditions (P < ) and (E < ) (see [4]).Here we study partially ordered monoids with axiomatizable, complete, model complete, and categorical classes of partially ordered polygons satisfying (P < ) or (E < ). The results obtained for axiomatizable, complete, model complete, and categorical classes of partially ordered projective polygons free over a set are similar to corresponding results for polygons (see [1,2]). Again, we describe partially ordered monoids with finitely many different right ideals over which a class of partially ordered polygons free over a poset is axiomatizable, and remark that there does not exist a partially ordered monoid for which this class is axiomatizable and complete.The results presented in Sec. 3 are due to Stepanova; Theorem 2.6 was proved jointly by the two authors; other results are due to Pervukhin.

Axiomatisability problems for S-posets

“…is either empty or is an S-subposet of the right S-poset S × S, and r ≤ (s, t) is either empty or is a right ideal of S. Note that in [15], R ≤ (s, t) and r ≤ (s, t) are written as R < (s, t) and r < (s, t).…”

“…Axiomatisability of Pr. The question of the axiomatisability of Pr was addressed in [15]. Without giving much detail, Pervukhin and Stepanova indicate that if every ultrapower of a pomonoid S is projective as a left S-poset, then it can be argued, following the corresponding proofs for S-acts, that S is poperfect, which here can be taken to mean SF = Pr in the class of left S-posets.…”

“…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. We note that many of the techniques of [15] follow those in the M-act case and, for this reason, we aim here to produce two general strategies that will deal with a number of axiomatisability questions for classes of S-posets (and, with minor adjustment, M-acts).…”

“…Just as many concepts of flatness that are equivalent for R-modules over a unital ring R are different for M-acts, so many concepts that coincide for M-acts split for S-posets. Thus [15] left a number of classes open; we address many of them here, with both our general techniques and ad hoc methods.…”

“…There are two kinds of results, both phrased in terms of 'replacement tossings'; we show how they may be applied to reproduce the results of [15] determining for which pomonoids PF or PWF are axiomatisable, together with a number of other applications. In Section 5 we then consider classes defined by flatness conditions that translate into so called 'interpolation conditions'.…”