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.
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.
We study into monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, the monoids S the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid, and that it is primitive connected iff S is a group. It is pointed out that there exists no monoid S with an additive class of all S-polygons.Additive and primitive connected theories were studied in [1,2]. These theories are a generalization of module theory. Like the latter, the theories in question admit quantifier elimination down to primitive formulas (see [1,2]). The class of additive theories is contained in the class of primitive connected ones. As distinct from the primitive connected theory, the additive theory has the property that on factors of any primitive copies w.r.t. some primitive equivalence, isomorphic Abelian groups can be defined using a primitive formula. This property of additive theories generalizes a known property of modules stating the following: in any module, primitive copies are conjugacy classes of some Abelian group. The additive and primitive connected theories are, by definition, primitive normal theories.In this paper, we look at monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, those monoids the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid (Thm. 1); there exists no monoid S with an additive class of all S-polygons (Thm. 2); the class of all S-polygons is primitive connected iff S is a group. PRELIMINARIESModel-theoretic information given under this section is borrowed from [1,2]. Let T be a complete theory of a language L. Fix some sufficiently large and sufficiently saturated model C of T , which we call a monster model, since it is assumed that all the models of T under consideration are elementary submodels of C. All elements, tuples of elements, and sets are taken from the monster model C. A tuple a 1 , . . . , a n of elements and a tuple x 1 , . . . , x n of variables are denoted byā andx, respectively. Instead ofā ∈ C n , we writē a ∈ C. Lets,t be tuples of elements or variables. We introduce the following notation: l(s) is the length of s; s is the set consisting of elements ofs;sˆt is a tuple obtained by ascribing the tuplet to the right of s. We write s ∈s in place of s ∈ s, and writes ∪t in place of s ∪ t . If Φ(x,ȳ) is a formula in L, A is a model of T ,ā is a tuple of elements of A, and l(ā) = l(ȳ), then Φ(A,ā) denotes the set {b | A |= Φ(b,ā)}.
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