On Acyclic Hypergraphs of Minimal Prime Models

We discuss the question of restoring the structural properties of theories from the hypergraphs of minimal prime models. We describe the spectrum and the main model-theoretic properties of acyclic complete theories with the property of extension of isomorphisms of families of minimal prime models.     

Topologies,Ranks, and Closures for Families of Theories. II

Algebra and Logic - We describe topological properties, ranks, and closures as well as their dynamics for families of theories. Closures for families of theories are introduced based on sentences...     

Complete theories with finitely many countable models. II

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin-Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin-Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories. Supported by RFBR grant Nos. 02-01-00258 and 05-01-00411. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 314–353, May–June, 2006.     

On the number of countable models of complete theories with finite Rudin-Keisler preorders

The aim of this article is to generalize the classification of complete theories with finitely many countable models with respect to two principal characteristics, Rudin-Keisler preorders and the distribution functions of the number of limit models, to an arbitrary case with a finite Rudin-Keisler preorder. We establish that the same characteristics play a crucial role in the case we consider. We prove the compatibility of arbitrary finite Rudin-Keisler preorders with arbitrary distribution functions f satisfying the condition rang f?ω∪{ω, 2^ω}.     

Powerful digraphs

We introduce the concept of a powerful digraph and establish that a powerful digraph structure is included into the saturated structure of each nonprincipal powerful type p possessing the global pairwise intersection property and the similarity property for the theories of graph structures of type p and some of its first-order definable restrictions (all powerful types in the available theories with finitely many (> 1) pairwise nonisomorphic countable models have this property). We describe the structures of the transitive closures of the saturated powerful digraphs that occur in the models of theories with nonprincipal powerful 1-types provided that the number of nonprincipal 1-types is finite. We prove that a powerful digraph structure, considered in a model of a simple theory, induces an infinite weight, which implies that the powerful digraphs do not occur in the structures of the available classes of the simple theories (like the supersimple or finitely based theories) that do not contain theories with finitely many (> 1) countable models.     

On expansions and extensions of powerful digraphs

We study the problem of expanding and extending the structure of a stable powerful digraph to the structure of a stable Ehrenfeucht theory. We define the concepts of type unstability and type strict order property. We establish the presence of the type strict order property for every acyclic graph structure with an infinite chain. The simplest form of expansion of a powerful digraph to the structure of an Ehrenfeucht theory is the expansion with a 1-inessential ordered coloring and locally graph ?-definable many-placed relations, which enable us to mutually realize nonprincipal types; we prove that this expansion is incapable of keeping the structure in the class of stable structures, and moreover, by the type strict order property it generates the first-order definable strict order property. We define the concept of a locally countably categorical theory (LCC theory) and prove that given the list p ₁(x), ..., p _n (x) of all nonprincipal 1-types in an LCC theory, if all types r(x ₁, ..., x _m ) containing \(p_{i_1 } \) (x ₁) ∪ ... ∪ \(p_{i_m } \)(x _m ) are dominated by some type q then q is a powerful type.     

Embedding in the class of trigonometries of groups

We give a characterization of embeddability of one group trigonometry in another and argue that there exists an isomorphism between the trigonometries in question. A criterion for a group trigonometry to exist on a faithful pseudoplane is proved and a criterion for one trigonometry to be embeddable in another trigonometry on a projective plane is established.Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 429–447, July-August, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1520.     

Type reduction and powerful types

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 150–159, January–February, 1992.