In the early days of the development of model theory it was considered natural and was certainly beneficial to assume that the theories under investigation were in a countable language. The primary advantage of this assumption was the presence of the Omitting Types Theorem of Grzegorczyk, Mostowski, and Ryll-Nardzewski [1], which generalized arguments of Henkin [3] and Orey [8]. Following this,...

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. Theorem 0.1 (Superstability from categoricity). Let K be a (< κ)-tame AEC with amalgamation. If κ = iκ > ...

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1. Let M0 ≺ N be models from K and A be a set. We say that the Galois-type of A over N does not fork over M0, written A^ M0 N , iff for all small a ∈ A and all small N− ≺ N , we have...

Abstract: In article is considered the problem development mathematical model of the mechanism of parallel structure type of hexapod. The received model describes kinematic and dynamic properties of object. At the kinematic analysis are formed the matrix of transformation which depends from linear and rotational moving of platform. As a result of transformations we receive expressions establish...

In this paper, first, we introduce the new concept of 2-inner product on Banach modules over a $C^*$-algebra. Next,  we present the concept of 2-linear operators over a $C^*$-algebra. Our result improve  the main result of the paper  Z. Lewandowska.  In the final of this paper, we define the notions 2-adjointable mappings between 2-pre Hilbert C*-modules and prove supperstability of them ...

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. Theorem 0.1 (Superstability from categoricity). Let K be a (< κ)-tame AEC with amalgamation. If κ = iκ > ...

Historically one of the great successes of model theory has been Shelah’s stability theory: a program, described in [17], of showing that the arrangement of first-order theories into complexity classes according to a priori set-theoretic criteria (e.g. counting types over sets) in fact pushes down to reveal a very rich and entirely model-theoretic structure theory for the classes involved: what...

The notion of a normal theory is introduced, and it is proved that for such a theory T, /(N„, 7") = 1 or 3e S(l. Wc also include a short proof of Lachlan's theorem that for superstable 7". /(X0, 7") = 1 or > N(1. (The property of normality is stronger than stability but incomparable to superstability.) 1. Preliminaries. We consider here theories T which are countable and complete (and of course...