PROPERTIES OF HYPERGRAPHS OF MODELS OF WEAKLY O-MINIMAL THEORIES

Abstract

In this paper, we study the notions of relative H-freedom and relative H-independence for hypergraphs of models of weakly o-minimal theories. Hypergraphs of models of a theory are derived objects that allow obtaining essential structural information both about the theories themselves and about related semantic objects. Recall that a hypergraph is any pair of sets (X, Y), where Y is some subset of the Boolean P(X) of a set X. In this case, the set X is called the support of the hypergraph (X, Y), and elements from Y are called edges of the hypergraph (X, Y). Weak o-minimality was originally deeply investigated by D. Macpherson, D. Marker, and C. Steinhorn. In the nineties of the last century, Kazakhstan scientists successfully joined the study of this concept, solving a number of problems posed by the authors. In this paper, we continue the study of model-theoretic properties of weakly o-minimal structures. A criterion for relative H-freedom of the set of realizations of non-algebraic 1-type in almost omega-categorical weakly o-minimal theories is obtained in terms of convexity rank. We also establish a criterion for relative H-independence of the sets of realizations of two non-algebraic 1-types in almost omega-categorical weakly o-minimal theories in terms of weak orthogonality of 1-types.