On freedom and independence in hypergraphs of models of quite o-minimal theories with few countable models

Abstract

We study properties of the concepts of
freedom and independence for hypergraphs of models of a quite
o-minimal theory with few countable models. Conditions for freedom
of sets of realizations of isolated and non-isolated types
are characterized in terms of the convexity rank. In terms of weak
orthogonality, characterizations of the
relative independence of sets of realizations of isolated and
non-isolated types of convexity rank 1 are obtained. Conditions
for freedom and independence of equivalence classes are
established, indicating the finite rank of convexity of a
non-algebraic isolated type of a given theory. In terms of
equivalence classes, the conditions for the relative freedom of
isolated and non-isolated types are characterized. In terms of
weak orthogonality, characterizations of the relative independence
of sets of realizations of isolated and non-isolated types over
given equivalence relations are obtained. The transfer of the
property of relative freedom of types under the action of
definable bijections is proved. It is shown that for the specified
conditions the non-maximality of the number of countable models of
the theory is essential.