Abstract
The present paper concerns the notion of weak o-minimality introduced by M. Dickmann and originally studied by D. Macpherson, D. Marker, and C. Steinhorn. Weak o-minimality is a generalization of the notion of o-minimality introduced by A. Pillay and C. Steinhorn in series of joint papers. As is known, the ordered field of real numbers is an example of an o-minimal structure. We continue studying properties of almost omega-categorical weakly o-minimal theories. Almost omega-categoricity is a notion generalizing the notion of omega-categoricity. Recently, a criterion for binarity of almost omega-categorical weakly o-minimal theories in terms of convexity rank has been obtained. Binary convexity rank is the convexity rank in which parametrically definable equivalence relations are replaced by ∅ - definable equivalence relations. (p, q)-splitting formulas express a connection between non-weakly orthogonal non-algebraic 1-types in weakly o-minimal theories. In many cases, the binary convexity ranks of non-weakly orthogonal non-algebraic 1-types are not equal. The main result of this paper is finding necessary and sufficient conditions for equality of the binary convexity ranks for non-weakly orthogonal non-algebraic 1-types in almost omega-categorical weakly o-minimal theories in terms of (p, q)-splitting formulas.
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