Weakly o-minimal types

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type p S (A) is weakly o-minimal with respect to <, a relatively A-definable linear order on p (C), if every relatively definable subset has finitely many convex components; we prove that in that case the latter holds for all orders. Notably, we prove: (i) a monotonicity theorem for relatively definable functions on the locus of a weakly o-minimal type; (ii) weakly o-minimal types are dp-minimal, and the weak and forking non-orthogonality are equivalence relations on weakly o-minimal types. For a weakly o-minimal pair p= (p, <), we introduce the notions of the left- and right- p-genericity of a p over B; the latter is denoted by B^ pa. We prove that ^ p behaves particularly well on realizations of p: the ^ p-incomparability and forking-dependence of x and y over the domain of p are the same equivalence relation and the quotient order is dense linear. We show that this naturally generalizes to the set of realizations of weakly o-minimal types from a fixed ʷ-class.