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 Bᵖa. 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.