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 $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every relatively $L_{\mathfrak{C}}$-definable subset of $p(\mathfrak{C})$