A criterion for binarity of almost $\omega$-categorical weakly $o$-minimal theories

Continuing the study of weak $o$-minimality, we prove a theorem on the behavior of a definable unary function on the set of realizations of a nonalgebraic $1$-type in an arbitrary weakly $o$-minimal theory. Under study are the properties of almost $\omega$-categorical weakly $o$-minimal theories. We find sufficient conditions both for weak orthogonality and orthogonality of any finite family of nonalgebraic $1$-types over the empty set. The main result of the paper is a criterion for binarity of almost $\omega$-categorical weakly $o$-minimal theories.