Generalized o-minimality for partial orders

We consider partially ordered models. We introduce the notions of a weakly (quasi-)$p.o.$-minimal model and a weakly (quasi-)$p.o.$-minimal theory. We prove that weakly quasi-$p.o.$-minimal theories of finite width lack the independence property, weakly $p.o.$-minimal directed groups are Abelian and divisible, weakly quasi-$p.o.$-minimal directed groups with unique roots are Abelian, and the direct product of a finite family of weakly $p.o.$-minimal models is a weakly $p.o.$-minimal model. We obtain results on existence of small extensions of models of weakly quasi-$p.o.$-minimal atomic theories. In particular, for such a theory of finite length, we find an upper estimate of the Hanf number for omitting a family of pure types. We also find an upper bound for the cardinalities of weakly quasi-$p.o.$-minimal absolutely homogeneous models of moderate width.