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Generalized o-minimality for partial orders
K. Zh. Kudaibergenov School of General Education, KIMEP, Almaty, Kazakhstan
Abstract:
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.
Key words:
weakly $p.o.$-minimal model, weakly quasi-$p.o.$-minimal model, weakly $p.o.$-minimal directed group, independence property, small extension of a model, Hanf number for omitting types, absolutely homogeneous model.
UDC:
510.67 Received: 22.10.2010