Generalized o-minimality for partial orders
Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 86-108
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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.
@article{MT_2012_15_1_a4,
author = {K. Zh. Kudaibergenov},
title = {Generalized o-minimality for partial orders},
journal = {Matemati\v{c}eskie trudy},
pages = {86--108},
publisher = {mathdoc},
volume = {15},
number = {1},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MT_2012_15_1_a4/}
}
K. Zh. Kudaibergenov. Generalized o-minimality for partial orders. Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 86-108. http://geodesic.mathdoc.fr/item/MT_2012_15_1_a4/