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.
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Original Russian Text © K. Zh. Kudaĭbergenov, 2012, published in Matematicheskie Trudy, 2012, Vol. 15, No. 1, pp. 86–108.
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Kudaĭbergenov, K.Z. Generalized o-minimality for partial orders. Sib. Adv. Math. 23, 47–60 (2013). https://doi.org/10.3103/S1055134413010045
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DOI: https://doi.org/10.3103/S1055134413010045