Geodesic
Generalized o-minimality for partial orders
Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 86-108.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - K. Zh. Kudaibergenov
TI  - Generalized o-minimality for partial orders
JO  - Matematičeskie trudy
PY  - 2012
SP  - 86
EP  - 108
VL  - 15
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2012_15_1_a4/
LA  - ru
ID  - MT_2012_15_1_a4
ER  - 
%0 Journal Article
%A K. Zh. Kudaibergenov
%T Generalized o-minimality for partial orders
%J Matematičeskie trudy
%D 2012
%P 86-108
%V 15
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2012_15_1_a4/
%G ru
%F 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/

[1] Ershov Yu. L., Problemy razreshimosti i konstruktivnye modeli, Nauka, M., 1980 | MR

[2] Kudaibergenov K. Zh., “Ob odnoi gipoteze Shelakha”, Algebra i logika, 26:2 (1987), 191–203 | MR

[3] Kudaibergenov K. Zh., “Malye rasshireniya modelei o-minimalnykh teorii i absolyutnaya odnorodnost”, Matem. tr., 10:1 (2007), 154–163 | MR | Zbl

[4] Kudaibergenov K. Zh., “Slabo kvazi-o-minimalnye modeli”, Matem. tr., 13:1 (2010), 156–168 | MR | Zbl

[5] Belegradek O., Peterzil Y., Wagner F., “Quasi-o-minimal structures”, J. Symbolic Logic, 65:3 (2000), 1115–1132 | DOI | MR | Zbl

[6] Bhattacharjee M., Macpherson D., Möller R. G., Neumann P. M., Notes on Infinite Permutation Groups, Lecture Notes in Math., 1698, Springer, Berlin, 1998 | MR | Zbl

[7] Dickmann M. A., “Elimination of quantifiers for ordered valuation rings”, Proc. of the 3rd Easter Conf. on Model Theory (Gross Koris, 1985), Humboldt Univ., Berlin, 1985, 64–88 | MR | Zbl

[8] Kudaibergenov K. Zh., “Homogeneous models of stable theories”, Siberian Adv. Math., 3:3 (1993), 56–88 | MR

[9] Macpherson H. D., Marker D., Steinhorn C., “Weakly o-minimal structures and real closed fields”, Trans. Amer. Math. Soc., 352:12 (2000), 5435–5483, (electronic) | DOI | MR | Zbl

[10] Marker D., “Omitting types in $\mathcal O$-minimal theories”, J. Symbolic Logic, 51:1 (1986), 63–74 | DOI | MR | Zbl

[11] Newelski L., Wencel R., “Definable sets in boolean-ordered o-minimal structures. I”, J. Symbolic Logic, 66:4 (2001), 1821–1836 | DOI | MR | Zbl

[12] Pillay A., Steinhorn C., “Definable sets in ordered structures. I”, Trans. Amer. Math. Soc., 295:2 (1986), 565–592 | DOI | MR | Zbl

[13] Poizat B., “Théories instables”, J. Symbolic Logic, 46 (1981), 513–522 | DOI | MR | Zbl

[14] Shelah S., “Finite diagrams stable in power”, Ann. Math. Logic, 2:1 (1970), 69–118 | DOI | MR | Zbl

[15] Shelah S., “Stability, the f.c.p., and superstability”, Ann. Math. Logic, 3:3 (1971), 271–362 | DOI | MR | Zbl

[16] Shelah S., Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, 92, North-Holland Publishing Co., Amsterdam–New York, 1978 | MR | Zbl

[17] Toffalori C., “Lattice ordered o-minimal structures”, Notre Dame J. Formal Logic, 39:4 (1998), 447–463 | DOI | MR | Zbl