Geodesic
Complete Theories with Finitely Many Countable Models. II
Algebra i logika, Tome 45 (2006) no. 3, pp. 314-353.

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

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin–Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin–Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories.
Keywords: complete theory, Ehrenfeucht theory, number of countable models, Rudin–Keisler quasiorder. 
@article{AL_2006_45_3_a2,
     author = {S. V. Sudoplatov},
     title = {Complete {Theories} with {Finitely} {Many} {Countable} {Models.} {II}},
     journal = {Algebra i logika},
     pages = {314--353},
     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2006_45_3_a2/}
}
TY  - JOUR
AU  - S. V. Sudoplatov
TI  - Complete Theories with Finitely Many Countable Models. II
JO  - Algebra i logika
PY  - 2006
SP  - 314
EP  - 353
VL  - 45
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2006_45_3_a2/
LA  - ru
ID  - AL_2006_45_3_a2
ER  - 
%0 Journal Article
%A S. V. Sudoplatov
%T Complete Theories with Finitely Many Countable Models. II
%J Algebra i logika
%D 2006
%P 314-353
%V 45
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2006_45_3_a2/
%G ru
%F AL_2006_45_3_a2
S. V. Sudoplatov. Complete Theories with Finitely Many Countable Models. II. Algebra i logika, Tome 45 (2006) no. 3, pp. 314-353. http://geodesic.mathdoc.fr/item/AL_2006_45_3_a2/

[1] S. V. Sudoplatov, “Polnye teorii s konechnym chislom schëtnykh modelei. I”, Algebra i logika, 43:1 (2004), 110–124 | MR | Zbl

[2] C. C. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauchnaya kniga (NII MIOO NGU), Novosibirsk, 1996

[3] C. B. Sudoplatov, “Vlastnye orgrafy”, Sib. matem. zhurn., 48:1 (2007), 205–213 | MR

[4] Dzh. Barvaisa (red.), Spravochnaya kniga po matematicheskoi logike. Ch. 1: Teoriya modelei, Nauka, M., 1982 | MR

[5] S. Shelah, Classification theory and the number of non-isomorphic models,, 2nd ed., Stud. Logic Found. Math., 92, North–Holland, Amsterdam a. o., 1990 | MR | Zbl

[6] F. O. Wagner, Simple theories, Kluwer Acad. Publ., Dordrecht–Boston–London, 2000 | MR

[7] V. A. Emelichev i dr., Lektsii po teorii grafov, Nauka, M., 1990 | MR | Zbl

[8] J. T. Baldwin, N. Shi, “Stable generic structures”, Ann. Pure App. Logic, 79 (1996), 1–35 | DOI | MR | Zbl

[9] S. B. Sudoplatov, “Tipovaya redutsirovannost i moschnye tipy”, Sib. matem. zh., 33:1 (1992), 150–159 | MR | Zbl