List of issues > Series «Mathematics». 2017. Vol. 22
Generations of generative classes
We study generating sets of diagrams for generative classes. The generative classes appeared solving a series of model-theoretic problems. They are divided into semantic and syntactic ones. The fists ones are witnessed by well-known Frasse constructions and Hrushovski constructions. Syntactic generative classes and syntactic generic constructions were introduced by the author. They allow to consider any ω-homogeneous structure as a generic limit of diagrams over finite sets. Therefore any elementary theory is represented by some their generic models. Moreover, an information written by diagrams is realized in these models.
We consider generic constructions both in general case and with some natural restrictions, in particular, with the self-sufficiency property. We study the dominating relation and domination-equivalence for generative classes. These relations allow to characterize the finiteness of generic structure reducing the construction of generic structures to maximal diagrams. We also have that a generic structure is finite if and only if given generative class is finitely generated, i.e., all diagrams of this class are reduced to copying of some finite set of diagrams.
It is shown that a generative class without maximal diagrams is countably generated, i.e., reduced to some at most countable set of diagrams if and only if there is a countable generic structure. And the uncountable generation is equivalent to the absence of generic structures or to the existence only uncountable generative structures.
For citation:
Sudoplatov S.V. Generations of generative classes. The Bulletin of Irkutsk State University. Series Mathematics, 2017, vol. 22, pp. 106-117. https://doi.org/10.26516/1997-7670.2017.22.106
1. Baldwin J.T., Shi N. Stable generic structures Ann. Pure and Appl. Logic, 1996, vol. 79, no 1, pp. 1-35. https://doi.org/10.1016/0168-0072(95)00027-5
2. Frasse R. Sur certaines relations qui generalisent l’ordre des nombres rationnels C.R. Acad. Sci. Paris, 1953, vol. 237, pp. 540-542.
3. Frasse R. Sur l’extension aux relations de quelques proprietes des ordres. Annales Scientifiques de l’Ecole Normale Superieure. Troisieme Serie, 1954, vol. 71, pp. 363-388.
4. Hodges W. Model theory. Cambridge, Cambridge University Press, 1993. https://doi.org/10.1017/CBO9780511551574
5. Hrushovski E. Strongly minimal expansions of algebraically closed fields. Israel J. Math., 1992, vol. 79, no 2-3, pp. 129-151. https://doi.org/10.1007/BF02808211
6. Hrushovski E. Extending partial isomorphisms of graphs.Combinatorica, 1992, vol. 12, no 4. pp. 204–218. https://doi.org/10.1007/BF01305233
7. Hrushovski E. A new strongly minimal set. Ann. Pure and Appl. Logic, 1993, vol. 62, no 2. pp. 147–166. ttps://doi.org/10.1016/0168-0072(93)90171-9
8. Kiouvrekis Y., Stefaneas P., Sudoplatov S. V. Definable sets in generic structures and their cardinalities. Siberian Advances in Mathematics, 2018. (to appear)
9. Sudoplatov S. V. Syntactic approach to constructions of generic models. Algebra and Logic. 2007, vol. 46, no 2, pp. 134–146. https://doi.org/10.1007/s10469-007-0012-4
10. Sudoplatov S.V. The Lachlan Problem. Novosibirsk: NSTU, 2009. (in Russian)
11. Sudoplatov S. V. Classification of Countable Models of Complete Theories. Novosibirsk, NSTU, 2014. (in Russian)
12. Sudoplatov S.V. Generative and pre-generative classes Proceedings of the 10th Panhellenic Logic Symposium, June 11-15, 2015, Samos, Greece. University of Aegean, University of Crete, and University of Athens, 2015, pp. 30-34.
13. Sudoplatov S.V. Generative classes generated by sets of diagrams.Algebra and Model Theory 10. Collection of papers. Edited by A.G. Pinus, K.N. Ponomaryov, S.V. Sudoplatov, E.I. Timoshenko. Novosibirsk, NSTU, 2015, pp. 163–174.
14. Sudoplatov S.V., Kiouvrekis Y., Stefaneas P. Generic constructions and generic limits. Algebraic Modeling of Topological and Computational Structures and Applications. Springer Proc. in Math. & Statist.,;219. S. Lambropoulou (ed.) et al. Berlin, Springer Int. Publ., 2017.