Closures and generating sets related to combinations of structures

S. V. Sudoplatov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 16 (2016), pp. 131-144

Résumé

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories including the negation of finite character and the exchange property. It is shown that closure operators for $E$-combinations correspond to the closures with respect to the ultraproduct operator forming Hausdorff topological spaces. It is also shown that closure operators for disjoint $P$-combinations form topological $T_0$-spaces, which can be not Hausdorff. Thus topologies for $E$-combinations and $P$-combinations are rather different. We prove, for $E$-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set: it is shown that elements of the least generating set are isolated and dense in its $E$-closure. Related properties for $P$-combinations are considered: it is proved that again the existence of a minimal generating set of theories is equivalent to the existence of the least generating set but it is not equivalent to the isolation of elements in the generating set. It is shown that $P$-closures with the least generating sets are connected with families which are not $\omega$-reconstructible, as well as with families having finite $e$-spectra. Two questions on the least generating sets for $E$-combinations and $P$-combinations are formulated and partial answers are suggested.

Détail

Keywords: $E$-combination, $P$-combination, closure operator, generating set.

Bibliographie

[1] J. T. Baldwin, J. M. Plotkin, “A topology for the space of countable models of a first order theory”, Zeitshrift Math. Logik and Grundlagen der Math., 20:8–12 (1974), 173–178 | DOI | MR | Zbl

[2] P. Bankston, “Ulptraproducts in topology”, General Topology and its Applications, 7:3 (1977), 283–308 | MR | Zbl

[3] P. Bankston, “A survey of ultraproduct constructions in general topology”, Topology Atlas Invited Contributions, 8:2 (2003), 1–32

[4] C. C. Chang, H. J. Keisler, Model theory, Studies in Logic and the Foundations of Mathematics, 73, Elsevier, Amsterdam, 1990, 650 pp. ; G. Keisler, Ch. Ch. Chen, Teoriya modelei, Mir, M., 1977, 616 pp. | MR | MR

[5] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989, 529 pp. [Р. Энгелькинг, Общая топология, Мир, М., 1986, 752 с. ] | MR | Zbl | MR

[6] Yu. L. Ershov, E. A. Palyutin, Mathematical logic, Fizmatlit, M., 2011, 356 pp.

[7] J. Gismatullin, D. Penazzi, A. Pillay, “On compactifications and the topological dynamics of definable groups”, Annals of Pure and Applied Logic, 165:2 (2014), 552–562 | DOI | MR | Zbl

[8] A. S. Kechris, V. G. Pestov, S. Todorcevic, “Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups”, Geometric and Functional Analysis, 15:1 (2005), 106–189 | DOI | MR | Zbl

[9] L. Newelski, “Topological dynamics of definable group actions”, J. Symbolic Logic, 74:1 (2009), 50–72 | DOI | MR | Zbl

[10] A. Pillay, “Topological dynamics and definable groups”, J. Symbolic Logic, 78:2 (2013), 657–666 | DOI | MR | Zbl

[11] S. V. Sudoplatov, Group polygonometries, NSTU, Novosibirsk, 2011; 2013, 302 pp. (in Russian) [С. В. Судоплатов, Полигонометрии групп, Изд-во НГТУ, Новосибирск, 2011]; 2013, 302 с. (in Russian)

[12] S. V. Sudoplatov, “Models of cubic theories”, Bulletin of the Section of Logic, 43:1–2 (2014), 19–34 | MR | Zbl

[13] S. V. Sudoplatov, “Classes of structures and their generic limits”, Lobachevskii J. Math., 36:4 (2015), 426–433 | DOI | MR | Zbl

[14] S. V. Sudoplatov, P. Tanović, “Semi-isolation and the strict order property”, Notre Dame Journal of Formal Logic, 56:4 (2015), 555–572 | DOI | MR | Zbl

[15] S. V. Sudoplatov, Combinations of structures, 2016, 19 pp., arXiv: 1601.00041v1 [math.LO]