ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2018. Vol. 24

Combinations of structures

S. V. Sudoplatov

We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving ω-categoricity and Ehrenfeuchtness under these combinations are characterized. The notions of e-spectra are introduced and possibilities for e-spectra are described.

It is shown that ω-categoricity for disjoint P-combinations means that there are finitely many indexes for new unary predicates and each structure in new unary predicate is either finite or ω-categorical. Similarly, the theory of E-combination is ω-categorical if and only if each given structure is either finite or ω-categorical and the set of indexes is either finite, or it is infinite and Ei-classes do not approximate infinitely many n-types for nω. The theory of disjoint P-combination is Ehrenfeucht if and only if the set of indexes is finite, each given structure is either finite, or ω-categorical, or Ehrenfeucht, and some given structure is Ehrenfeucht.

Variations of structures related to combinations and E-representability are considered.

We introduce e-spectra for P-combinations and E-combinations, and show that these e-spectra can have arbitrary cardinalities.

The property of Ehrenfeuchtness for E-combinations is characterized in terms of e-spectra.

About the Authors

Sergey V. Sudoplatov, Dr. Sci. (Phys.–Math.), Assoc. Prof., Leading Researcher, Sobolev Institute of Mathematics SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090, Russian Federation Head of Chair, Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073, Russian Federation Prof., Novosibirsk State University, 1, Pirogov st., Novosibirsk, 630090, Russian Federation, e-mail: sudoplat@math.nsc.ru

For citation:
Sudoplatov S. V. Combinations of Structures. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 24, pp. 82-101. https://doi.org/10.26516/1997-7670.2018.24.82
combination of structures, P-combination, E-combination, e-spectrum


03C30, 03C15, 03C50




1. Andrews U., Keisler H.J. Separable models of randomizations. J. Symbolic Logic, 2015, vol. 80, no. 4, pp. 1149-1181.https://doi.org/10.1017/jsl.2015.33

2. Baldwin J.T., Plotkin J.M. A topology for the space of countable models of a first order theory. Zeitshrift Math. Logik and Grundlagen der Math., 1974, vol. 20, no. 8–12, pp. 173-178.https://doi.org/10.1002/malq.19740200806

3. Bankston P. Ulptraproducts in topology. General Topology and its Applications, 1977, vol. 7, no. 3, pp. 283–308.

4. Bankston P. A survey of ultraproduct constructions in general topology. Topology Atlas Invited Contributions, 2003, vol. 8, no. 2, pp. 1-32.

5. Benda M. Remarks on countable models. Fund. Math., 1974, vol. 81, no. 2, pp. 107-119.https://doi.org/10.4064/fm-81-2-107-119

6. Henkin L. Relativization with respect to formulas and its use in proofs of independence. Composito Mathematica, 1968, vol. 20, pp. 88-106.

7. Newelski L. Topological dynamics of definable group actions. J. Symbolic Logic, 2009, vol. 74, no. 1, pp. 50-72.https://doi.org/10.2178/jsl/1231082302

8. Pillay A. Topological dynamics and definable groups. J. Symbolic Logic, 2013, vol. 78, no. 2, pp. 657-666.https://doi.org/10.2178/jsl.7802170

9. Sudoplatov S.V. Transitive arrangements of algebraic systems. Siberian Math. J., 1999, vol. 40, no. 6, pp. 1142-1145. https://doi.org/10.1007/BF02677538

10. Sudoplatov S.V. Inessential combinations and colorings of models. Siberian Math. J., 2003, vol. 44, no. 5, pp. 883–890.https://doi.org/10.1023/A:1025901223496

11. Sudoplatov S.V. Powerful digraphs. Siberian Math. J., 2007, vol. 48, no. 1, pp. 165–171.https://doi.org/10.1007/s11202-007-0017-1

12. Sudoplatov S.V. Klassifikatsiya schetnykh modeley polnykh teoriy [Classification of Countable Models of Complete Theories]. Novosibirsk, NSTU Publ., 2018.(in Russian)

13. Vaught R. Denumerable models of complete theories. Infinistic Methods, London, Pergamon, 1961, pp. 303-321.

14. Woodrow R.E. Theories with a finite number of countable models and a small language. Ph. D. Thesis. Simon Fraser University, 1976, 99 p.

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