Geodesic
Topologies, ranks, and closures for families of theories.~I
Algebra i logika, Tome 59 (2020) no. 6, pp. 649-679.

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

We describe topological properties, ranks, closures, and their dynamics for families of theories. Types of topologies for families of theories are characterized. A relationship is established between ranks and topologies for families of theories. Boolean combinations of $s$-definable families of theories are treated, ranks and degrees with respect to these families are found, and values of the characteristics in question are described. We study closures of families of theories with respect to $s$-definable subfamilies and their Boolean combinations, properties of closure operators, and also a condition for the existence of a least generating set. Rank values for families of theories are specified in terms of algebras of definable subfamilies.
Keywords: topology, rank, closure, family of theories. 
@article{AL_2020_59_6_a2,
     author = {N. D. Markhabatov and S. V. Sudoplatov},
     title = {Topologies, ranks, and closures for families of {theories.~I}},
     journal = {Algebra i logika},
     pages = {649--679},
     publisher = {mathdoc},
     volume = {59},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2020_59_6_a2/}
}
TY  - JOUR
AU  - N. D. Markhabatov
AU  - S. V. Sudoplatov
TI  - Topologies, ranks, and closures for families of theories.~I
JO  - Algebra i logika
PY  - 2020
SP  - 649
EP  - 679
VL  - 59
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2020_59_6_a2/
LA  - ru
ID  - AL_2020_59_6_a2
ER  - 
%0 Journal Article
%A N. D. Markhabatov
%A S. V. Sudoplatov
%T Topologies, ranks, and closures for families of theories.~I
%J Algebra i logika
%D 2020
%P 649-679
%V 59
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2020_59_6_a2/
%G ru
%F AL_2020_59_6_a2
N. D. Markhabatov; S. V. Sudoplatov. Topologies, ranks, and closures for families of theories.~I. Algebra i logika, Tome 59 (2020) no. 6, pp. 649-679. http://geodesic.mathdoc.fr/item/AL_2020_59_6_a2/

[1] S. V. Sudoplatov, Approximations of theories, 17 (2020), 715–725 http://semr.math.nsc.ru/v17/p715-725.pdf | MR | Zbl

[2] S. Sudoplatov, Ranks for families of theories and their spectra, 2019, arXiv: 1901.08464v1 | MR

[3] N. Markhabatov, S. Sudoplatov, “Ranks for families of all theories of given languages”, Eurasian Math. J. (to appear) , arXiv: 1901.09903v1

[4] N. D. Markhabatov, S. V. Sudoplatov, Definable families of theories, related calculi and ranks, 17 (2020), 700–714 http://semr.math.nsc.ru/v17/p700-714.pdf | MR | Zbl

[5] N. D. Markhabatov, S. V. Sudoplatov, Algebras for definable families of theories, 16 (2019), 600–608 http://semr.math.nsc.ru/v16/p600-608.pdf | MR | Zbl

[6] N. D. Markhabatov, Ranks for families of permutation theories, 28 (2019), 85–94 | MR | Zbl

[7] In. I. Pavlyuk, S. V. Sudoplatov, Ranks for families of theories of abelian groups, 28 (2019), 95–112 | MR | Zbl

[8] B. Sh. Kulpeshov, S. V. Sudoplatov, Ranks and approximations for families of ordered theories, Algebra and model theory, 12, eds. A. G. Pinus et al., Novosibirsk State Tech. Univ., Novosibirsk, 2019, 32–40 | MR

[9] In. I. Pavlyuk, S. V. Sudoplatov, “Approximations for theories of abelian groups”, Math. Stat., 8:2 (2020), 220–224 | DOI | MR

[10] Yu. L. Ershov, E. A. Palyutin, Matematicheskaya logika, 6-e izd., Fizmatlit, M., 2011 | MR

[11] B. Poizat, A. Yeshkeyev, “Positive Jonsson theories”, Log. Univers., 12:1/2 (2018), 101–127 | DOI | MR | Zbl

[12] R. Engelking, Obschaya topologiya, Mir, M., 1986 | MR

[13] S. V. Sudoplatov, On the separability of elements and sets in hypergraphs of models of a theory, 82:2 (2016), 113–120 | MR

[14] B. Sh. Kulpeshov, S. V. Sudoplatov, On relative separability in hypergraphs of models of theories, 9:4 (2018), 68–78 | MR | Zbl

[15] S. V. Sudoplatov, Closures and generating sets related to combinations of structures, 16 (2016), 131–144 | MR | Zbl

[16] W. J. Blok, D. Pigozzi, Algebraizable logics, Mem. Am. Math. Soc., 77, no. 396, 1989 | MR

[17] P. G. Hinman, Fundamentals of mathematical logic, A K Peters, Wellesley, MA, 2005 | MR | Zbl