Ranks for Families of Theories of Abelian Groups
The rank for families of theories is similar to Morley rank and can be considered as a measure for complexity or richness of these families. Increasing the rank by extensions of families we produce more rich families and obtaining families with the infinite rank that can be considered as “rich enough”. In the paper, we realize ranks for families of theories of abelian groups. In particular, we study ranks and closures for families of theories of finite abelian groups observing that the set of theories of finite abelian groups in not totally transcendental, i.e., its rank equals infinity. We characterize pseudofinite abelian groups in terms of Szmielew invariants. Besides we characterize e-minimal families of theories of abelian groups both in terms of dimension, i.e., the number of independent limits for Szmielew invariants, and in terms of inequalities for Szmielew invariants. These characterizations are obtained both for finite abelian groups and in general case. Furthermore we give characterizations for approximability of theories of abelian groups and show the possibility to count Szmielew invariants via these parameters for approximations. We describe possibilities to form d-definable families of theories of abelian groups having given countable rank and degree.
Inessa Pavlyuk, Cand. Sci. (Phys.–Math.), docent, Novosibirsk State Pedagogical University, 28, Vilyuiskaya street, Novosibirsk, 630126, Russian Federation; tel. (383)2441586, e-mail: firstname.lastname@example.org
Sergey Sudoplatov, Dr. Sci. (Phys.–Math.), docent; Leading researcher, Sobolev Institute of Mathematics of SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090, tel.: (383)3297586; Head of Chair, Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073, tel.: (383)3461166; Professor, Novosibirsk State University, 1, Pirogova street, Novosibirsk, 630090, Russian Federation, tel.: (383)3634020, e-mail: email@example.com
Pavlyuk In.I., Sudoplatov S.V. Ranks for Families of Theories of Abelian Groups. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 95-112. https://doi.org/10.26516/1997-7670.2019.28.95
- Eklof P.C., Fischer E.R. The elementary theory of abelian groups. Annals of Mathematical Logic, 1972, vol. 4, pp. 115-171. https://doi.org/10.1016/0003-4843(72)90013-7
- Ershov Yu.L., Palyutin E.A. Mathematical logic. Moscow, Fizmatlit Publ., 2011.
- Koppelberg S. Handbook of Boolean Algebras. Vol. 1, Monk J.D., Bonnet R. (eds.). Amsterdam, New York, Oxford, Tokyo, North-Holland, 1989, 342 p.
- Macpherson D. Model theory of finite and pseudofinite groups. Archive for Mathematical Logic, 2018, vol. 57, no. 1-2, pp. 159-184. https://doi.org/10.1007/s00153-017-0584-1
- Markhabatov N.D., Sudoplatov S.V. Ranks for families of all theories of given languages. arXiv:1901.09903v1 [math.LO], 2019, 9 p.
- Markhabatov N.D., Sudoplatov S.V. Definable subfamilies of theories and related calculi. arXiv:1901.08961v1 [math.LO], 2019, 20 p.
- Morley M. Categoricity in Power. Transactions of the American Mathematical Society, 1965, vol. 114, no. 2, pp. 514-538. https://doi.org/10.2307/1994188
- Pavlyuk In.I., Sudoplatov S.V. Families of theories of abelian groups and their closures. Bulletin of Karaganda University. Mathematics, 2018, vol. 92, no. 4, pp. 72-78. https://doi.org/10.31489/2018M4/72-78
- Popkov R.A. Distribution of countable models for the theory of the group of integers. Siberian. Math. J., 2015, vol. 56, no. 1, pp. 185-191. https://doi.org/10.1134/S0037446615010152
- Rosen E. Some Aspects of Model Theory and Finite Structures. The Bulletin of Symbolic Logic, 2002, vol. 8, no. 3, pp. 380-403. https://doi.org/10.2307/3062205
- Sudoplatov S. V. Ranks for families of theories and their spectra. arXiv:1901.08464v1 [math.LO], 2019, 17 p.
- Sudoplatov S.V. Approximations of theories. arXiv:1901.08961v1 [math.LO], 2019, 16 p.
- Szmielew W. Elementary properties of Abelian groups. Fund. Math., 1955, vol. 41, pp. 203-271. https://doi.org/10.4064/fm-41-2-203-271