Ranks for Families of Permutation Theories
The notion of rank for families of theories, similar to Morley rank for fixed theories, serves as a measure of complexity for given families. There arises a natural problem of describing a rank hierarchy for a series of families of theories.
In this article, we answer the question posed and describe the ranks and degrees for families of theories of permutations with different numbers of cycles of a certain length. A number examples of families of permutation theories that have a finite rank are given, and it is constructed a family of permutation theories having a specified countable rank and degree n. It is proved that in the family of permutation theories any theory equals a theory of a finite structure or it is approximated by finite structures, i.e. any permutation theory on an infinite set is pseudofinite. Topological properties of the families under consideration were studied.
Nurlan Markhabatov, Postgraduate Student, Novosibirsk State Technical University, 20, K. Marx Ave., 630073, Novosibirsk, Russian Federation; tel.: (383)3461166, e-mail: firstname.lastname@example.org
Markhabatov N.D. Ranks for Families of Permutation Theories. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 85-94. https://doi.org/10.26516/1997-7670.2019.28.85
- Ivanov A.A. Complete theories of unars. Algebra and Logic, 1984, vol. 23, no. 1, pp. 36-55.
- Marcus L. The number of countable models of a theory of one unary function. Fundamenta Mathematicae, 1980, vol. CVIII, issue 3, pp. 171-181.
- Markhabatov N.D., Sudoplatov S.V. Ranks for families of all theories of given languages. arXiv:1901.09903v1 [math.LO], 2019, available at: https://arxiv.org/abs/1901.09903v1
- Morley M. Categoricity in Power. Transactions of the American Mathematical Society, 1965, vol. 114, no. 2, pp. 514-538.
- Popkov R.A. Klassifikacija schjotnyh modelej polnyh teorij odnomestnyh predikatov s podstanovkoj ogranichennogo porjadka [Classification of countable models of complete theories of unary predicates with permutation of bounded order], Algebra and Model Theory 8. Collection of papers, NSTU, Novosibirsk, 2011, pp. 73-82.
- 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/10.2178/bsl/1182353894
- Ryaskin A.N. The number of models of complete theories of unars. Model Theory and Its Applications, Tr. Inst. Mat. SO AN SSSR, 1988, vol. 8, pp. 162-182.
- Shishmarev Yu.E. On categorical theories of one function. Mat. Zametki, 1972, vol. 11, no. 1, pp. 89-98.
- Sudoplatov S.V. Ranks for families of theories and their spectra. arXiv:1901.08464v1 [math.LO], 2019, available at: https://arxiv.org/abs/1901.08464
- Sudoplatov S.V. Approximations of theories. arXiv:1901.08961v1 [math.LO], 2019, available at: https://arxiv.org/abs/1901.08961