Geodesic
Ranks for families of permutation theories
The Bulletin of Irkutsk State University. Series Mathematics, Tome 28 (2019), pp. 85-94

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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.
Keywords: family of theories, pseudofinite theory, rank, degree.
Mots-clés : permutation 
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     author = {N. D. Markhabatov},
     title = {Ranks for families of permutation theories},
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N. D. Markhabatov. Ranks for families of permutation theories. The Bulletin of Irkutsk State University. Series Mathematics, Tome 28 (2019), pp. 85-94. http://geodesic.mathdoc.fr/item/IIGUM_2019_28_a5/