Geodesic
Uniform $m$-equivalences and numberings of classical systems
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 49-65.

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The paper considers the representability of algebraic structures (groups, lattices, semigroups, etc.) over equivalence relations on natural numbers. The concept of a (uniform) $m$-equivalence is studied. It is proved that the numbering equivalence of any numbered group is a uniform $m$-equivalence. On the other hand, we construct an example of a uniform $m$-equivalence over which no group is representable. Additionally we show that there exists a positive equivalence over which no upper (lower) semilattice is representable.
Keywords: uniform $m$-equivalence, lattice, field.
Mots-clés : group 
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N. Kh. Kasymov; R. N. Dadazhanov; S. K. Zhavliev. Uniform $m$-equivalences and numberings of classical systems. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 49-65. http://geodesic.mathdoc.fr/item/SEMR_2022_19_1_a2/

[1] A.I. Mal'tsev, “Strongly related models and recursively complete algebras”, Sov. Math., Dokl., 3 (1962), 987–991 | Zbl

[2] A.I. Mal'tsev, “On recursive abelian groups”, Sov. Math., Dokl., 32 (1962), 1431–1434 | Zbl

[3] A.S. Morozov, “Computable groups of automorphisms of models”, Algebra Logic, 25 (1986), 261–266 | DOI | MR | Zbl

[4] A.S. Morozov, “Elementary properties of groups of recursive permutations”, Sov. Math., Dokl., 39:2 (1989), 282–284 | MR | Zbl

[5] A.S. Morozov, “On the theories of classes of recursive permutations groups”, Sib. Adv. Math., 1:1 (1991), 138–153 | MR | Zbl

[6] A.S. Morozov, “Groups of computable automorphisms”, Handbook of recursive mathematics, v. 1, Stud. Log. Found. Math., 138, Recursive model theory, eds. Yu.L. Ershov et al., Elsevier, Amsterdam, 1998, 311–345 | DOI | MR | Zbl

[7] A.S. Morozov, J.K. Truss, “On computable automorphisms of the rational numbers”, J. Symb. Log., 66:3 (2001), 1458–1470 | DOI | MR | Zbl

[8] V. Harizanov, R. Miller, A.S. Morozov, “Simple structures with complex symmetry”, Algebra Logic, 49:1 (2010), 68–90 | DOI | MR | Zbl

[9] A.S. Morozov, “Higman's question revisited”, Algebra Logic, 39:2 (2000), 78–83 | DOI | MR | Zbl

[10] N.G. Khisamiev, “On positive and constructive groups”, Sib. Math. J., 53:5 (2012), 906–917 | DOI | MR | Zbl

[11] N.G. Khisamiev, “Arithmetic hierarchy of abelian groups”, Sib. Math. J., 29:6 (1988), 987–999 | DOI | MR | Zbl

[12] S.S. Goncharov, Yu.L. Ershov, Constructive models, Siberian School of Algebra and Logic, Consultants Bureau, New York, 2000 | MR | Zbl

[13] Yu.L. Ershov, Theory of numerations, Nauka, M., 1977 | MR

[14] S.S. Goncharov, “Computability and computable models”, Mathematical Problems from Applied Logic, v. II, Logics for the XXIst Century, eds. Gabbay Dov M. et al., Springer, New York, 2007, 99–216 | DOI | MR | Zbl

[15] N.Kh. Kasymov, A.S. Morozov, “Definability of linear orders over negative equivalences”, Algebra Logic, 55:1 (2016), 24–37 | DOI | MR | Zbl

[16] N.Kh. Kasymov, “Recursively separable enumerated algebras”, Russ. Math. Surv., 51:3 (1996), 509–538 | DOI | MR | Zbl

[17] N.Kh. Kasymov, “Homomorphisms onto effectively separable algebras”, Sib. Math. J., 57:1 (2016), 36–50 | DOI | MR | Zbl

[18] N.Kh. Kasymov, A.S. Morozov, I.A. Khodzhamuratova, “$T_1$-separable enumarations of subdirectly indecomposable algebras”, Algebra Logika, 60:4 (2021), 400–424 | DOI | Zbl

[19] B.M. Khoussainov, T. Slaman, P. Semukhin, “$\prod_1^0$-presentasions of algebras”, Arch. Math. Logic, 45:6 (2006), 769–781 | DOI | MR | Zbl

[20] N.Kh. Kasymov, B.M. Khoussainov, “Finitely generated, enumerable and absolutely locally finite algebras”, Vychisl. Sist., 116 (1986), 3–15 | MR | Zbl

[21] N.Kh. Kasymov, R.N. Dadazhanov, “Negative dense linear orders”, Sib. Math. J., 58:6 (2017), 1015–1033 | DOI | MR | Zbl

[22] N.Kh. Kasymov, “Algebras over negative equivalences”, Algebra Logic, 33:1 (1994), 46–48 | DOI | MR | Zbl

[23] N.Kh. Kasymov, F.N. Ibragimov, “Separable enumerations of division rings and effective embeddability of rings therin”, Sib. Math. J., 60:1 (2019), 62–70 | DOI | MR | Zbl

[24] E.B. Fokina, B.M. Khoussainov, P. Semukhin, D. Turetskiy, “Linear orders realized by c.e. equivalence relations”, J. Symb. Log., 81:2 (2016), 463–482 | DOI | MR | Zbl