Geodesic
Effectively infinite classes of numberings and fixed point theorems
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1519-1536.

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

In this paper, we prove a sufficient condition for the effective infinity of classes of complete and precomplete numberings, as well as numberings satisfying the recursion theorem, of computable families. A sufficient condition for the effective infinity of classes of non-precomplete numberings of computable families satisfying the recursion theorem is also obtained. These conditions are satisfied by the family of all c.e. sets and the family of graphs of all partially computable functions. For finite families of c.e. sets, we prove a criterion for the effective infinity of classes of their numberings that satisfy the recursion theorem. Finally, it is established that the classes of complete and precomplete numberings of finite families of c.e. sets are not effectively infinite.
Keywords: computable numbering, complete numbering, precomplete numbering, recursion theorem, effective infinity. 
@article{SEMR_2023_20_2_a22,
     author = {M. Kh. Faizrahmanov},
     title = {Effectively infinite classes of numberings and fixed point theorems},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1519--1536},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a22/}
}
TY  - JOUR
AU  - M. Kh. Faizrahmanov
TI  - Effectively infinite classes of numberings and fixed point theorems
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2023
SP  - 1519
EP  - 1536
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a22/
LA  - ru
ID  - SEMR_2023_20_2_a22
ER  - 
%0 Journal Article
%A M. Kh. Faizrahmanov
%T Effectively infinite classes of numberings and fixed point theorems
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2023
%P 1519-1536
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a22/
%G ru
%F SEMR_2023_20_2_a22
M. Kh. Faizrahmanov. Effectively infinite classes of numberings and fixed point theorems. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1519-1536. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a22/

[1] S.S. Goncharov, A. Yakhnis, V. Yakhnis, “Some effectively infinite classes of enumerations”, Ann. Pure App. Logic, 60:3 (1993), 207–235 | DOI | MR | Zbl

[2] A.I. Mal'tsev, “Completely enumerated sets”, Algebra Logika, 2:2 (1963), 4–29 | MR | Zbl

[3] A.I. Mal'tsev, The metamathematics of algebraic systems, North-Holland, Amsterdam–London, 1971 https://lib.ugent.be/catalog/rug01:000004761 | Zbl

[4] Yu.L. Ershov, “Theorie der numerierungen I”, Z. Math. Log. Grundl. Math., 19 (1973), 289–388 | DOI | MR | Zbl

[5] Yu.L. Ershov, Theory of numberings, Nauka, M., 1977 | MR

[6] Yu.L. Ershov, “Completely enumerated sets”, Sib. Math. J., 10 (1970), 773–784 | DOI | MR | Zbl

[7] Yu.L. Ershov, “On inseparable pairs”, Algebra Logic, 9:6 (1972), 396–399 | DOI | MR | Zbl

[8] V.L. Selivanov, “Index sets of quotient objects of the Post numeration”, Algebra Logic, 27:3 (1988), 215–224 | DOI | MR | Zbl

[9] S.A. Badaev, S.S. Goncharov, A. Sorbi, “Completeness and universality of arithmetical numberings”, Computability and Models, Univ. Ser. Math., Kluwer Academic/Plenum Publishers, New York, 2003, 11–44 | DOI | MR

[10] H. Barendregt, S.A. Terwijn, “Fixed point theorems for precomplete numberings”, Ann. Pure App. Logic, 170:10 (2019), 1151–1161 | DOI | MR | Zbl

[11] M.M. Arslanov, “Fixed-point selection functions”, Lobachevskii J. Math., 42:4 (2021), 685–692 | DOI | MR | Zbl

[12] V.L. Selivanov, “Precomplete numberings”, J. Math. Sci., New York, 256:1 (2021), 96–124 | DOI | MR | Zbl

[13] T.H. Payne, “Effective extendability and fixed-points”, Notre Dame J. Formal Logic, 14:1 (1973), 123–124 | DOI | MR | Zbl

[14] S.A. Badaev, “On weakly pre-complete positive equivalences”, Sib. Math. J., 32:2 (1991), 321–323 | DOI | MR | Zbl

[15] M. Faizrahmanov, “Extremal numberings and fixed point theorems”, Math. Log. Q., 68:4 (2022), 398–408 | DOI | MR | Zbl

[16] Yu.L. Ershov, “Theory of numberings”, Handbook of computability theory, Elsevier. Stud. Logic Found. Math., 140, ed. Griffor Edward R., Elsevier, Amsterdam, 1999, 473–503 | DOI | MR | Zbl

[17] R.I. Soare, Recursively enumerable sets and degrees. A study of computable functions and computably generated sets, Perspectives in Mathematical Logic, Springer-Verlag, Berlin etc., 1987 https://link.springer.com/book/9783540666813 | DOI | MR | Zbl

[18] R.I. Soare, Turing computability. Theory and applications, Springer, Berlin, 2016 | DOI | MR | Zbl

[19] Yu.L. Ershov, “Theorie der numerierungen II”, Z. Math. Log. Grundl. Math., 21:1 (1975), 473–584 | DOI | MR | Zbl

[20] R.M. Friedberg, “Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication”, J. Symb. Log., 23:3 (1959), 309–316 | DOI | MR | Zbl