Geodesic
Weakly precomplete equivalence relations in the Ershov hierarchy
Algebra i logika, Tome 58 (2019) no. 3, pp. 297-319

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

We study the computable reducibility $\leq_c$ for equivalence relations in the Ershov hierarchy. For an arbitrary notation $a$ for a nonzero computable ordinal, it is stated that there exist a $\Pi^{-1}_a$-universal equivalence relation and a weakly precomplete $\Sigma^{-1}_a$-universal equivalence relation. We prove that for any $\Sigma^{-1}_a$ equivalence relation $E$, there is a weakly precomplete $\Sigma^{-1}_a$ equivalence relation $F$ such that $E\leq_c F$. For finite levels $\Sigma^{-1}_m$ in the Ershov hierarchy at which $m=4k+1$ or $m=4k+2$, it is shown that there exist infinitely many $\leq_c$-degrees containing weakly precomplete, proper $\Sigma^{-1}_m$ equivalence relations.
Keywords: Ershov hierarchy, computable reducibility, universal equivalence relation, weakly precomplete equivalence relation.
Mots-clés : equivalence relation 
@article{AL_2019_58_3_a0,
     author = {N. A. Bazhenov and B. S. Kalmurzaev},
     title = {Weakly precomplete equivalence relations in the {Ershov} hierarchy},
     journal = {Algebra i logika},
     pages = {297--319},
     publisher = {mathdoc},
     volume = {58},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2019_58_3_a0/}
}
TY  - JOUR
AU  - N. A. Bazhenov
AU  - B. S. Kalmurzaev
TI  - Weakly precomplete equivalence relations in the Ershov hierarchy
JO  - Algebra i logika
PY  - 2019
SP  - 297
EP  - 319
VL  - 58
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2019_58_3_a0/
LA  - ru
ID  - AL_2019_58_3_a0
ER  - 
%0 Journal Article
%A N. A. Bazhenov
%A B. S. Kalmurzaev
%T Weakly precomplete equivalence relations in the Ershov hierarchy
%J Algebra i logika
%D 2019
%P 297-319
%V 58
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2019_58_3_a0/
%G ru
%F AL_2019_58_3_a0
N. A. Bazhenov; B. S. Kalmurzaev. Weakly precomplete equivalence relations in the Ershov hierarchy. Algebra i logika, Tome 58 (2019) no. 3, pp. 297-319. http://geodesic.mathdoc.fr/item/AL_2019_58_3_a0/