Geodesic
Degree spectra of the block relation of $1$-computable linear orders
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 3, pp. 296-305

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In this paper, we study the intrinsically computably enumerable relations on linear orderings, such as the successor relation on computable linear orderings and the block relation on $1$-computable linear orderings. For ease of reading, the linear ordering , in which the signature is enriched with the successor relation, is called $1$-computable linear ordering. This notion is consistent with the known results. We have proved that for any $\mathbf0'$-computable linear ordering $L$ there exists a $1$-computable linear ordering, in which the degree spectrum of the block relation coincides with the $\Sigma_1^0$-spectrum of the linear ordering $L$. The degree spectrum of the block relation of a linear ordering $R$ is called the class of Turing degrees of the images of the block relation on computable presentations of $R$; and $\Sigma_1^0$-spectrum of a linear ordering $L$ is called the class of Turing enumerable degrees of $L$. This obtained result provides a number of examples of the spectra of the block relation of $1$-computable linear orderings. In particular, the class of all enumerable high$_n$ degrees and the class of all enumerable non-low$_n$ degrees are realized by the spectra of the block relation of some $1$-computable linear orderings.
Keywords: linear orders, $1$-computability, block relation, successivity relation, intrinsically computably enumerable relations.
Mots-clés : spectra of relations 
@article{UZKU_2017_159_3_a2,
     author = {R. I. Bikmukhametov and M. S. Eryashkin and A. N. Frolov},
     title = {Degree spectra of the block relation of $1$-computable linear orders},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {296--305},
     publisher = {mathdoc},
     volume = {159},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2017_159_3_a2/}
}
TY  - JOUR
AU  - R. I. Bikmukhametov
AU  - M. S. Eryashkin
AU  - A. N. Frolov
TI  - Degree spectra of the block relation of $1$-computable linear orders
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2017
SP  - 296
EP  - 305
VL  - 159
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UZKU_2017_159_3_a2/
LA  - ru
ID  - UZKU_2017_159_3_a2
ER  - 
%0 Journal Article
%A R. I. Bikmukhametov
%A M. S. Eryashkin
%A A. N. Frolov
%T Degree spectra of the block relation of $1$-computable linear orders
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2017
%P 296-305
%V 159
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UZKU_2017_159_3_a2/
%G ru
%F UZKU_2017_159_3_a2
R. I. Bikmukhametov; M. S. Eryashkin; A. N. Frolov. Degree spectra of the block relation of $1$-computable linear orders. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 3, pp. 296-305. http://geodesic.mathdoc.fr/item/UZKU_2017_159_3_a2/