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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2023, Volume 20, Issue 1, Pages 1–16
DOI: https://doi.org/10.33048/semi.2023.20.001
(Mi semr1565)
 

Mathematical logic, algebra and number theory

A note on joins and meets for positive linear preorders

N. Bazhenova, B. Kalmurzayevbac, M. Zubkovad

a Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, 630090, Novosibirsk, Russia
b Kazakh-British Technical University, 59 Tole bi St., 050000, Almaty, Kazakhstan
c Al-Farabi Kazakh National University, 71 Al Farabi Avenue, 050040, Almaty, Kazakhstan
d Kazan (Volga Region) Federal University, 35 Kremlevskaya St., 420008, Kazan, Russia
References:
Abstract: A positive preorder $R$ is linear if the corresponding quotient poset is linearly ordered. Following the recent advances in the studies of positive preorders, the paper investigates the degree structure Celps of positive linear preorders under computable reducibility. We prove that if a positive linear preorder $L$ is non-universal and the quotient poset of $L$ is infinite, then $L$ is a part of an infinite antichain inside Celps.
For a pair $L,R$ from Celps, we obtain sufficient conditions for when the pair has neither join, nor meet (with respect to computable reducibility). We give an example of a pair from Celps that has a meet. Inside the substructure $\Omega$ of Celps containing only computable linear orders of order-type $\omega$, we build a pair that has a join inside $\Omega$.
Keywords: computable reducibility, computably enumerable preorder, positive linear preorder.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-15-2022-281
Ministry of Education and Science of the Republic of Kazakhstan AP08856493
The work of the authors is supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. N. Bazhenov and B. Kalmurzyev were supported by the Ministry of Education and Science of the Republic of Kazakhstan, grant AP08856493 “Positive graphs and computable reducibility on them as mathematical model of databases”.
Received June 20, 2022, published January 23, 2023
Document Type: Article
UDC: 510.5
MSC: 03C57, 03D30
Language: English
Citation: N. Bazhenov, B. Kalmurzayev, M. Zubkov, “A note on joins and meets for positive linear preorders”, Sib. Èlektron. Mat. Izv., 20:1 (2023), 1–16
Citation in format AMSBIB
\Bibitem{BazKalZub23}
\by N.~Bazhenov, B.~Kalmurzayev, M.~Zubkov
\paper A note on joins and meets for positive linear preorders
\jour Sib. \`Elektron. Mat. Izv.
\yr 2023
\vol 20
\issue 1
\pages 1--16
\mathnet{http://mi.mathnet.ru/semr1565}
\crossref{https://doi.org/10.33048/semi.2023.20.001}
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