The structure of computably enumerable preorder relations

Badaev, S. A.; Bazhenov, N. A.; Kalmurzaev, B. S.

doi:10.33048/alglog.2020.59.301

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“…Our choice of reducibility c in Definition 1.1 has a twofold motivation. First, computable reducibility proved to be a useful tool for classifying positive preorders (see, for example, [7,12,13]). Second, we can observe the following.…”

Section: Pr(s ) = ({Deg Pr (A )mentioning

confidence: 99%

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Computable Reducibility for Computable Linear Orders of Type ω

2022

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We study computable reducibility for computable isomorphic copies of the standard ordering of natural numbers. Following Andrews and Sorbi, we isolate the class of self-full degrees inside the induced degree structure Ω. We show that, over an arbitrary degree from Ω, there exists an infinite antichain of self-full degrees. This fact implies that the poset Ω has continuum many automorphisms. We prove that any non-self-full degree from Ω has no minimal covers, which implies that, inside Ω, the self-full degrees are precisely those elements that have a minimal cover. Bibliography: 18 titles.The paper studies computable reducibility for computable linear orders. Let R and S be binary relations on the set of natural numbers ω. The relation R is computably reducible to S (denoted by R c S) if there is a total computable function f (x) such thatIn this case, one also says that the function f is a computable reduction from R to S.The systematic study of computable reducibility for positive (computably enumerable) equivalence relations was initiated by Ershov [1] who constructed one of the first examples of a universal positive equivalence. Further developments led to sophisticated classifications for various * To whom the correspondence should be addressed.

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Section: Pr(s ) = ({Deg Pr (A )mentioning

confidence: 99%

“…Andrews and Sorbi [6] developed intricate methods for working with the poset of c-degrees, i.e., degrees with respect to computable reducibility, of positive equivalences. The methods of [6] were used in [7] to investigate c-degrees of positive preorders.…”

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confidence: 99%