Computable Linear Orders and Limitwise Monotonic Functions

Frolov, A. N.; Zubkov, M. V.

doi:10.1007/s10958-021-05421-3

Cited by 2 publications

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“…These functions have become one of the standard tools for computability theorists. We refer the reader to, e.g., the surveys [11,21] for the discussion of applications for limitwise monotonic sets and functions. In what follows, we use the acronym l.m.…”

Section: Introductionmentioning

confidence: 99%

Rogers semilattices of limitwise monotonic numberings

Bazhenov

Mustafa

Tleuliyeva

2022

Mathematical Logic Qtrly

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Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S⊂Pfalse(ωfalse)$S\subset P(\omega )$ is limitwise monotonic (l.m.) if every set νfalse(kfalse)$\nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice Rlm(S)$R_{lm}(S)$. The semilattices Rlm(S)$R_{lm}(S)$ exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of normalΣ20$\Sigma ^0_2$‐computable families. We show that every Rogers semilattice of a normalΣ20$\Sigma ^0_2$‐computable family is isomorphic to some semilattice Rlm(S)$R_{lm}(S)$. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices Rlm(S)$R_{lm}(S)$. In particular, there is an l.m. family S such that Rlm(S)$R_{lm}(S)$ is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset Rlm(S)$R_{lm}(S)$ is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all normalΣ20$\Sigma ^0_2$‐computable numberings. We prove that inside this class, the index set of l.m. numberings is normalΣ40$\Sigma ^0_4$‐complete.

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Section: Introductionmentioning

confidence: 99%