arXiv subjects

Mars Yamaleev

Publications and source records attributed to Mars Yamaleev.

3 recordsLinked to original sources

arXiv2019

Isomorphism types of Rogers semilattices in the analytical hierarchy

A numbering of a countable family $S$ is a surjective map from the set of natural numbers $\omega$ onto $S$. A numbering $\nu$ is reducible to a numbering $\mu$ if there is an effective procedure which given a $\nu$-index of an object from $S$, computes a $\mu$-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families $S \subset P(\omega)$ belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers $m\neq n$, any non-trivial Rogers semilattice of a $\Pi^1_m$-computable family cannot be isomorphic to a Rogers semilattice of a $\Pi^1_n$-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.

math.LO↗

arXiv2019

Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies

A standard tool for classifying the complexity of equivalence relations on $ω$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $Γ$ be one of the following classes: $Σ^0_α$, $Π^0_α$, $Σ^1_n$, or $Π^1_n$, where $α\geq 2$ is a computable ordinal and $n$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $Γ$.

math.LO↗

arXiv2018

Classifying equivalence relations in the Ershov hierarchy

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $\leq_c$. This gives rise to a rich degree-structure. In this paper, we lift the study of $c$-degrees to the $Δ^0_2$ case. In doing so, we rely on the Ershov hierarchy. For any notation $a$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $\leq_c$ on the $Σ^{-1}_{a}\smallsetminus Π^{-1}_a$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of $c$-degrees.

math.LO↗