Notes on conjunctive and Quasi degrees

Chitaia, Irakli O.; Omanadze, R. Sh.; Sorbi, Andrea

doi:10.1093/logcom/exab026

Journal of Logic and Computation

2021

DOI: 10.1093/logcom/exab026

|View full text |Cite

Notes on conjunctive and Quasi degrees

Irakli O. Chitaia

R. Sh. Omanadze

Andrea Sorbi

Abstract: In this article we prove the following results: (i) Every hemimaximal set has minimal $c_{1}$-degree, i.e. if $B$ is hemimaximal and $A$ is a c.e. set such that $A \le _{c_{1}} B$ then either $B \leq _{{c}_{1}} A$ or $A$ is computable. (ii) The $sQ$-degree of a c.e. set contains either only one or infinitely many c.e. $c$-degrees. (iii) If $A,B$ are c.e. cylinders in the same $sQ_{1}$-degree and $A<_{c_{1}} B$, then this $sQ_{1}$-degree contains infinitely many c.e. $c_{1}$-degrees.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2021

2024

Publication Types

Select...

Article4

Relationship

Self Cite0

Independent4

Authors

Journals

Cited by 4 publications

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Non-empty open intervals of computably enumerable sQ1-degrees

Omanadze,

Chitaia

2024

Logic Journal of the IGPL

View full text Add to dashboard Cite

We prove that if $A$, $B$ are noncomputable c.e. sets, $A<_{sQ_{1}}B$ and [($B$ is not simple and $A\oplus B\leq _{sQ_{1}}B$) or $B\equiv _{sQ_{1}}B\times \omega $], then there exist infinitely many pairwise $sQ_{1}$-incomparable c.e. sets $\{C_{i}\}_{i\in \omega }$ such that $A<_{sQ_{1}}C_{i}<_{sQ_{1}}B$, for all $i\in \omega $. We also show that there exist infinite collections of $sQ_{1}$-degrees $\{\boldsymbol {a_{i}}\}_{i\in \omega }$ and $\{\boldsymbol {b_{i}}\}_{i\in \omega }$ such that for every $i, j,$ (1) $\boldsymbol {a_{i}}<_{sQ_{1}}\boldsymbol {a_{i+1}}$, $\boldsymbol {b_{j+1}}<_{sQ_{1}}\boldsymbol {b_{j}}$ and $\boldsymbol {a_{i}}<_{sQ_{1}}\boldsymbol {b_{j}}$; (2) every c.e set in $\boldsymbol {a_{i}}$ is a maximal set; and (3) every c.e. set in $\boldsymbol {b_{j}}$ is a non-maximal hyperhypersimple set. We prove that a noncomputable c.e. $sQ$-degree contains either only one or infinitely many c.e. $sQ_{1}$-degrees. The $sQ_{1}$-degrees of c.e. cylinders are dense upper semilattice.

show abstract

Non-empty open intervals of computably enumerable sQ1-degrees

Omanadze,

Chitaia

2024

Logic Journal of the IGPL

View full text Add to dashboard Cite

show abstract

Minimal degrees and downwards density in some strong positive reducibilities and quasi-reducibilities

Chitaia

Sorbi

et al. 2022

Journal of Logic and Computation

View full text Add to dashboard Cite

We consider three strong reducibilities, $s_{1}, s_{2}, Q_{1}$ (where we identify a reducibility $\leqslant _r$ with its index $r$). The first two reducibilities can be viewed as injective versions of $s$-reducibility, whereas $Q_1$-reducibility can be viewed as an injective version of $Q$-reducibility. We have, with proper inclusions, $s_{1} \subset s_{2} \subset s$. It is well known that there is no minimal $s$-degree, and there is no minimal $Q$-degree. We show on the contrary that there exist minimal $\varDelta ^{0}_{2}$$s_{2}$-degrees and minimal $\varDelta ^{0}_{2}$$s_{1}$-degrees. On the other hand, both the $\varPi ^{0}_{1}$$s_{2}$-degrees and the $\varPi ^{0}_{1}$$s_{1}$-degrees are downwards dense. By the isomorphism of the $s_1$-degrees with the $Q_1$-degrees induced by complementation of sets, it follows that there exist minimal $\varDelta ^0_2$$Q_1$-degrees, but the c.e. $Q_{1}$-degrees are downwards dense.

show abstract

Conjunctive degrees and cylinders

Chitaia,

Omanadze,

Sorbi

2023

Journal of Logic and Computation

View full text Add to dashboard Cite

In this article, we define and study the notion of a $(c,c_{1})$-cylinder, which turns out to be very useful instrument for investigating the relationships between conjunctive reducibility ($c$-reducibility) and its injective version $c_{1}$-reducibility. Using this notion, we prove the following results: (i) Neither hypersimple sets nor hemimaximal sets can be $(c,c_{1})$-cylinders; (ii) The $c$-degree of a noncomputable c.e. set contains either only one or infinitely many noncomputable $c_{1}$-degrees; (iii) the $c$-degree of either a hemimaximal set or a hypersimple set contains infinitely many noncomputable $c_{1}$-degrees.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Notes on conjunctive and Quasi degrees

Cited by 4 publications

References 12 publications

Non-empty open intervals of computably enumerable sQ1-degrees

Non-empty open intervals of computably enumerable sQ1-degrees

Minimal degrees and downwards density in some strong positive reducibilities and quasi-reducibilities

Conjunctive degrees and cylinders

Contact Info

Product

Resources

About