التفاصيل البيبلوغرافية
| العنوان: |
Minimal degrees and downwards density in some strong positive reducibilities and quasi-reducibilities. |
| المؤلفون: |
Chitaia, Irakli, Ng, Keng Meng, Sorbi, Andrea, Yang, Yue |
| المصدر: |
Journal of Logic & Computation; Jul2023, Vol. 33 Issue 5, p1060-1088, 29p |
| مصطلحات موضوعية: |
ISOMORPHISM (Mathematics) |
| مستخلص: |
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. [ABSTRACT FROM AUTHOR] |
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