Some properties of maximal sets

We introduce agreement reducibility and highlight its major features. Given subsets A and B of N, we write A ≤ agree B if there is a total computable function f : N → N satisfying for all e, e , W e ∩ A = W e ∩ A if and only if W f (e) ∩ B = W f (e) ∩ B. We shall discuss the central role N plays in this reducibility and its connection to strong-hyper-hyper-immunity. We shall also compare agreement reducibility to other well-known reducibilities, in particular s 1-and s-reducibility. We came upon this reducibility while studying the computable reducibility of a class of equivalence relations on N based on set-agreement. We end by describing the origin of agreement reducibility and presenting some results in that context.

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

Chitaia

Sorbi

et al. 2022

Journal of Logic and Computation

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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.

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Some properties of maximal sets

Cited by 3 publications

References 7 publications

r‐Maximal sets and Q1,N‐reducibility

r‐Maximal sets and Q1,N‐reducibility

Agreement reducibility

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

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