On Kalimullin pairs

Mingzhong Cai, Steffen Lempp, Joseph S. Miller, Mariya I. Soskova
2016 Computability - The Journal of the Assosiation  
We study Kalimullin pairs, a definable class (of pairs) of enumeration degrees that has been used to give first-order definitions of other important classes and relations, including the enumeration jump and the total enumeration degrees. We show that the global definition of Kalimullin pairs is also valid in the local structure of the enumeration degrees, giving a simpler local definition than was previously known. We prove that the typical enumeration degree is not half of a nontrivial
more » ... in pair, both in the sense of category and measure. Using genericity, we show that not all members of nontrivial Kalimullin pairs are half of a maximal Kalimullin pair. Finally, we construct such a set that has no maximal Kalimullin partner, making it qualitatively different from half of a maximal Kalimullin pair. Definition 2.10 (Ganchev, Soskova [7]). A K-pair of enumeration degrees {a, b} is a maximal K-pair if for every K-pair of enumeration degrees {c, d}, such that a ≤ c and b ≤ d, we have that a = c and b = d. Cai, Ganchev, Lempp, Miller and Soskova [2] show that every maximal K-pair is of the form {d e (A), d e (A)} for some semi-computable set A. Thus the nonzero total enumeration degrees can be defined as the least upper bounds of maximal K-pairs. K-pairs in D e (≤ 0 ) K-pairs in the local structure of the enumeration degrees have particularly nice properties. It is immediate from Proposition 2.6 that if {a, b} is a nontrivial K-pair in D e (≤ 0 ), then a and b are ∆ 0 2 -enumeration degrees and, in fact, even low enumeration degrees, i.e., a = b = 0 e . The main tool for constructing K-pairs in D e (≤ 0 ) is given by the following dynamic characterization:
doi:10.3233/com-150046 fatcat:ycbsja6hxrgwzpfgsdg7c64244