Affiliations: [a] Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA. [email protected] | [b] Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA. [email protected] | [c] Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA. [email protected] | [d] Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria. [email protected]
Abstract: 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 Kalimullin 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.