We show that the jump operator in the enumeration degrees is order-theoretic definable in the enumeration degrees. This solves a question posed by S. B. Cooper.
Namely, for all e-degrees u and x the following is equivalent: 1) x ≥ u', 2) x ≥ u and for all e-degrees a > u, b > u and c > u if each of pairs (a, b) and (a, c) is u-e-ideal, then bUx = cUx.
(We say that a pair of e-degrees (a,b) is u-e-ideal iff every e-degreel z ≥ u is the greatest lower bound of e-degrees aUz and bUz.)
Thus, the class of e-degrees above u' is definable by an AEA-formula (in the language of ordering).
Furthermore, we show that for all e-degrees u and x the following is equivalent: 1) x ≤ u' 2) there exists a triple of e-degrees a > u, b > u and c > u, such that each of pairs (a, b), (a, c), (b, c) is u-e-ideal, and x ≤ aUb Uc.
Thus, the class of e-degrees below u' is definable by an EAE-formula (in the language of ordering).
Hence, the map u onto u' is definable in the e-degrees by a formula which is a conjunction of an AEA-formula and an EAE-formula. Also, by Friedberg Completeness Criterion we can define the class of all total e-degrees above 0' as the image of the jump operator.