The jump operator on the -enumeration degrees

The jump operator on the ω -enumeration degrees was introduced in [I.N. Soskov, The ω -enumeration degrees, J. Logic Computat. 17 (2007) 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure D ω ′ of the ω -enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of D ω ′ and of the enumeration degrees are isomorphic. second part of the paper we study the jumps of the ω -enumeration degrees below 0 ω ′ . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω -enumeration degrees below 0 ω ′ which are high n for some n and of the class L of the ω -enumeration degrees below 0 ω ′ which are low n for some n .