Enumeration 1-Genericity in the Local Enumeration Degrees

We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Delta such that, for any A, the set Delta(A) is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such that a' = b', but also that, if a is nonzero, then we can find such b satisfying 0(e) < b < a. We conclude by proving the existence of both a nonzero low and a properly Sigma(0)(2) nonsplittable enumeration 1-generic degree, hence proving that the class of 1-generic degrees is properly subsumed by the class of enumeration 1-generic degrees.