Bounded enumeration reducibility and its degree structure

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤_be, which is a natural extension of s-reducibility ≤_s. We show that ≤_s, ≤_be, and enumeration reducibility do not coincide on the ${Pi^0_1}$ –sets, and the structure ${boldsymbol{mathcal{D}_{rm be}}}$ of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of ${boldsymbol{mathcal{D}_{rm be}}}$ is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987).