Bounded enumeration reducibility and its degree structure

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by $\leq_{be}$, which is a natural extension of $s$-reducibility $\leq_{s}$. We show that $\leq_{s}$, $\leq_{be}$, and enumeration reducibility do not coincide on the $\Pi^0_1$-sets, and the structure $\mathcal{D}_{be}$ of the $\be$-degrees is not elementarily equivalent to the structure of the $s$--degrees. We show also that the first order theory of $\mathcal{D}_{be}$ is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper.