We introduce agreement reducibility and highlight its major features. Given subsets A and B of, we write if there is a total computable function satisfying for all,.We shall discuss the central role plays in this reducibility and its connection to strong‐hyper‐hyper‐immunity. We shall also compare agreement reducibility to other well‐known reducibilities, in particular s1‐ and s‐reducibility. We came upon this reducibility while studying the computable reducibility of a class of equivalence relations on based on set‐agreement. We end by describing the (...) origin of agreement reducibility and presenting some results in that context. (shrink)

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b' = 0'). This extends Csima's result that every CAD theory has a low prime model. We (...) also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low₂ (c" > 0"), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b' is any degree c.e. in c with d' ≤ b'. As a corollary, we show that for any degree c with 0 < c < 0', every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models. (shrink)