On the Quotient Structure of Computably Enumerable Degrees Modulo the Noncuppable Ideal

We show that minimal pairs exist in the quotient structure of R modulo the ideal of noncuppable degrees. In the study of mathematical structures it is very common to form quotient structures by identifying elements in some equivalence classes. By varying the equivalence relations, the corresponding quotient structures often reveal certain hidden features of the original structure. In this paper, we focus on of the upper semi-lattice of computably enumerable degrees and the equivalence relations are induced by definable ideals. We begin with introducing some notations and terminologies. Let R be the class of computably enumerable degrees or simply c.e. degrees. Definition 1. We say that a nonempty subset I of R is an ideal of R if I is downward closed and closed under join. In other words, the following conditions are satisfied. (a) If a is in I and b ≤ a then b is in I; (b) If a and b are in I, then their least upper bound, denoted by a ∨ b, is in I. We say that an ideal I is definable if there is a first-order formula φ(x) over the partial order language L = {≤} such that a c.e. degree a ∈ I if and only if R |= φ(a). Each ideal I of R naturally induced an equivalence relation ≡I as follows. For any two c.e. degrees a and b, define a ≤I b if and only if ∃x ∈ I(a ≤T b ∨ x), 1991 Mathematics Subject Classification. 03D25. A. Li is partially supported by National Distinguished Young Investigator Award no. 60325206 (China). Y. Yang is partially supported by NUS Academic Research Grant R-146-000-078-112 “Enumerability and Reducibility” (Singapore) and R252-000-212-112. G. Wu is partially supported by a start-up grant from Nanyang Technological University (Singapore). All three authors are partially supported by NSFC grant no. 60310213 “New Directions in Theory and Applications of Models of Computation” (China). The work was done partially while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore in 2005. The visit was supported by the Institute.