Maximal Pairs of Computably Enumerable Sets in the Computably Lipschitz Degrees

Abstract

A set A is computably Lipschitz or cl-reducible, for short, to a set B if A is Turing reducible to B by an oracle Turing machine with use function ? such that ? is bounded by the identity function up to an additive constant, i.e., ?(n)?n+O(1). In this paper we study maximal pairs of computably enumerable (c.e.) cl-degrees or maximal pairs, for short, i.e., pairs of c.e. cl-degrees such that there is no c.e. cl-degree that is above both cl-degrees in this pair. Our main results are as follows. (1) A c.e. Turing degree contains a c.e. cl-degree that is half of a maximal pair if and only if this Turing degree contains a maximal pair if and only if this Turing degree is array noncomputable. (2) The cl-degrees of all weak truth-table complete sets are halves of maximal pairs while there is a Turing complete set A such that the cl-degree of A is not half of any maximal pair. In fact, any high c.e. Turing degree contains a c.e. cl-degree that is not half of a maximal pair. (3) Above any c.e. cl-degree there is a maximal pair. (4) There is a maximal pair which at the same time is a minimal pair. (5) There is a pair of c.e. cl-degrees that is not maximal and does not possess a least upper bound. Moreover, we make some observations on the structure of the c.e. cl-degrees in general. For instance, we give a very simple proof of the fact that there are no maximal c.e. cl-degrees.