An Inhomogeneity in the Structure of Karp Degrees

We show that there is a recursive nonzero Karp (polynomial time many-one) degree which is not supremum of a minimal pair, i.e. of an incomparable pair of degrees with infimum 0, the degree of polynomial time computable sets. By existence of minimal pairs, this implies that there are nonisomorphic initial segments of Karp degrees.