A Minimal Pair of Turing Degrees

Abstract

Let $P(A)$ be the following property, where $A$ is any infinite set of natural numbers: \begin{displaymath} (\forall X)[X\subseteq A\wedge |A-X|=\infty\Rightarrow A\not\le_m X]. \end{displaymath} Let $({\bf R}, \le)$ be the partial ordering of all the r.e. Turing degrees. We propose the study of the order theoretic properties of the substructure $({\bf S}_{m},\le_{{\bf S}_m})$, where ${\bf S}_{m}=_{\rm dfn}\{{\bf a}\in {\bf R}$: ${\bf a}$ contains an infinite set $A$ such that P(A) is true$\}$, and $\le_{{\bf S}_m}$ is the restriction of $\le$ to ${\bf S}_m$. In this paper we start by studying the existence of minimal pairs in ${\bf S}_{m}$.