A note on the join property

A Turing degree a satisfies the join property if, for every non-zero b<a, there exists c<a with b V c = a. It was observed by Downey, Greenberg, Lewis and Montalbán that all degrees which are non-GL2 satisfy the join property. This, however, leaves open many questions. Do all a.n.r. degrees satisfy the join property? What about the PA degrees or the Martin-Löf random degrees? A degree b satisfies the cupping property if, for every a>b, there exists c<a with b V c = a. Is satisfying the cupping property equivalent to all degrees above satisfying join? We answer all of these questions by showing that above every low degree there is a low degree which does not satisfy join. We show, in fact, that all low fixed point free degrees a fail to satisfy join and, moreover, that the non-zero degree below a without any joining partner can be chosen to be a c.e. degree.