A computably enumerable vector space with the strong antibasis property

Abstract

Downey and Remmel have completely characterized the degrees of c.e. bases for c.e. vector spaces (and c.e. fields) in terms of weak truth table degrees. In this paper we obtain a structural result concerning the interaction between the c.e. Turing degrees and the c.e. weak truth table degrees, which by Downey and Remmel's classification, establishes the existence of c.e. vector spaces (and fields) with the strong antibasis property (a question which they raised). Namely, we construct c.e. sets \(B<_{\rm T}A\) such that the c.e. W-degrees below that of A are disjoint from the nonzero c.e. T-degrees below that of A and comparable to that of B.