Convex subsets of 2n and bounded truth-table reducibility

Let 2ⁿ be the set of n-tuples of 0's and 1's, partially ordered componentwise. A characterization is given of the possible decompositions of arbitrary subsets of 2ⁿ as disjoint unions of sets which are convex in this ordering; this result is used to obtain a decomposition theorem for Boolean functions in terms of monotone functions. The second half of the paper contains applications to recursion theory; in particular, canonical forms for certain minimum-norm bounded-truth-table reductions are obtained.