@TECHREPORT{Schaefer97simplesets, author = {Marcus Schaefer}, title = {Simple Sets Are Not Btt-Cuppable}, institution = {}, year = {1997} }

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Abstract

We extend Post's result that a simple set cannot be btt-complete by showing that in fact it cannot be btt-cuppable, i.e. if the join of a c.e. set and a simple set is btt-complete, then the non-simple set is btt-complete itself. The proof also yields that simple sets are not d-cuppable (i.e. not cuppable with regard to disjunctive reductions). Post showed that a simple set cannot be btt-complete. In a paper by Stephen Fenner and the author [3] this was generalized to non-c.e. sets by isolating the immunity property which is responsible for the incompleteness. Another approach to the btt-incompleteness of simple sets would have been through degrees. How incomplete are simple sets? Putting it dioeerently: can the join of a btt-incomplete degree with a simple degree be btt-complete? We will show that the answer is no. Deønition 1 A set A is called r-cuppable, if there is a c.e. set B such that ; 0 r A \Phi B and ; 0 6 r B, where r is a class of reductions (like m, 1, btt, c, d, tt...