Conjunctive reducibilities and completeness

In this article we study the notion of completeness for conjunctive reducibilities. We investigate the relationship between c-completeness and r-completeness of computably enumerable (c.e.) sets with respect to various strong reducibilities ¤r. By using simplicity properties of sets, we prove that there exist c.e. sets that are simultaneously Q-complete and bd-complete, yet fail to be c-complete. Similarly, there exist c.e. sets that are simultaneously Q-complete and bwttcomplete (respectively, btt-complete) but not c-complete. Furthermore, we study two restrictions of c-reducibility, namely c1- and c1,N-reducibility, and show that they are distinct on the c.e. sets. Nevertheless, we prove that the notions of completeness for c, c1, and c1,N coincide.