Conjunctive reducibilities and completeness

Abstract

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 $\le _{r}$. 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 $bwtt$-complete (respectively, $btt$-complete) but not $c$-complete. Furthermore, we study two restrictions of $c$-reducibility, namely $c_{1}$- and $c_{1,N}$-reducibility, and show that they are distinct on the c.e. sets. Nevertheless, we prove that the notions of completeness for $c$, $c_{1}$, and $c_{1,N}$ coincide.