Notes on conjunctive and Quasi degrees

Abstract

Abstract In this article we prove the following results: (i) Every hemimaximal set has minimal $c_{1}$-degree, i.e. if $B$ is hemimaximal and $A$ is a c.e. set such that $A \le _{c_{1}} B$ then either $B \leq _{{c}_{1}} A$ or $A$ is computable. (ii) The $sQ$-degree of a c.e. set contains either only one or infinitely many c.e. $c$-degrees. (iii) If $A,B$ are c.e. cylinders in the same $sQ_{1}$-degree and $A<_{c_{1}} B$, then this $sQ_{1}$-degree contains infinitely many c.e. $c_{1}$-degrees.