Some properties of precompletely and positively numbered sets

In this paper, we prove a joint generalization of Arslanov's completeness criterion and Visser's ADN theorem for precomplete numberings which, for the Gödel numbering $x\mapsto W_x$, has been proved by Terwijn (2018). The question of whether this joint generalization takes place in each precomplete numbering has been raised in his joint paper with Barendregt in 2019. Then we consider the properties of completeness and precompleteness of numberings in the context of the positivity property. We show that no completion of a positive numbering is a minimal cover of that numbering, and that the Turing completeness of any set A is equivalent to the existence of a positive precomplete A-computable numbering of any infinite family with positive A-computable numbering. In addition, we prove that each ${\Sigma }_n^0$-computable numbering ($n⩾2$) of a ${\Sigma }_n^0$-computable non-principal family has a ${\Sigma }_n^0$-computable minimal cover ν such that for every computable function f there exists an integer n with $\nu \left(f\right(n\left)\right)=\nu \left(n\right)$.