Positive reducibilities, extreme numberings, and completeness

In the present article, we study universal, minimal, and complete numberings of families of arithmetic sets. We show that, for every $m\in\mathbb{N}$ and every nontrivial $\Sigma^0_{m+2}$-computable family $\mathcal{S}$, there exists a $\Sigma^0_{m+2}$-computable numbering that is not universal with respect to positive reducibilities and is complete with respect to each element $B\in\mathcal{S}$. For finite families of computably enumerable sets, we obtain necessary and sufficient conditions for existence of numberings that are complete, computable, and not universal with respect to positive reducibilities. For every infinite $\Sigma^0_{m+2}$-computable family $\mathcal{S}$ and every element $B\in\mathcal{S}$, we construct a $\Sigma^0_{m+2}$-computable numbering that is complete with respect to $B$ and minimal with respect to classical and positive reducibilities.