On $p$-universal and $p$-minimal numberings

We study the $p$-reducibility of numberings which was introduced and first studied by Degtev. $p$-Reducibility is an effectively bounded version of the $e$-reducibility of numberings. Also, we prove that for every set $A$ there exists an $A$-computable family without universal numberings but admitting $p$-universal numberings and obtain a criterion for the existence of $p$-universal numberings of finite families of $A$-c.e. sets. Finally, we show that every $A$-computable family, with $\emptyset''\leq _TA$, has infinitely many pairwise non-$p$-equivalent $p$-minimal $A$-computable numberings.