The paper studies \(\varvec{\Sigma ^0_n}\)-computable families (\(\varvec{n\geqslant 2}\)) and their numberings. It is proved that any non-trivial \(\varvec{\Sigma ^0_n}\)-computable family has a complete with respect to any of its elements \(\varvec{\Sigma ^0_n}\)-computable non-principal numbering. It is established that if a \(\varvec{\Sigma ^0_n}\)-computable family is not principal, then any of its \(\varvec{\Sigma ^0_n}\)-computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal \(\varvec{\Sigma ^0_n}\)-computable numberings. It is also shown that for any \(\varvec{\Sigma ^0_n}\)-computable numbering \(\varvec{\nu }\) of a \(\varvec{\Sigma ^0_n}\)-computable non-principal family there exists its \(\varvec{\Sigma ^0_n}\)-computable numbering that is incomparable with \(\varvec{\nu }\). If a non-trivial \(\varvec{\Sigma ^0_n}\)-computable family contains the least and greatest elements under inclusion, then for any of its \(\varvec{\Sigma ^0_n}\)-computable non-principal non-least numberings \(\varvec{\nu }\) there exists a \(\varvec{\Sigma ^0_n}\)-computable numbering of the family incomparable with \(\varvec{\nu }\). In particular, this is true for the family of all \(\varvec{\Sigma ^0_n}\)-sets and for the families consisting of two inclusion-comparable \(\varvec{\Sigma ^0_n}\)-sets (semilattices of the \(\varvec{\Sigma ^0_n}\)-computable numberings of such families are isomorphic to the semilattice of \(\varvec{m}\)-degrees of \(\varvec{\Sigma ^0_n}\)-sets).

本文研究了\(\varvec{\Sigma ^0_n}\)可计算族 ( \(\varvec{n\geqslant 2}\) ) 及其编号。证明任何非平凡的\(\varvec{\Sigma ^0_n}\) -可计算族对于其任何元素都有一个完整的\(\varvec{\Sigma ^0_n}\) -可计算非主函数编号。可以确定，如果一个\(\varvec{\Sigma ^0_n}\)可计算族不是主要的，则其任何一个\(\varvec{\Sigma ^0_n}\)可计算编号都具有最小覆盖范围，并且，如果该族是无限的，则与其最小的\(\varvec{\Sigma ^0_n}\)可计算编号之一无法比较。还表明，对于任何\(\varvec{\Sigma ^0_n}\)可计算编号\(\varvec{\nu }\)的\(\varvec{\Sigma ^0_n}\)可计算非主要族存在其\(\varvec{\Sigma ^0_n}\)可计算编号，与\(\varvec{\nu }\)无法比较。如果一个非平凡的\(\varvec{\Sigma ^0_n}\)可计算族包含包含的最小和最大元素，则对于其任何一个\(\varvec{\Sigma ^0_n}\)可计算非主要非最小编号\(\varvec{\nu }\)存在一个\(\varvec{\Sigma ^0_n}\)与\(\varvec{\nu }\)无法比较的族的可计算编号。特别是，对于所有\(\varvec{\Sigma ^0_n}\)集合的族以及由两个包含可比较的\(\varvec{\Sigma ^0_n}\)集合组成的族（这些族的\(\varvec{\Sigma ^0_n}\)可计算编号的半格与\(\varvec{m}\)度\(\varvec{\Sigma ^0_n}\)的半格同构） -套）。