Initial Segments in Rogers Semilattices of $\Sigma^0_n$-Computable Numberings

S. Goncharov and S. Badaev showed that for $n\geqslant 2$, there exist infinite families whose Rogers semilattices contain ideals without minimal elements. In this connection, the question was posed as to whether there are examples of families that lack this property. We answer this question in the negative. It is proved that independently of a family chosen, the class of semilattices that are principal ideals of the Rogers semilattice of that family is rather wide: it includes both a factor lattice of the lattice of recursively enumerable sets modulo finite sets and a family of initial segments in the semilattice of $m$-degrees generated by immune sets.