On numberings for classes of families of total functions

In the present article, we study computable numberings and the Rogers semilattices for classes of families of everywhere defined (total) computable functions. We prove that the isomorphism type of the Rogers semilattice for a finite class $\mathfrak{F}$ of computable families of total functions depends only on the order with respect to inclusion on the class $\mathfrak{F}$ itself and the class $C(\mathfrak{F})$ of the closures of its elements regarded as subsets of a Baire space. We obtain necessary and sufficient conditions for existence of universal numberings for finite classes of computable families of total functions. We also consider a question whether a class of families of total functions admitting a universal numbering is closed under the union of computable increasing sequences of its elements. For a computable class $\mathfrak{F}$ such that $C(\mathfrak{F})$ is finite, we prove that the Rogers semilattice is either trivial or infinite; moreover, in the latter case, this semilattice is not a lattice.