Arithmetical numberings are computable numberings of the families of sets from a fixed class of the arithmetical hierarchy. Classical computable numberings of the families of c.e. sets provide very important special case of arithmetical numberings.
According to the concepts of the general theory of numberings, arithmetical numberings are considered with respect to relation of reducibility of numberings. Arithmetical numberings of a family treated up to equivalence form an algebraic structure called Rogers semilattice of that family.
Our talk is devoted to the problems conserning the extremal elements of Rogers semilattices. We investigate also completions of arithmetical numberings and show interconnections between complete numberings, and universal numberings and meetreducibilty property of Rogers semilattices.