Friedberg Numberings of Families of $n$-Computably Enumerable Sets

We establish a number of results on numberings, in particular, on Friedberg numberings, of families of d.c.e. sets. First, it is proved that there exists a Friedberg numbering of the family of all d.c.e. sets. We also show that this result, patterned on Friedberg's famous theorem for the family of all c.e. sets, holds for the family of all $n$-c.e. sets for any $n>2$. Second, it is stated that there exists an infinite family of d. c. e. sets without a Friedberg numbering. Third, it is shown that there exists an infinite family of c. e. sets (treated as a family of d. c. e. sets) with a numbering which is unique up to equivalence. Fourth, it is proved that there exists a family of d. c. e. sets with a least numbering (under reducibility) which is Friedberg but is not the only numbering (modulo reducibility).