Some properties of classes of minimal numberings of arithmetical set families

We prove that for each u ⩾ 2 the class of all single-valued Σ0 u computable numberings of any infinite family of total functions is effectively infinite and the class of all its Σ0 u - 1-computable numberings is generated by the downward closure with respect to the reducibility of the set of all infinite direct sums of uniformly Σ0 u - 1-computable sequences of its single-valued numberings. It is established that if u > 2, then the class of all Σ0 u -computable numberings of any infinite family is generated by infinite direct sums of uniformly Σ0 u -computable and uniformly Σ0 u -minimal sequences of its numberings.