Effective Inseparability for Sequences of Sets

The notions of effective inseparability (for pairs of sets) and of double productivity are defined in [5]. In this paper are considered some extensions of these notions for sequences of sets, and also certain alternative formulations of them in the spirit of Mucnik [3]. Let M be a nonempty recursively enumerable (r.e.) set which will be fixed in each context. By a sequence of sets we mean a sequence of sets of natural numbers indexed by M; by a disjoint sequence we mean one whose members are pairwise disjoint. Our suffixes will range over M, thus for a sequence of sets we shall use a notation such as \Ax} ; A* will be used to denote the set of numbers which belong to Ax but not to any other member of the sequence \Ax\. The complement of a set A will be denoted by A'. We use Vx and Ax as universal and existential quantifiers over M. Superscripts will range over the set N of all natural numbers; variables left free should be interpreted as universally quantified over N, or over M for suffixes. A sequence of sets {Ax\ is called r.e. if there is a binary r.e. relation A (x, y) such that