§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters (or, dually, maximal ideals). There is also a substantial interest in nicely definable (Borel, analytic) ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space (Rosenthal compacta), see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.
1. The partial order induced by I on P(ω): X ≥I Y iff X \ Y ϵ I () and the partial order (I, ⊂)().
2. Boolean algebras of the form P(ω)/I and their automorphisms ([6, 5, 19, 20]).
3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I ([4, 14, 15,9]).
In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3.