Classifying pairs of equivalence relations (Q1182704)

This paper is a follow up of the author's ``Stability for pairs of equivalence relations'' [ibid. 32, No. 1, 112--128 (1991; Zbl 0728.03027)], where the problem of determining the models of the theory \(T\) (of pairs of equivalence relations) which are categorical in some infinite cardinal is dealt with. In the present article the author is concerned with the stability questions of the models of \(T\). It is known that \(T\) is undecidable [\textit{H. Rogers jun.}, Ann. Math. (2) 64, 264--284 (1956; Zbl 0074.01403)] and therefore the problem is complicated in the general setting. This forces the author to impose a further condition: If \(E\) denotes the smallest equivalence relation whose classes contain the \(E_0\) and \(E_1\)-classes, then there is an integer \(n\) such that any \(E\)-class contains at most \(n E_0\) or \(E_1\)-classes. Let \(T_+\) denote the theory of two equivalence relations together with the above axiom. The author shows that \(T_+\) is superstable, monadically stable, presentable, shallow with depth \(\le 3\) and satisfies the existence property, and hence is classifiable in the sense of Shelah.