Stability for pairs of equivalence relations (Q804568)

In the context of stability, the theory of one equivalence relation is well understood; in particular, characterizations of \(\aleph_ 0\)- and \(\aleph_ 1\)-categoricity, and of bounded dimensionality are given in rather simple terms. The author achieves these three characterizations for the theory of two equivalence relations \(E_ 0\) and \(E_ 1\) under the assumption: their (lattice-theoretic) join E has the property that there is an \(h\in \omega\) such that each equivalence class of E includes at most h classes of \(E_ 0\) or of \(E_ 1\). First he uses as additional assumption that \(E_ 0\) and \(E_ 1\) commute, and eventually shows that it is innocuous. Here are samples of characterizations. The theory is \(\aleph_ 0\)-categorical iff it does not satisfy the finite cover property. For it to be \(\aleph_ 1\)-categorical or of bounded dimension, it must satisfy rather complicated conditions besides \(\aleph_ 0\)- categoricity. In somewhat different direction, the author shows that every theory under consideration is superstable. He employs a number of interesting model constructions throughout the paper.