The Failure of Stable Composition for Equivalence Relations in Simple Theories

Abstract

Casanovas and Potier proved that algebraic quantification preserves stability of formulas. They also gave a nonsimple example, answering a question of Laskowski, showing that the algebraicity hypothesis cannot simply be replaced by NFCP, and asked whether a similar example exists in a simple theory. We give such an example in elementary form. The edge-set structure of the random bipartite graph has two definable equivalence relations, both stable and NFCP, whose relational composition has the order property. The resulting theory is simple and $\aleph_0$-categorical. We also prove a formal sharpness observation: every formula in every first-order theory is an existential composition of two stable NFCP formulas algebraic in the two outer variables. Consequently, closure of stable NFCP formulas under this composition characterizes stability of the whole theory.