The failure of stable composition for equivalence relations in simple theories

@misc{mirabi2026failure,
  title = {{The failure of stable composition for equivalence relations in simple theories}},
  author = {Mostafa Mirabi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.29087},
  eprint = {2606.29087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.29087},
}

TY  - UNPB
TI  - The failure of stable composition for equivalence relations in simple theories
AU  - Mostafa Mirabi
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2606.29087
UR  - https://arxiv.org/abs/2606.29087
AB  - 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 ℵ_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.
ER  -