Uniqueness typing for intersection types

Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Veneri [1981], we prove several facts about sets of terms having a given intersection type. Our main result is that every strongly normalizing term $M$ admits a uniqueness typing, which is a pair $(\mathrm{\Gamma },A)$ such that

1. $\mathrm{\Gamma }⊢M:A$
2. $\mathrm{\Gamma }⊢N:A⟹M{=}_{\beta \eta }N$ We also discuss several presentations of intersection type algebras, and the corresponding choices of type assignment rules.