Semidecidable least-representative transversal ($S=\{n:\forall m<n(m\nsim n)\}$)

For a co-computably enumerable equivalence relation, the least element of each equivalence class forms a complete transversal of a set family. Membership is semidecidable: run the finitely many inequivalence tests against smaller numbers in parallel and accept when all succeed. Infinitely many classes are needed only to make the resulting transversal infinite.