 Commuting equivalence relations

For two equivalence relations on the same set, their composite contains each and is reflexive. It is symmetric exactly when the relations commute. It is transitive exactly when $S\circ R\subseteq R\circ S$, which by inversion is also equivalent to commutation. Under these conditions the composite is the smallest equivalence relation containing both. For congruences modulo $m,n$ on the integers, their composite is congruence modulo $\gcd (m,n)$.

// Target: combinatorics.bigb