Commuting equivalence relations ($R\circ S=S\circ R$)

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