Ring don’t require multiplication to commute, only addition. (Square) matrices make just fine rings despite noncommutativity. But the semiring of types does hav

Ring don’t require multiplication to commute, only addition. (Square) matrices make just fine rings despite noncommutativity. But the semiring of types does have commutative multiplication (only a semiring, or "rig", because the lack of additive inverses). Those two structs are equivalent (isomorphic) in the algebraic sense – names don’t matter, and order (indices are just a kind of names) doesn’t either. (a, b) = a x b = b x a = (b, a).