Example

(1) Addition and multiplication of natural numbers are commutative laws on $\mathbf{N}$ (Set Theory, III, § 3, no. 3, Corollary to Proposition 5).

(2) On a lattice the laws sup and inf are commutative; so, in particular, are the laws $\cup$ and $\cap$ between subsets of a set E.

(3) Let E be a set of cardinal $> 1$. The law $(f, g) \mapsto f \circ g$ between mappings of E into E is not commutative as is seen by taking f and g to be distinct constant mappings, but the identity mapping is permutable with every mapping.

(4) Let $(x, y) \mapsto x \top y$ be a commutative law on $E$; the law

between subsets of $E$ is commutative.