Example

(1) The mappings (X, Y) ↦ X ∪ Y and (X, Y) ↦ X ∩ Y are laws of composition on the set of subsets of a set E.

(2) On the set N of natural numbers, addition, multiplication and exponentiation are laws of composition (the compositions of x ∈ N and y ∈ N under these laws being denoted respectively by x + y, xy or x.y and x^y) (Set Theory, III, § 3, no. 4).

(3) Let E be a set; the mapping (X, Y) ↦ X ⋅ Y is a law of composition on the set of subsets of E × E (Set Theory, II, § 3, no. 3, Definition 6); the mapping (f, g) ↦ f ⋅ g is a law of composition on the set of mappings of E into E (Set Theory, II, § 5, no. 2).

(4) Let E be a lattice (Set Theory, III, § 1, no. 11); if sup(x, y) denotes the least upper bound of the set {x, y}, the mapping (x, y) ↦ sup(x, y) is a law of composition on E. Similarly for the greatest lower bound inf(x, y). Example 1 above is a particular case of this with $\mathfrak{P}(E)$ ordered by inclusion.

(5) Let $(E_t)_{t \in I}$ be a family of magmas. Let $\tau_t$ denote the law of composition on $E_t$. The mapping

is a law of composition on the product $E = \prod_{t \in I} E_t$, called the product of the laws $\tau_t$. The set E with this law is called the product magma of the magmas $E_t$. In particular, if all the magmas $E_t$ are equal to the same magma M, we obtain the magma of mappings of I from M.

Let $(x, y) \mapsto x \tau y$ be a law of composition on a set E. Given any two subsets X, Y of E, $X \tau Y$ (provided this notation does not lead to confusion†) will denote the set of elements $x \tau y$ in E, where $x \in X, y \in Y$ (in other words, the image of $X \times Y$ under the mapping $(x, y) \mapsto x \tau y$).

If $a \in E$ we usually write $a \tau Y$ instead of $\{a\} \tau Y$ and $X \tau a$ instead of $X \tau \{a\}$. The mapping $(X, Y) \mapsto X \tau Y$ is a law of composition on the set of subsets of E.