Definition 1

Let E be a set. A mapping f of E × E into E is called a law of composition on E. The value f(x, y) of f for an ordered pair (x, y) ∈ E × E is called the composition of x and y under this law. A set with a law of composition is called a magma.

The composition of x and y is usually denoted by writing x and y in a definite order and separating them by a characteristic symbol of the law in question (a symbol which it may be agreed to omit). Among the symbols most often used are + and ., the usual convection being to omit the latter if desired; with these symbols the composition of x and y is written respectively as x + y and x.y or xy. A law denoted by the symbol + is usually called addition (the composition x + y being called the sum of x and y) and we say that it is written additively; a law denoted by the symbol . is usually called multiplication (the composition x.y = xy being called the product of x and y) and we say that it is written multiplicatively. In the general arguments of paragraphs 1 to 3 of this chapter we shall generally use the symbols ⊔ and ⊓ to denote arbitrary laws of composition.

By an abuse of language, a mapping of a subset of E × E into E is sometimes called a law of composition not everywhere defined on E.