Binary operation

From Online Dictionary of Crystallography

Loi de composition (Fr). Zweistellige Verknüpfung (Ge). Operazione binaria (It). 二項演算 (Ja). Бинарная операция (Ru). Operación binaria (Sp).

A binary operation on a set S is a mapping f from the Cartesian product S × S to S. A mapping from K x S to S, where K need not be S, is called an external binary operation.

Many binary operations are commutative [i.e. f(a,b) = f(b,a) holds for all a, b in S] or associative [i.e. f(f(a,b), c) = f(a, f(b,c)) holds for all a,b,c in S]. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (*) of numbers and matrices as well as composition of functions or symmetry operations.

Examples of binary operations that are not commutative are subtraction (-), division (/), exponentiation(^), super-exponentiation(@) and composition.

Binary operations are often written using infix notation such as a * b, a + b or a · b, rather than by functional notation of the form f(a,b). Sometimes they are even written just by concatenation: ab.