Binary operation

From Online Dictionary of Crystallography

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

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

Many binary operations are commutative or associative. 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 on a single set.

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

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 juxtaposition: ab.