From Online Dictionary of Crystallography
Revision as of 16:00, 9 November 2017 by BrianMcMahon (Tidied translations.)
Loi de composition (Fr). Zweistellige Verknüpfung (Ge). Operazione binaria (It). Бинарная операция (Ru). 二項演算 (Ja). Operación binaria (Sp).
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.