1.10: Binary Operation
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 .