Antikommutativity

Antikommutativity is a property of a binary operation. An operation denoted by * is said to be anticommutative if for any elements a and b in the set on which the operation is defined, the result of applying the operation to a and b is the negative of the result of applying the operation to b and a. This can be expressed mathematically as a * b = -(b * a). A special case of this property occurs when a = b. In this situation, if an operation is anticommutative, then a * a = -(a * a). This implies that 2(a * a) = 0, which means that a * a must be the additive identity (usually denoted as 0) if the underlying algebraic structure has characteristic not equal to 2. The cross product of vectors in three-dimensional Euclidean space is a well-known example of an anticommutative operation. For any two vectors u and v, their cross product satisfies u × v = -(v × u). Another example is the Lie bracket in Lie algebra, denoted by [a, b], which satisfies [a, b] = -[b, a]. In contrast, standard addition and multiplication of numbers are commutative, meaning a + b = b + a and a * b = b * a, respectively, and are not anticommutative.