Anticommutative

Anticommutative is a property of a binary operation. A binary operation * on a set S is called anticommutative if for all elements a and b in S, the equation a * b = -(b * a) holds. This property is closely related to the concept of commutativity, where a * b = b * a. If an operation is anticommutative, it means that swapping the order of the operands results in the negation of the original result.

A common example of an anticommutative operation is the cross product in three-dimensional Euclidean space. For any two vectors u and v, their cross product satisfies u × v = -(v × u). This means that the direction of the resulting vector is reversed when the order of the operands is swapped, while the magnitude remains the same.

Another context where anticommutativity appears is in abstract algebra, particularly with Lie brackets. The Lie bracket [x, y] of two elements x and y in a Lie algebra satisfies [x, y] = -[y, x]. This anticommutativity is a fundamental property of Lie algebras and plays a crucial role in their structure and applications in physics and mathematics.

It is important to note that an operation cannot be both commutative and anticommutative unless the result is always zero. If a * b = b * a (commutative) and a * b = -(b * a) (anticommutative), then b * a = -(b * a), which implies 2(b * a) = 0. If the underlying set allows for division by 2 or if the characteristic of the ring is not 2, this leads to b * a = 0 for all a and b. Therefore, for non-trivial operations, anticommutativity and commutativity are mutually exclusive.