In mathematics, anticommutativity is the property of an operation with two or more arguments wherein swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric operations.
An -ary operation is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation ∗ is anti-commutative if for all x and y,
- x ∗ y = −(y ∗ x).
- the value of the operation is unchanged, when applied to all ordered tuples constructed by even permutation of the elements of a fixed one.
- the value of the operation is the inverse of its value on a fixed tuple, when applied to all ordered tuples constructed by odd permutation to the elements of the fixed one. The need for the existence of this inverse element is the main reason for requiring the codomain of the operation to be at least a group.
This means that x1 ∗ x2 is the inverse of the element x2 ∗ x1 in .
If the group is such that
In the case this means
Examples of anticommutative binary operations include:
- Exterior algebra
- Operation (mathematics)
- Symmetry in mathematics
- Particle statistics (for anticommutativity in physics).
- Bourbaki, Nicolas (1989), "Chapter III. Tensor algebras, exterior algebras, symmetric algebras", Algebra. Chapters 1–3, Elements of Mathematics (2nd printing ed.), Berlin-Heidelberg-New York: Springer-Verlag, pp. xxiii+709, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001.
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