Inmathematics,anticommutativity is a specific property of some non-commutative mathematicaloperations. Swapping the position oftwo arguments of an antisymmetric operation yields a result which is theinverse of the result with unswapped arguments. The notioninverse refers to agroup structure on the operation'scodomain, possibly with another operation.Subtraction is an anticommutative operation because commuting the operands ofa −b givesb −a = −(a −b); for example,2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is theLie bracket.
Inmathematical physics, wheresymmetry is of central importance, or even just inmultilinear algebra these operations are mostly (multilinear with respect to somevector structures and then) calledantisymmetric operations, and when they are not already ofarity greater than two, extended in anassociative setting to cover more than twoarguments.
If are twoabelian groups, abilinear map isanticommutative if for all we have
More generally, amultilinear map is anticommutative if for all we have
where is thesign of thepermutation.
If the abelian group has no 2-torsion, implying that if then, then any anticommutative bilinear map satisfies
More generally, bytransposing two elements, any anticommutative multilinear map satisfies
if any of the are equal; such a map is said to bealternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if is alternating then by bilinearity we have
and the proof in the multilinear case is the same but in only two of the inputs.
Examples of anticommutative binary operations include:
