If ${\displaystyle A,B}$  are two abelian groups, a bilinear map ${\displaystyle f\colon A^{2}\to B}$  is anticommutative if for all ${\displaystyle x,y\in A}$  we have

${\displaystyle f(x,y)=-f(y,x).}$ 

More generally, a multilinear map ${\displaystyle g:A^{n}\to B}$  is anticommutative if for all ${\displaystyle x_{1},\dots x_{n}\in A}$  we have

${\displaystyle g(x_{1},x_{2},\dots x_{n})={\text{sgn}}(\sigma )g(x_{\sigma (1)},x_{\sigma (2)},\dots x_{\sigma (n)})}$ 

where ${\displaystyle {\text{sgn}}(\sigma )}$  is the sign of the permutation ${\displaystyle \sigma }$ .

If the abelian group ${\displaystyle B}$  has no 2-torsion, implying that if ${\displaystyle x=-x}$  then ${\displaystyle x=0}$ , then any anticommutative bilinear map ${\displaystyle f\colon A^{2}\to B}$  satisfies

${\displaystyle f(x,x)=0.}$ 

More generally, by transposing two elements, any anticommutative multilinear map ${\displaystyle g\colon A^{n}\to B}$  satisfies

${\displaystyle g(x_{1},x_{2},\dots x_{n})=0}$ 

if any of the ${\displaystyle x_{i}}$  are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if ${\displaystyle f\colon A^{2}\to B}$  is alternating then by bilinearity we have

${\displaystyle f(x+y,x+y)=f(x,x)+f(x,y)+f(y,x)+f(y,y)=f(x,y)+f(y,x)=0}$ 

and the proof in the multilinear case is the same but in only two of the inputs.

Examples of anticommutative binary operations include:

• Cross product
• Lie bracket of a Lie algebra
• Lie bracket of a Lie ring
• Subtraction

• Commutativity
• Commutator
• Exterior algebra
• Graded-commutative ring
• 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 City: Springer-Verlag, ISBN 3-540-64243-9, MR 0979982, Zbl 0904.00001.

• Gainov, A.T. (2001) [1994], "Anti-commutative algebra", Encyclopedia of Mathematics, EMS Press. Which references the Original Russian work
• Weisstein, Eric W. "Anticommutative". MathWorld.