Definition:Anticommutative

Definition
Let $\circ$ be a binary operation.

Let $S$ be an algebraic structure.

Structures with One Operation
Suppose $S$ has one binary operation.

Then $\circ$ is anticommutative on $S$ iff


 * $\forall x, y \in S: x \circ y = y \circ x \iff x = y$

Equivalently, it can be defined as:


 * $\forall x, y \in S: x \ne y \iff x \circ y \ne y \circ x$

Structures with Two Operations
Suppose $S$ has two or more binary operations, one of which is addition.

Suppose every element in $S$ has an additive inverse.

Then we say that $\circ$ is anticommutative on $S$ iff:


 * $\forall x, y \in S: x \circ y = -\left({y \circ x}\right)$

Also see

 * Subtraction on Numbers is Anticommutative
 * Cross Product is Anticommutative