Definition:Anticommutative/Structure with One Operation

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure

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:


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

The word can also be applied to the structure itself: $\left({S, \circ}\right)$ is referred to as an anticommutative structure, or described as being anticommutative.

Also see

 * Subtraction on Numbers is Anticommutative