# Definition:Anticommutative

## Definition

### Structure with One Operation

Let $\struct {S, \circ}$ be an algebraic structure

Then $\circ$ is anticommutative on $S$ if and only if:

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

Equivalently, it can be defined as:

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

### Structure with Two Operations

Let $\struct {S, +, \circ}$ be an algebraic structure.

Let every element $x$ in $\struct {S, +}$ have an inverse element $-x$.

Then $\circ$ is anticommutative on $S$ with respect to $+$ if and only if:

$\forall x, y \in S: x \circ y = -\paren {y \circ x}$

## Also see

• Results about anticommutativity can be found here.