# Category:Anticommutativity

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This category contains results about **Anticommutativity**.

Definitions specific to this category can be found in Definitions/Anticommutativity.

### 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:

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

## Subcategories

This category has only the following subcategory.

### E

## Pages in category "Anticommutativity"

The following 6 pages are in this category, out of 6 total.