Definition:Anticommutative/Structure with One Operation

Definition

Let $\left({S, \circ}\right)$ 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:

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

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $2.17$

• This article incorporates material from Anticommutative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.