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

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $2.17$
• This article incorporates material from Anticommutative on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.