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Structure with One Operation

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:

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

Structure with Two Operations

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

Suppose every element $x$ in $\left({S, +}\right)$ has an inverse element $-x$.

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

$\forall x, y \in S: x \circ y = -\left({y \circ x}\right)$

Also see


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