Definition:Anticommutative

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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.


Sources

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