Definition:Anticommutative/Structure with One Operation

Definition

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:

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

The word can also be applied to the structure itself: $\struct {S, \circ}$ is referred to as an anticommutative structure, or described as being anticommutative.

Also see

• Subtraction on Numbers is Anticommutative

• Results about anticommutativity can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.17$