Definition:Anticommutative/Structure with One Operation
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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
- Results about anticommutativity can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.17$