Definition:Alternative Operation
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Definition
Let $\circ$ be a binary operation.
Then $\circ$ is defined as being alternative on $S$ if and only if:
- $\forall T := \set {x, y} \subseteq S: \forall x, y, z \in T: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$
That is, $\circ$ is associative over any two elements of $S$.
For example, for any $x, y \in S$:
- $\paren {x \circ y} \circ x = x \circ \paren {y \circ x}$
- $\paren {x \circ x} \circ y = x \circ \paren {x \circ y}$
and so on.
Also see