# Definition:Alternative Operation

## 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.