Definition:Alternative

Definition

Let $\circ$ be a binary operation.

Then $\circ$ is defined as being alternative on $S$ iff:

$\forall T = \left\{{x, y}\right\} \subseteq S: \forall x, y, z \in T: \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right)$

That is, $\circ$ is associative over any two elements of $S$.

For example, for any $x, y \in S$:

$\left({x \circ y}\right) \circ x = x \circ \left({y \circ x}\right)$

$\left({x \circ x}\right) \circ y = x \circ \left({x \circ y}\right)$

and so on.

Also see

• Power-Associative

• Associative

Sources

• John C. Baez: The Octonions (2002): 1.1 Preliminaries