\begin{definition}[Definition:Alternative Operation]
Let $\circ$ be a binary operation.
Then $\circ$ is defined as being '''alternative on $S$''' {{iff}}:
:$\forall T := \set {x, y} \subseteq S: \forall x, y, z \in T: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$
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.
\end{definition}