Self-Distributive Structure/Examples/Arithmetic Mean

Example of Self-Distributive Structure
Let $\Q$ denote the set of rational numbers.

Let $\circ$ be the operation defined on $\Q$ as:
 * $\forall x, y \in \Q: x \circ y := \dfrac {x + y} 2$

That is, $x \circ y$ is the arithmetic mean of $x$ and $y$ in $\Q$.

Then the algebraic structure $\struct {\Q, \circ}$ so formed is a self-distributive quasigroup.

Proof
As Rational Addition is Commutative, it follows immediately from Left Distributive and Commutative implies Distributive that:


 * $\forall a, b, c \in \Q: \paren {a \circ b} \circ c = \paren {a \circ c} \circ \paren {b \circ c}$

To demonstrate that $\struct {\Q, \circ}$ is a quasigroup, it remains to be shown that:
 * $\forall a, b \in \Q: \exists ! x \in \Q: x \circ a = b$
 * $\forall a, b \in \Q: \exists ! y \in \Q: a \circ y = b$

We have:

and similarly:

Hence both $x$ and $y$ are determined uniquely by $a$ and $b$.

Hence by definition $\struct {\Q, \circ}$ is a quasigroup.