Self-Distributive Structure/Examples

Examples of Self-Distributive Structures

Arithmetic Mean

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.