Definition:Self-Distributive Structure

Definition

Let $\struct {S, \circ}$ be an algebraic structure such that $\circ$ is a self-distributive operation.

Then $\struct {S, \circ}$ is known as a self-distributive structure.

Also known as

Some sources refer to such a structure as a distributive structure.

Examples

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.

Also see

• Results about self-distributive operations can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.21$