Definition:Self-Distributive Structure
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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 $16.21$