Condition for Algebraic Structure to be Self-Distributive Quasigroup

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Theorem

Let $\struct {S, \odot}$ be an algebraic structure.

Then:

$\struct {S, \odot}$ is a self-distributive quasigroup

if and only if:

for every $a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ on $S$ are automorphisms of $\struct {S, \odot}$.


Proof

Sufficient Condition

Let $\struct {S, \odot}$ be a self-distributive quasigroup.

By definition of quasigroup, we have that $\lambda_a$ and $\rho_a$ are both permutations on $S$.


It remains for the morphism property to be demonstrated.


Indeed:

\(\ds \forall x, y \in S: \, \) \(\ds \map {\lambda_a} x \odot \map {\lambda_a} y\) \(=\) \(\ds \paren {a \odot x} \odot \paren {a \odot y}\) Definition of Left Regular Representation
\(\ds \) \(=\) \(\ds a \odot \paren {x \odot y}\) Definition of Self-Distributive Structure
\(\ds \) \(=\) \(\ds \map {\lambda_a} {x \odot y}\) Definition of Left Regular Representation

and:

\(\ds \forall x, y \in S: \, \) \(\ds \map {\rho_a} x \odot \map {\rho_a} y\) \(=\) \(\ds \paren {x \odot a} \odot \paren {y \odot a}\) Definition of Right Regular Representation
\(\ds \) \(=\) \(\ds \paren {x \odot y} \odot a\) Definition of Self-Distributive Structure
\(\ds \) \(=\) \(\ds \map {\rho_a} {x \odot y}\) Definition of Right Regular Representation


Thus $\lambda_a$ and $\rho_a$ have been shown to be automorphisms of $\struct {S, \odot}$.

$\Box$


Necessary Condition

Let $\odot$ be such that for every $a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ on $S$ are automorphisms of $\struct {S, \odot}$.

We have a fortiori that:

$\lambda_a$ and $\rho_a$ are both permutations on $S$
$\lambda_a$ and $\rho_a$ are both homomorphisms of $S$.

Hence by definition $\struct {S, \odot}$ is a quasigroup.


It remains to demonstrate self-distributivity.


Indeed:

\(\ds \forall x, y, z \in S: \, \) \(\ds x \odot \paren {y \odot z}\) \(=\) \(\ds \map {\lambda_x} {y \odot z}\) Definition of Left Regular Representation
\(\ds \) \(=\) \(\ds \map {\lambda_x} y \odot \map {\lambda_x} z\) as $\lambda_x$ is a homomorphism on $S$
\(\ds \) \(=\) \(\ds \paren {x \odot y} \odot \paren {x \odot z}\) Definition of Left Regular Representation

and:

\(\ds \forall x, y, z \in S: \, \) \(\ds \paren {x \odot y} \odot z\) \(=\) \(\ds \map {\rho_z} {x \odot y}\) Definition of Right Regular Representation
\(\ds \) \(=\) \(\ds \map {\rho_z} x \odot \map {\rho_z} y\) as $\rho_z$ is a homomorphism on $S$
\(\ds \) \(=\) \(\ds \paren {x \odot z} \odot \paren {y \odot z}\) Definition of Right Regular Representation

Thus $\struct {S, \odot}$ is a self-distributive quasigroup.

$\blacksquare$


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