Endomorphisms on Entropic Structure whose Pointwise Product is Identity Automorphism

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

Let $\alpha$ and $\beta$ be endomorphisms on $S$ such that:
 * $\alpha \odot \beta$ is the identity automorphism on $S$

where $\alpha \odot \beta$ denotes the pointwise product of $\alpha$ with $\beta$:
 * $\forall x \in S: \map {\paren {\alpha \odot \beta} } x = \map \alpha x \odot \map \beta x$

Let $\otimes$ be the operation on $S$ defined as:
 * $\forall x, y \in S: x \otimes y := \map \alpha x \odot \map \beta y$

Then $\struct {S, \otimes}$ is an entropic idempotent structure, and hence self-distributive.

Proof
Let $a, b, c, d \in S$ be arbitrary.

Hence $\struct {S, \otimes}$ is an idempotent structure.