Endomorphisms on Entropic Structure whose Pointwise Product is Identity Automorphism

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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.


\(\ds a \otimes a\) \(=\) \(\ds \map \alpha a \odot \map \beta a\) Definition of $\otimes$
\(\ds \) \(=\) \(\ds \map {\paren {\alpha \odot \beta} } a\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds a\) Definition of Identity Mapping

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


\(\ds \paren {a \otimes b} \otimes \paren {c \otimes d}\) \(=\) \(\ds \map \alpha {a \otimes b} \odot \map \beta {c \otimes d}\)
\(\ds \) \(=\) \(\ds \map \alpha {\map \alpha a \odot \map \beta b} \odot \map \beta {\map \alpha c \odot \map \beta d}\)
\(\ds \) \(=\) \(\ds \paren {\map \alpha {\map \alpha a} \odot \map \alpha {\map \beta b} } \odot \paren {\map \beta {\map \alpha c} \odot \map \beta {\map \beta d} }\) Definition of Endomorphism
\(\ds \) \(=\) \(\ds \paren {\map \alpha {\map \alpha a} \odot \map \beta {\map \alpha c} } \odot \paren {\map \alpha {\map \beta b} \odot \map \beta {\map \beta d} }\) Definition of Entropic Structure
\(\ds \) \(=\) \(\ds \paren {\map \alpha a \otimes \map \alpha c} \odot \paren {\map \beta b \otimes \map \beta d}\) Definition of $\otimes$


This needs considerable tedious hard slog to complete it.
In particular: Can't see where to go from here. If we can prove $\map \alpha a \otimes \map \alpha c = \map \alpha {a \otimes c}$ we're home and dry, but it's not clear that $\alpha$ and $\beta$ are homomorphisms for $\struct {S, \otimes}$ like they are for $\struct {S, \odot}$.
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Sources

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