# 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.

 $\ds a \otimes a$ $=$ $\ds \map \alpha a \odot \map \beta a$ 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 $\otimes$