Condition for Isomorphism between Structures Induced by Permutations

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Theorem

Let $S$ be a set.

Let $\oplus$ and $\otimes$ be closed operations on $S$ such that both $\oplus$ and $\otimes$ have the same identity.

Let $\sigma$ and $\tau$ be permutations on $S$.


Let $\struct {S, \oplus_\sigma}$ and $\struct {S, \otimes_\tau}$ be the operations on $S$ induced on $\oplus$ by $\sigma$ and on $\otimes$ by $\tau$ respectively:

$\forall x, y \in S: x \oplus_\sigma y := \map \sigma {x \oplus y}$
$\forall x, y \in S: x \otimes_\tau y := \map \tau {x \otimes y}$

Let $f: S \to S$ be a mapping.


Then:

$f$ is an isomorphism from $\struct {S, \oplus_\sigma}$ to $\struct {S, \otimes_\tau}$

if and only if:

$f$ is an isomorphism from $\struct {S, \oplus}$ to $\struct {S, \otimes}$ satisfying the condition:
$f \circ \sigma = \tau \circ f$
where $\circ$ denotes composition of mappings.


Proof




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