Quine Transform of a Model

Suppose that $\mathcal{A}=(A,\overrightarrow{R},\overrightarrow{f})$ is a model, and let $\pi :A\to A$ be a bijection (permutation of $A$ to itself). Next, define the following notion:

Definition [Quine Transform]
The Quine transform of $\mathcal{A}$ under $\pi$, written ${\mathcal{A}}^{\pi }$, is given by:

${\mathcal{A}}^{\pi }:=(A,\pi [\overrightarrow{R}],\pi [\overrightarrow{f}])$. 

For example, suppose the model $\mathcal{A}=(A,R)$ is specified as follows:

$A=\{0,1,2\}$
$R=\{(0,1),(0,2),(1,2)\}$. 

Let $\pi :A\to A$ be the transposition that swaps $0$ to $1$. Then,

$\pi [R]=\{(1,0),(1,2),(0,2)\}$

Consequently, $R$ and $\pi [R]$ are extensionally distinct. However, $\mathcal{A}$ and ${\mathcal{A}}^{\pi }$ are isomorphic under $\pi$. More generally, one can see that:

Lemma ["Quine Transform Lemma"]
Let $\pi :A\to A$ be any bijection. Then: ${\mathcal{A}}^{\pi }\cong \mathcal{A}$.

This is all quite simple discrete mathematics. But it has interesting applications.