Transplant (Abstract Algebra)/Examples/Multiplication on Integers under Doubling

Example of Transplant
Let $\struct {\Z, \times}$ be the set of integers under multiplication.

Let $E$ be the set of even integers.

Let $f: \Z \to E$ be the mapping from $\Z$ to $E$ defined as:
 * $\forall n \in \Z: \map f n = 2 n$

The transplant $\otimes$ of $\times$ on $\Z$ under $f$ is given by:
 * $\forall n, m \in E: n \otimes m = \dfrac {n m} 2$

Proof
From Bijection between Integers and Even Integers, $f$ is a bijection.

The inverse of $f$ is given as:
 * $\forall n \in E: \map {f^{-1} } n = \dfrac n 2$

Hence from the Transplanting Theorem: