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:

\(\ds \forall n, m \in E: \, \)

\(\ds n \otimes m\)

\(=\)

\(\ds \map f {\map {f^{-1} } n \times \map {f^{-1} } m}\)

\(\ds \)

\(=\)

\(\ds \map f {\dfrac n 2 \times \dfrac m 2}\)

\(\ds \)

\(=\)

\(\ds \map f {\dfrac {n m} 4}\)

\(\ds \)

\(=\)

\(\ds 2 \paren {\dfrac {n m} 4}\)

\(\ds \)

\(=\)

\(\ds \dfrac {n m} 2\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures