Transplant (Abstract Algebra)/Examples

Examples of Transplants

Multiplication on $\Z$ under Doubling

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:

$\forall n, m \in E: n \otimes m = \dfrac {n m} 2$

Multiplication on $\R$ under $10^x$

Let $\struct {\R, \times}$ be the set of real numbers under multiplication.

Let $\R_{>0}$ be the set of strictly positive real numbers.

Let $f: \R \to \R_{>0}$ be the mapping from $\R$ to $\R_{>0}$ defined as:

$\forall x \in \R: \map f x = 10^x$

The transplant $\otimes$ of $\times$ on $\R$ under $f$ is given by:

$\forall x, y \in \R_{>0}: x \otimes y = x^{\log_{10} y}$

Multiplication on $\R$ under $1 - x$

Let $\struct {\R, \times}$ be the set of real numbers under multiplication.

Let $f: \R \to \R$ be the permutation defined as:

$\forall x \in \R: \map f x = 1 - x$

The transplant $\otimes$ of $\times$ under $f$ is given by:

$x \otimes y = x + y - x y$

Addition on $\R_{>0}$ under $x^2$

Let $\struct {\R_{>0}, +}$ be the set of strictly positive real numbers under addition.

Let $f: \R_{>0} \to \R_{>0}$ be the permutation defined as:

$\forall x \in \R_{>0}: \map f x = x^2$

The transplant $\oplus$ of $+$ under $f$ is given by:

$x \oplus y = x + y + 2 \sqrt {x y}$

Addition on $\R_{>0}$ under $\log_{10} x$

Let $\struct {\R_{>0}, +}$ be the set of strictly positive real numbers under addition.

Let $f: \R_{>0} \to \R$ be the permutation defined as:

$\forall x \in \R_{>0}: \map f x = \log_{10} x$

The transplant $\oplus$ of $+$ under $f$ is given by:

$x \oplus y = \map {\log_{10} } {10^x + 10^y}$