Definition:Transplant (Abstract Algebra)

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $f: S \to T$ be a bijection.

Let $\oplus$ be the one and only one operation such that $f: \struct {S, \circ} \to \struct {T, \oplus}$ is an isomorphism.

The operation $\oplus$ is called the transplant of $\circ$ under $f$.

Examples

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:

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

Also see

• Transplanting Theorem where it is shown that:

$\forall x, y \in T: x \oplus y = \map f {\map {f^{-1} } x \circ \map {f^{-1} } y}$

• Results about transplants can be found here.

Sources

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