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