Transplant (Abstract Algebra)/Examples/Addition on Positive Reals under Squaring

Example of Transplant

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:

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

Proof

From Restriction of Real Square Mapping to Positive Reals is Bijection, $f$ is a bijection.

The inverse of $f$ is given as:

$\forall x \in \R_{>0}: \map {f^{-1} } x = \sqrt x$

Hence from the Transplanting Theorem:

\(\ds \forall x, y \in \R_{>0}: \, \)

\(\ds x \oplus y\)

\(=\)

\(\ds \map f {\map {f^{-1} } x + \map {f^{-1} } y}\)

\(\ds \)

\(=\)

\(\ds \map f {\sqrt x + \sqrt y}\)

\(\ds \)

\(=\)

\(\ds \paren {\sqrt x + \sqrt y}^2\)

\(\ds \)

\(=\)

\(\ds x + y + 2 \sqrt {x y}\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.8 \ \text {(b)}$