# Max and Min Operations on Real Numbers are Isomorphic

## Theorem

Let $\R$ denote the set of real numbers.

Let $\vee$ and $\wedge$ denote the max operation and min operation respectively.

Let $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ denote the algebraic structures formed from the above.

Then $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are isomorphic.

## Proof

First we note that from:

Min Operation on Toset is Semigroup

and:

Max Operation on Toset is Semigroup

both $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are semigroups.

Let $\phi: \R \to \R$ defined as:

$\forall x \in \R: \map \phi x = -x$

We have that:

$-x = -y \iff x = y$

which demonstrates that $\phi$ is a bijection.

Then we have:

 $\ds \forall x, y \in \R: \,$ $\ds \map \phi {x \vee y}$ $=$ $\ds -\paren {x \vee y}$ $\ds$ $=$ $\ds \paren {-x} \wedge \paren {-y}$ $\ds$ $=$ $\ds \map \phi x \wedge \map \phi y$

which demonstrates that $\phi$ is a (semigroup) homomorphism.

The result follows by definition of (semigroup) isomorphism.

$\blacksquare$