Max and Min Operations on Real Numbers are Isomorphic

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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.

$\blacksquare$


Sources