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.

First we note that from:

and:

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.

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.6 \ \text {(a)}$