Ordered Set with Order Type of Natural Numbers plus Dual has Minimum Element

Theorem

Let $\struct {S, \preccurlyeq}$ be an ordered structure such that:

$\map \ot {S, \preccurlyeq} = \omega + \omega^*$

where:

$\ot$ denotes order type

$\omega$ denotes the order type of the natural numbers $\N$

$\omega^*$ denotes the dual of $\omega$

$+$ denotes addition of order types.

Then $\struct {S, \preccurlyeq}$ has a smallest element.

Proof

By definition of order type addition:

$\struct {S, \preccurlyeq}$ is isomorphic to $\struct {\N, \le} \oplus \struct {\N, \ge}$

where:

$\cong$ denotes order isomorphism

$\oplus$ denotes order sum.

By the Well-Ordering Principle, $\struct {\N, \le}$ has a smallest element.

By definition of order sum, every element of $\struct {\N, \le}$ precedes every element of $\struct {\N, \ge}$.

Hence the smallest element of $\struct {\N, \le}$ is also the smallest element of $\struct {\N, \le} \oplus \struct {\N, \ge}$.

Hence the result.

$\blacksquare$

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $28 \ \text {(b)}$