Definition:Producer of Dedekind Cut

Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $S' \subseteq S$.

Let $\tuple {L, R}$ be a Dedekind cut of $S'$.

An $\alpha \in S$ is referred to as a producer of $\tuple {L, R}$ if and only if:

$l \preceq \alpha$ for all $l \in L$

$\alpha \preceq r$ for all $r \in R$.

Let $L = \set {l \in \Q: l \le 2}$ and $R = \set {r \in \Q: 2 < r}$.

Then $\tuple {L, R}$ is a dedekind cut of $\Q$, and $2$ is a producer of $\tuple {L, R}$

Let $L = \set {l \in \Q: l < \sqrt 2}$ and $R = \set {r \in \Q: \sqrt 2 < r}$.

Then $\tuple {L, R}$ is a Dedekind cut of $\Q$, and $\sqrt 2 \notin \Q$ is a producer of $\tuple {L, R}$.