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- Neither $L$ nor $R$ is empty.
- $L \cap R = \emptyset$.
- $L \cup R = X$.
- If $\ell \in L$, and $x \in X$, and $x \leq \ell$, then $x \in L$. ($L$ is "closed downwards").
- If $r \in R$, and $x \in X$, and $x \geq r$, then $x \in R$. ($R$ is "closed upwards").
- The set $L$ does not have a maximal element.
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