Equivalence of Definitions of Lower Set

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Theorem

The following definitions of the concept of Lower Set are equivalent:

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $U \subseteq S$.

Then the following are equivalent:

\((1)\)	$:$			\(\ds \forall u \in U: \forall s \in S: s \preceq u \implies s \in U \)
\((2)\)	$:$			\(\ds U^\preceq \subseteq U \)
\((3)\)	$:$			\(\ds U^\preceq = U \)

where $U^\preceq$ is the lower closure of $U$.

Proof

By the Duality Principle, it suffices to prove that:

$(1^*)$, $(2^*)$ and $(3^*)$ are equivalent

where these are the dual statements of $(1)$, $(2)$ and $(3)$, respectively.

By Dual Pairs, it can be seen that these dual statements are as follows:

\((1^*)\)	$:$			\(\ds \forall u \in U: \forall s \in S: u \preceq s \implies s \in U \)
\((2^*)\)	$:$			\(\ds U^\succeq \subseteq U \)
\((3^*)\)	$:$			\(\ds U^\succeq = U \)

Their equivalence is proved on Equivalence of Definitions of Upper Set.

$\blacksquare$