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:

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

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:

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

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

$\blacksquare$