# Equivalence of Definitions of Lower Set

## 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.

$\blacksquare$