Prime Ideal is Prime Filter in Dual Lattice

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Let $X$ be a subset of $S$.

Then
 * $X$ is a prime ideal in $L$


 * $X$ is a prime filter in $L^{-1}$

where $L^{-1} = \left({S, \succeq}\right)$ denotes the dual of $L$.

Sufficient Condition
Let $X$ be a prime ideal in $L$.

Then
 * $X$ is an ideal in $L$.

By Ideal is Filter in Dual Ordered Set:
 * $X$ is filter in $L^{-1}$.