Prime Filter is Prime Ideal in Dual Lattice

Theorem
Let $L = \struct {S, \preceq}$ be a lattice.

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

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


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

where $L^{-1} = \struct {S, \succeq}$ denotes the dual of $L$.

Proof
By Dual of Dual Ordering:
 * dual of $L^{-1}$ is $L$.

Hence by Prime Ideal is Prime Filter in Dual Lattice:
 * the result follows.