Filter is Ideal in Dual Ordered Set

Theorem

Let $P = \struct {S, \preceq}$ be an ordered set.

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

Then

$X$ is filter in $P$

if and only if

$X$ is ideal in $P^{-1}$

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

Proof

By Dual of Dual Ordering:

dual of $P^{-1}$ is $P$.

Hence by Ideal is Filter in Dual Ordered Set:

the result fallows.

$\blacksquare$

Sources

• Mizar article YELLOW_7:29

• Mizar article YELLOW_7:27