Filter is Ideal in Dual Ordered Set

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $P = \left({S, \preceq}\right)$ 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 $P^{-1} = \left({S, \succeq}\right)$ 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