Editing Maximal and minimal elements (section)

== Related notions ==
A subset <math>Q</math> of a partially ordered set <math>P</math> is said to be [[Cofinal (mathematics)|cofinal]] if for every <math>x \in P</math> there exists some <math>y \in Q</math> such that <math>x \leq y.</math> Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements.

A subset <math>L</math> of a partially ordered set <math>P</math> is said to be a [[lower set]] of <math>P</math> if it is downward closed: if <math>y \in L</math> and <math>x \leq y</math> then <math>x \in L.</math> Every lower set <math>L</math> of a finite ordered set <math>P</math> is equal to the smallest lower set containing all maximal elements of <math>L.</math>