Subsets of a poset

There are certain special subsets of a partially ordered set:

• Directed Sets: these are subsets whose every pair of elements contains an upper or lower bound.
• Centered Sets: these are subsets whose every finite subset of elements contain an upper or lower bound.
• Upper/Lower Sets: subsets that contain all elements less then or greater then values that they contain. The lattice of these sets is a sufficient basis for representing distributive lattices.
• Dedekind Cuts: subsets whose set of lower bounds of its set of upper bounds is equal to itself. The lattice of these is the dedekind completion.

The analysis of such subsets is important to our understanding of order theory.