Upper Set - Properties

Properties

• Every partially ordered set is an upper set of itself.
• The intersection and the union of upper sets is again an upper set.
• The complement of any upper set is a lower set, and vice versa.
• Given a partially ordered set (X,≤), the family of lower sets of X ordered with the inclusion relation is a complete lattice, the down-set lattice O(X).
• Given an arbitrary subset Y of an ordered set X, the smallest upper set containing Y is denoted using an up arrow as ↑Y.
  • Dually, the smallest lower set containing Y is denoted using a down arrow as ↓Y.
• A lower set is called principal if it is of the form ↓{x} where x is an element of X.
• Every lower set Y of a finite ordered set X is equal to the smallest lower set containing all maximal elements of Y: Y = ↓Max(Y) where Max(Y) denotes the set containing the maximal elements of Y.
• A directed lower set is called an order ideal.
• The minimal elements of any upper set form an antichain.
  • Conversely any antichain A determines an upper set {x: for some y in A, x ≥ y}. For partial orders satisfying the descending chain condition this correspondence between antichains and upper sets is 1-1, but for more general partial orders this is not true.