In mathematics, an upper set (also called an upward closed set or just an upset) of a partially ordered set (X,≤) is a subset U with the property that, if x is in U and x≤y, then y is in U.
The dual notion is lower set (alternatively, down set, decreasing set, initial segment, semi-ideal; the set is downward closed), which is a subset L with the property that, if x is in L and y≤x, then y is in L.
The terms order ideal or ideal are sometimes used as synonyms for lower set. This choice of terminology fails to reflect the notion of an ideal of a lattice because a lower set of a lattice is not necessarily a sublattice.
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.
An ordinal number is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion.
