Upper set

In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X)[1] of a partially ordered set (X, ≤) is a subset SX with the following property: if s is in S and if x in X is larger than s (i.e. if sx), then x is in S. In words, this means that any x element of X that is ≥ to some element of S is necessarily also an element of S. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is ≤ to some element of S is necessarily also an element of S.

The terms order ideal or ideal are sometimes used as synonyms for lower set.[2][3][4] 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.[2]

The upper and lower closures, when viewed as function from the power set of X to itself, are examples of closure operators since they satisfy all of the Kuratowski closure axioms. As a result, the upper closure of a set is equal to the intersection of all upper sets containing it, and similarly for lower sets. Indeed, this is a general phenomenon of closure operators. For example, the topological closure of a set is the intersection of all closed sets containing it; the span of a set of vectors is the intersection of all subspaces containing it; the subgroup generated by a subset of a group is the intersection of all subgroups containing it; the ideal generated by a subset of a ring is the intersection of all ideals containing it; and so on.

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.