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

is a subset

with the following property: if s is in S and if x in X is larger than s (that is, if

), then x is in S. In other 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. Let

be a preordered set.

An upper set in

(also called an upward closed set, an upset, or an isotone set)[1] is a subset

that is "closed under going up", in the sense that The dual notion is a lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal), which is a subset

that is "closed under going down", in the sense that 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.

Given an element

of a partially ordered set

the upper closure or upward closure of

denoted by

while the lower closure or downward closure of

are, respectively, the smallest upper and lower sets containing

More generally, given a subset

define the upper/upward closure and the lower/downward closure of

where upper sets and lower sets of this form are called principal.

The upper closure and lower closure of a set are, respectively, the smallest upper set and lower set containing it.

The upper and lower closures, when viewed as functions from the power set of

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.