# 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 *S* ⊆ *X* with the following property: if *s* is in *S* and if *x* in *X* is larger than *s* (i.e. if *s* ≤ *x*), 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

**are sometimes used as synonyms for lower set.**

*ideal*^{[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.