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- Upper set

In mathematics, an **upper set** (also called an **upward closed set**, an **upset**, or an **isotone** set in *X*) 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*.

Let

*(X,**\leq)*

*X*

*U**\subseteq**X*

*u**\in**U*

*x**\in**X*

*u**\leq**x,*

*x**\in**U.*

*U*

for all

*u**\in**U*

*x**\in**X,*

*u**\leq**x*

*x**\in**U.*

The dual notion is a (also called a , , , , or ), which is a subset

*L**\subseteq**X*

*l**\in**L*

*x**\in**X*

*x**\leq**l,*

*x**\in**L.*

*L*

for all

*l**\in**L*

*x**\in**X,*

*x**\leq**l*

*x**\in**L.*

The terms or are sometimes used as synonyms for lower set.^{[1]} ^{[2]} 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.^{[3]}

- Every partially ordered set is an upper set of itself.
- The intersection and the union of any family 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 upper sets of*X*ordered with the inclusion relation is a complete lattice, the**upper set lattice**. - Given an arbitrary subset
*Y*of a partially ordered set*X*, the smallest upper set containing*Y*is denoted using an up arrow as ↑*Y*(see upper closure and lower closure).- Dually, the smallest lower set containing
*Y*is denoted using a down arrow as ↓*Y*.

- Dually, the smallest lower set containing
- A lower set is called
**principal**if it is of the form ↓ where*x*is an element of*X*. - Every lower set
*Y*of a finite partially 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 . 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.

- Conversely any antichain

Given an element

*x*

*(X,**\leq),*

*x,*

*x*^{\uparrow}*,*

*x*^{\uparrow}*,*

*\uparrow**x,*

*x*^{\uparrow}=*\uparrow**x*=*\left\{**u**\in**X**:**x**\leq**u**\right\}*

while the **lower closure** or **downward closure** of *x*, denoted by

*x*^{\downarrow}*,*

*x*^{\downarrow}*,*

*\downarrow**x,*

*x*^{\downarrow}=*\downarrow**x*=*\left\{**l**\in**X**:**l**\leq**x**\right\}.*

The sets

*\uparrow**x*

*\downarrow**x*

*x*

*A**\subseteq**X,*

*A*^{\uparrow}

*A*^{\downarrow}

*A*^{\uparrow}=*A*^{\uparrow}=c*up*_{a}*\uparrowa*

*A*^{\downarrow}=*A*^{\downarrow}=c*up*_{a}*\downarrowa.*

In this way, ↑*x* = ↑ and ↓*x* = ↓, where upper sets and lower sets of this form are called **principal**. The upper closures and lower closures of a set are, respectively, the smallest upper set and lower set containing it.

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.

One can also speak of the **strict upper closure** of an element

*x**\in**X*

*A**\subseteq**X,*

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.

- Cofinal set – a subset
*U*of a partially ordered set (*X*, ≤) that contains for every element

*x**\in**X,*

*x**\leq**y.*

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*The low separation axioms (T*_{0}) and (T_{1})

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