upper set
Let $P$ be a poset and $A$ a subset of $P$. The upper set of $A$ is defined to be the set
$$\{b\in P\mid a\le b\text{for some}a\in A\},$$ 
and is denoted by $\uparrow A$. In other words, $\uparrow A$ is the set of all upper bounds^{} of elements of $A$.
$\uparrow $ can be viewed as a unary operator on the power set^{} ${2}^{P}$ sending $A\in {2}^{P}$ to $\uparrow A\in {2}^{P}$. $\uparrow $ has the following properties

1.
$\uparrow \mathrm{\varnothing}=\mathrm{\varnothing}$,

2.
$A\subseteq \uparrow A$,

3.
$\uparrow \uparrow A=\uparrow A$, and

4.
if $A\subseteq B$, $\uparrow A\subseteq \uparrow B$.
So $\uparrow $ is a closure operator^{}.
An upper set in $P$ is a subset $A$ such that its upper set is itself: $\uparrow A=A$. In other words, $A$ is closed with respect to $\le $ in the sense that if $a\in A$ and $a\le b$, then $b\in A$. An upper set is also said to be upper closed. For this reason, for any subset $A$ of $P$, the $\uparrow A$ is also called the upper closure of $A$.
Dually, the lower set (or lower closure) of $A$ is the set of all lower bounds of elements of $A$. The lower set of $A$ is denoted by $\downarrow A$. If the lower set of $A$ is $A$ itself, then $A$ is a called a lower set, or a lower closed set.
Remarks.

•
$\uparrow A$ is not the same as the set of upper bounds of $A$, commonly denoted by ${A}^{u}$, which is defined as the set $\{b\in P\mid a\le b\text{for}\text{all}\text{}a\in A\}$. Similarly, $\downarrow A\ne {A}^{\mathrm{\ell}}$ in general, where ${A}^{\mathrm{\ell}}$ is the set of lower bounds of $A$.

•
When $A=\{x\}$, we write $\uparrow x$ for $\uparrow A$ and $\downarrow x$ for $\downarrow A$. $\uparrow x={\{x\}}^{u}$ and $\downarrow x={\{x\}}^{d}$.

•
If $P$ is a lattice^{} and $x\in P$, then $\uparrow x$ is the principal filter^{} generated by $x$, and $\downarrow x$ is the principal ideal^{} generated by $x$.

•
If $A$ is a lower set of $P$, then its set complement^{} ${A}^{\mathrm{\complement}}$ is an upper set: if $a\in {A}^{\mathrm{\complement}}$ and $a\le b$, then $b\in {A}^{\mathrm{\complement}}$ by a contrapositive argument^{}.

•
Let $P$ be a poset. The set of all lower sets of $P$ is denoted by $\mathcal{O}(P)$. It is easy to see that $\mathcal{O}(P)$ is a poset (ordered by inclusion), and $\mathcal{O}{(P)}^{\partial}=\mathcal{O}({P}^{\partial})$, where ${}^{\partial}$ is the dualization operation^{} (meaning that ${P}^{\partial}$ is the dual poset of $P$).
Title  upper set 
Canonical name  UpperSet 
Date of creation  20130322 15:49:50 
Last modified on  20130322 15:49:50 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  20 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06A06 
Synonym  up set 
Synonym  down set 
Synonym  upper closure 
Synonym  lower closure 
Related topic  LatticeIdeal 
Related topic  LatticeFilter 
Related topic  Filter 
Defines  lower set 
Defines  upper closed 
Defines  lower closed 