Let $P$ be a poset and $A$ a subset of $P$ . The upper set of $A$ is defined to be the set $$\lbrace b\in P\mid a\le b \mbox{ for some } a\in A\rbrace,$$ and is denoted by $\up A$ . In other words, $\up A$ 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 $\up A \in 2^P$ . $\uparrow$ has the following properties

1. $\up \varnothing=\varnothing$ ,
2. $A\subseteq \up A$ ,
3. $\uparrow \up A=\up A$ , and
4. if $A\subseteq B$ , $\up A\subseteq \up B$ .

An upper set in $P$ is a subset $A$ such that its upper set is itself: $\up A=A$ . 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 $\up A$ 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 $\down A$ . If the lower set of $A$ is $A$ itself, then $A$ is a called a lower set, or a lower closed set.

Remarks.

• $\up A$ is not the same as the set of upper bounds of $A$ , commonly denoted by $A^u$ , which is defined as the set $\lbrace b\in P\mid a\le b\mbox{ for \emph{all} }a\in A\rbrace$ . Similarly, $\down A\neq A^{\ell}$ in general, where $A^{\ell}$ is the set of lower bounds of $A$ .
• When $A=\lbrace x\rbrace$ , we write $\up x$ for $\up A$ and $\down x$ for $\down A$ . $\up x = \lbrace x\rbrace ^u$ and $\down x=\lbrace x\rbrace ^d$ .
• If $P$ is a lattice and $x\in P$ , then $\up x$ is the principal filter generated by $x$ , and $\down x$ is the principal ideal generated by $x$ .
• If $A$ is a lower set of $P$ , then its set complement $A^{\complement}$ is an upper set: if $a\in A^{\complement}$ and $a\le b$ , then $b\in A^{\complement}$ 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 $P$ ).