PlanetMath: upper set

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upper set

(Definition)

Let be a poset and a subset of . The upper set of is defined to be the set

$\displaystyle \lbrace b\in P\mid a\le b$ for some $\displaystyle a\in A\rbrace,$

and is denoted by $\uparrow\!\!A$ . In other words, $\uparrow\!\!A$ is the set of all upper bounds of elements of

$\uparrow$ can be viewed as a unary operator on the power set sending $A\in 2^P$ to $\uparrow\!\!A \in 2^P$ . $\uparrow$ has the following properties

$\uparrow\!\!\varnothing=\varnothing$ ,
$A\subseteq \uparrow\!\!A$ ,
$\uparrow \uparrow\!\!A=\uparrow\!\!A$ , and
if $A\subseteq B$ , $\uparrow\!\!A\subseteq \uparrow\!\!B$ .

So $\uparrow$ is a closure operator.

An upper set in is a subset such that its upper set is itself: $\uparrow\!\!A=A$ . In other words, 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 of , the $\uparrow\!\!A$ is also called the upper closure of .

Dually, the lower set (or lower closure) of is the set of all lower bounds of elements of . The lower set of is denoted by $\downarrow\!\!A$ . If the lower set of is itself, then 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 , commonly denoted by , which is defined as the set $\lbrace b\in P\mid a\le b$ for all $a\in A\rbrace$ . Similarly, $\downarrow\!\!A\neq A^{\ell}$ in general, where $A^{\ell}$ is the set of lower bounds of .
When $A=\lbrace x\rbrace$ , we write $\uparrow\!\!x$ for $\uparrow\!\!A$ and $\downarrow\!\!x$ for $\downarrow\!\!A$ . $\uparrow\!\!x = \lbrace x\rbrace ^u$ and $\downarrow\!\!x=\lbrace x\rbrace ^d$ .
If is a lattice and $x\in P$ , then $\uparrow\!\!x$ is the principal filter generated by , and $\downarrow\!\!x$ is the principal ideal generated by .
If is a lower set of , 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 be a poset. The set of all lower sets of 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 ).