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 $\{b\in P\mid a\le b\text{ for }\text{<em class="ltx\_emph ltx\_font\_italic">all</em>}a\in A\}$. 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 $𝒪(P)$. It is easy to see that $𝒪(P)$ is a poset (ordered by inclusion), and $𝒪{(P)}^{\partial }=𝒪(P^{\partial })$, 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	2013-03-22 15:49:50
Last modified on	2013-03-22 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