PlanetMath: continuous poset

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

continuous poset

(Definition)

A poset is said to be continuous if for every $a\in P$

the set $\operatorname{wb}(a)=\lbrace u\in P\mid u\ll a\rbrace$ is a directed set,
$\bigvee \operatorname{wb}(a)$ exists, and
$a=\bigvee \operatorname{wb}(a)$ .

In the first condition, $\ll$ indicates the way below relation on

. It is true that in any poset, if $b:=\bigvee \operatorname{wb}(a)$ exists, then $b\le a$ . So for a poset to be continuous, we require that $a\le b$ .

A continuous lattice is a complete lattice whose underlying poset is continuous. Note that if is a complete lattice, condition 1 above is automatically satisfied: suppose $u,v\ll a$ and $D\subseteq P$ with $a\le \bigvee D$ , then there are finite subsets of with $u\le \bigvee F$ and $v\le \bigvee G$ . Then $H:=F\cup G\subseteq D$ is finite and $u\vee v\le \big(\bigvee F\big)\vee \big(\bigvee G\big)=\bigvee H$ , or $u\vee v\ll a$ , implying that $\operatorname{wb}(a)$ is directed.

Examples.

Any finite poset is continuous, and so is any finite lattice (since it is complete).
A chain is continuous iff it is complete.
The lattice of ideals of a ring is continuous.
The set of all lower semicontinuous functions from a fixed compact topological space into the extended real numbers is a continuous lattice.
The set of all closed convex subsets of a compact convex subset of $\mathbb{R}^n$ ordered by reverse inclusion is a continuous lattice.

Remarks.

Every algebraic lattice is continuous.
Every continuous meet semilattice is meet continuous.

Bibliography

1: G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).

"continuous poset" is owned by CWoo.

(view preamble)

Also defines:

continuous lattice

Log in to rate this entry.
(view current ratings)

Cross-references: meet continuous, meet semilattice, algebraic lattice, inclusion, convex subsets, closed, extended real numbers, topological space, compact, fixed, functions, lower semicontinuous, ring, lattice of ideals, iff, chain, complete, lattice, subsets, finite, complete lattice, way below relation, directed set, poset
There are 2 references to this entry.

This is version 3 of continuous poset, born on 2007-02-21, modified 2007-03-11.
Object id is 8942, canonical name is ContinuousPoset.
Accessed 701 times total.

Classification:

AMS MSC:

06B35 (Order, lattices, ordered algebraic structures :: Lattices :: Continuous lattices and posets, applications)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact