meet continuous

Let $L$ be a meet semilattice. We say that $L$ is meet continuous if

1.

for any monotone net $D=\{x_{i}\mid i\in I\}$ in $L$ , its supremum $\bigvee D$ exists, and
2.

for any $a\in L$ and any monotone net $\{x_{i}\mid i\in I\}$ ,

$a\wedge\bigvee\{x_{i}\mid i\in I\}=\bigvee\{a\wedge x_{i}\mid i\in I\}.$

A monotone net $\{x_{i}\mid i\in I\}$ is a net $x:I\to L$ such that $x$ is a non-decreasing function; that is, for any $i\leq j$ in $I$ , $x_{i}\leq x_{j}$ in $L$ .

Note that we could have replaced the first condition by saying simply that $D\subseteq L$ is a directed set. (A monotone net is a directed set, and a directed set is a trivially a monotone net, by considering the identity function as the net). It’s not hard to see that if $D$ is a directed subset of $L$ , then $a\wedge D:=\{a\wedge x\mid x\in D\}$ is also directed, so that the right hand side of the second condition makes sense.

Dually, a join semilattice $L$ is join continuous if its dual (as a meet semilattice) is meet continuous. In other words, for any antitone net $D=\{x_{i}\mid i\in I\}$ , its infimum $\bigwedge D$ exists and that

a\vee\bigwedge\{x_{i}\mid i\in I\}=\bigwedge\{a\vee x_{i}\mid i\in I\}.

An antitone net is just a net $x:I\to L$ such that for $i\leq j$ in $I$ , $x_{j}\leq x_{i}$ in $L$ .

Remarks.

•

A meet continuous lattice is a complete lattice, since a poset such that finite joins and directed joins exist is a complete lattice (see the link below for a proof of this).
•

Let a lattice $L$ be both meet continuous and join continuous. Let $\{x_{i}\mid i\in I\}$ be any net in $L$ . We define the following:

$\overline{\lim}\ x_{i}=\bigwedge_{j\in I}\{\bigvee_{j\leq i}x_{i}\}\qquad\mbox% { and }\qquad\underline{\lim}\ x_{i}=\bigvee_{j\in I}\{\bigwedge_{i\leq j}x_{i}\}$

If there is an $x\in L$ such that $\overline{\lim}\ x_{i}=x=\underline{\lim}\ x_{i}$ , then we say that the net $\{x_{i}\}$ order converges to $x$ , and we write $x_{i}\to x$ , or $x=\lim\ x_{i}$ . Now, define a subset $C\subseteq L$ to be closed (in $L$ ) if for any net $\{x_{i}\}$ in $C$ such that $x_{i}\to x$ implies that $x\in C$ , and open if its set complement is closed, then $L$ becomes a topological lattice. With respect to this topology, meet $\wedge$ and join $\vee$ are easily seen to be continuous.

References

1 G. Birkhoff, Lattice Theory, 3rd Edition, Volume 25, AMS, Providence (1967).
2 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).
3 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).

Title	meet continuous
Canonical name	MeetContinuous
Date of creation	2013-03-22 16:36:41
Last modified on	2013-03-22 16:36:41
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06A12
Classification	msc 06B35
Synonym	order convergence
Related topic	CriteriaForAPosetToBeACompleteLattice
Related topic	JoinInfiniteDistributive
Defines	join continuous
Defines	order converges