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meet continuous (Definition)

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

  1. for any monotone net $ D=\lbrace x_i \mid i\in I\rbrace$ in $ L$, its supremum $ \bigvee D$ exists, and
  2. for any $ a\in L$ and any monotone net $ \lbrace x_i\mid i\in I\rbrace$,
    $\displaystyle a\wedge \bigvee \lbrace x_i \mid i\in I \rbrace = \bigvee \lbrace a\wedge x_i\mid i\in I \rbrace.$

A monotone net $ \lbrace x_i\mid i\in I\rbrace$ is a net $ x:I\to L$ such that $ x$ is a non-decreasing function; that is, for any $ i\le j$ in $ I$, $ x_i\le 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:=\lbrace a\wedge x\mid x\in D\rbrace$ 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=\lbrace x_i\mid i\in I\rbrace$, its infimum $ \bigwedge D$ exists and that

$\displaystyle a\vee \bigwedge \lbrace x_i\mid i\in I\rbrace =\bigwedge \lbrace a\vee x_i\mid i\in I\rbrace.$
An antitone net is just a net $ x:I\to L$ such that for $ i\le j$ in $ I$, $ x_j\le 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 $ \lbrace x_i\mid i\in I\rbrace$ be any net in $ L$. We define the following:
    $\displaystyle \overline{\lim}\ x_i = \bigwedge_{j\in I} \lbrace \bigvee_{j\le i} x_i\rbrace$    and $\displaystyle \qquad\underline{\lim}\ x_i = \bigvee_{j\in I} \lbrace \bigwedge_{i\le j} x_i\rbrace$
    If there is an $ x\in L$ such that $ \overline{\lim}\ x_i=x=\underline{\lim}\ x_i$, then we say that the net $ \lbrace x_i\rbrace$ 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 $ \lbrace x_i\rbrace$ 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.

Bibliography

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).



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See Also: criteria for a poset to be a complete lattice

Other names:  order convergence
Also defines:  join continuous, order converges
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Cross-references: continuous, meet, topology, topological lattice, complement, open, implies, closed, proof, link, joins, finite, poset, complete lattice, lattice, infimum, join semilattice, right hand side, subset, identity function, directed set, function, supremum, net, monotone, meet semilattice
There are 4 references to this entry.

This is version 9 of meet continuous, born on 2007-01-22, modified 2007-02-17.
Object id is 8808, canonical name is MeetContinuous.
Accessed 1670 times total.

Classification:
AMS MSC: 06B35 (Order, lattices, ordered algebraic structures :: Lattices :: Continuous lattices and posets, applications)
 06A12 (Order, lattices, ordered algebraic structures :: Ordered sets :: Semilattices)
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