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 $\lbrace x_i\mid i\in I\rbrace$ , $$a\wedge \bigvee \lbrace x_i \mid i\in I \rbrace = \bigvee \lbrace a\wedge x_i\mid i\in I \rbrace.$$

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 $$a\vee \bigwedge \lbrace x_i\mid i\in I\rbrace =\bigwedge \lbrace a\vee x_i\mid i\in I\rbrace.$$ 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: $$\overline{\lim}\ x_i = \bigwedge_{j\in I} \lbrace \bigvee_{j\le i} x_i\rbrace\qquad\mbox{ and }\qquad\underline{\lim}\ x_i = \bigvee_{j\in I} \lbrace \bigwedge_{i\le j} x_i\rbrace$$ 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).