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

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).
Mathematics Subject Classification
06A12 no label found06B35 no label found Forums
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