for any and any monotone net ,
A monotone net is a net such that is a non-decreasing function; that is, for any in , in .
Note that we could have replaced the first condition by saying simply that 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 is a directed subset of , then is also directed, so that the right hand side of the second condition makes sense.
An antitone net is just a net such that for in , in .
Let a lattice be both meet continuous and join continuous. Let be any net in . We define the following:
If there is an such that , then we say that the net order converges to , and we write , or . Now, define a subset to be closed (in ) if for any net in such that implies that , and open if its set complement is closed, then becomes a topological lattice. With respect to this topology, meet and join are easily seen to be continuous.
- 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).
|Date of creation||2013-03-22 16:36:41|
|Last modified on||2013-03-22 16:36:41|
|Last modified by||CWoo (3771)|