meet continuous

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

  1. 1.

    for any monotoneMathworldPlanetmathPlanetmath net D={xiiI} in L, its supremumMathworldPlanetmathPlanetmath D exists, and

  2. 2.

    for any aL and any monotone net {xiiI},


A monotone net {xiiI} is a net x:IL such that x is a non-decreasing function; that is, for any ij in I, xixj in L.

Note that we could have replaced the first condition by saying simply that DL is a directed setMathworldPlanetmath. (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 aD:={axxD} 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={xiiI}, its infimumMathworldPlanetmath D exists and that


An antitone net is just a net x:IL such that for ij in I, xjxi in L.


  • A meet continuous latticeMathworldPlanetmath is a complete latticeMathworldPlanetmath, 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 {xiiI} be any net in L. We define the following:

    lim¯xi=jI{jixi}   and   lim¯xi=jI{ijxi}

    If there is an xL such that lim¯xi=x=lim¯xi, then we say that the net {xi} order converges to x, and we write xix, or x=limxi. Now, define a subset CL to be closed (in L) if for any net {xi} in C such that xix implies that xC, and open if its set complementPlanetmathPlanetmath is closed, then L becomes a topological lattice. With respect to this topologyMathworldPlanetmath, meet and join are easily seen to be continuousMathworldPlanetmathPlanetmath.


  • 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