A complete lattice is a poset which has all small joins and meets (as opposed to just finite joins and meets).
In particular, it is a lattice.
Complete lattices and complete lattice homomorphisms form a concrete category CompLat.
By the adjoint functor theorem for posets, having either all joins or all meets is sufficient for the other. However, a suplattice morphism may preserve only joins, while dually an inflattice morphism may preserve only meets. Furthermore, a large poset with all small joins or meets need not have the other.
Regarded as a small category, a complete lattice is complete. Conversely, in classical logic and in any Grothendieck topos, any complete small category is in fact a preorder, and hence a complete lattice.
Complete lattices are harder to come by in constructive mathematics and nearly impossible in predicative mathematics (at least if they are to be small). In particular, one must use the Mac Neille reals (and be a bit careful about infinity) for the analytic examples to work constructively.
Last revised on March 15, 2012 at 15:39:49. See the history of this page for a list of all contributions to it.