complete lattice

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.

- Any power set;
- Any finite inhabited toset;
- The ordered set of real numbers, if a top element $\infty$ and a bottom element $-\infty$ are added;
- The unit interval $[0,1]$.

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.

Revised on March 15, 2012 15:39:49
by Urs Schreiber
(82.172.178.200)