complete lattice

Complete lattices

A complete latticeMathworldPlanetmath is a poset P such that every subset of P has both a supremumMathworldPlanetmathPlanetmath and an infimumMathworldPlanetmath in P.

For a complete lattice L, the supremum of L is denoted by 1, and the infimum of L is denoted by 0. Thus L is a bounded latticeMathworldPlanetmath, with 1 as its greatest element and 0 as its least element. Moreover, 1 is the infimum of the empty setMathworldPlanetmath, and 0 is the supremum of the empty set.


A countably complete lattice is a poset P such that every countableMathworldPlanetmath subset of P has both a supremum and an infimum in P.

Let κ be an infiniteMathworldPlanetmath cardinal. A κ-complete lattice is a latticeMathworldPlanetmath L such that for every subset AL with |A|κ, both A and A exist. (Note that an 0-complete lattice is the same as a countably complete lattice.)

Every complete lattice is a for every infinite cardinal κ, and in particular is a countably complete lattice. Every countably complete lattice is a bounded lattice.

Title complete lattice
Canonical name CompleteLattice
Date of creation 2013-03-22 12:56:44
Last modified on 2013-03-22 12:56:44
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Definition
Classification msc 06B23
Classification msc 03G10
Related topic TarskiKnasterTheorem
Related topic CompleteLatticeHomomorphism
Related topic Domain6
Related topic CompleteSemilattice
Related topic InfiniteAssociativityOfSupremumAndInfimumRegardingItself
Related topic CompleteBooleanAlgebra
Related topic ArbitraryJoin
Defines countably complete lattice
Defines countably-complete lattice
Defines κ-completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath
Defines κ-complete lattice