complete lattice
Complete lattices
A complete lattice is a poset P such that every subset of P has both a supremum and an infimum 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 lattice, with 1 as its greatest element and 0 as its least element. Moreover, 1 is the infimum of the empty set, and 0 is the supremum of the empty set.
Generalizations
A countably complete lattice is a poset P such that every countable subset of P has both a supremum and an infimum in P.
Let κ be an infinite cardinal. A κ-complete lattice is a lattice L such that for every subset A⊆L with |A|≤κ, both ⋁A and ⋀A exist. (Note that an ℵ0-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 |
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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-complete |
Defines | κ-complete lattice |