complete lattice
Complete lattices
A complete lattice is a poset
such that every subset of has both a supremum
and an infimum
in .
For a complete lattice ,
the supremum of is denoted by ,
and the infimum of is denoted by .
Thus is a bounded lattice,
with as its greatest element and as its least element.
Moreover, is the infimum of the empty set
,
and is the supremum of the empty set.
Generalizations
A countably complete lattice is a poset
such that every countable subset of
has both a supremum and an infimum in .
Let be an infinite cardinal.
A -complete lattice is a lattice
such that for every subset
with , both and exist.
(Note that an -complete lattice
is the same as a countably complete lattice.)
Every complete lattice is a for every infinite cardinal , 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 |
-complete |
Defines | -complete lattice |