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 $\kappa$ be an infinite cardinal. A $\kappa$-complete lattice is a lattice $L$ such that for every subset $A\subseteq L$ with $|A|\le \kappa$, both $\bigvee A$ and $\bigwedge A$ exist. (Note that an ${\mathrm{\aleph }}_0$-complete lattice is the same as a countably complete lattice.)

Every complete lattice is a for every infinite cardinal $\kappa$, 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	$\kappa$-complete
Defines	$\kappa$-complete lattice