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 $|A|\leq\kappa$, both $\bigvee A$ and $\bigwedge A$ exist. (Note that an $\aleph_{0}$-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 CompleteLattice 2013-03-22 12:56:44 2013-03-22 12:56:44 yark (2760) yark (2760) 10 yark (2760) Definition msc 06B23 msc 03G10 TarskiKnasterTheorem CompleteLatticeHomomorphism Domain6 CompleteSemilattice InfiniteAssociativityOfSupremumAndInfimumRegardingItself CompleteBooleanAlgebra ArbitraryJoin countably complete lattice countably-complete lattice $\kappa$-complete $\kappa$-complete lattice