complete latticeCompleteLattice2002-08-17 23:15:372008-02-20 14:24:33Definitioncountably complete latticecountably-complete lattice$\kappa$-complete$\kappa$-complete lattice\section*{Complete lattices}
A \emph{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.
\section*{Generalizations}
A \emph{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 $\aleph_0$-complete lattice
is the same as a countably complete lattice.)
Every complete lattice is a \PMlinkescapetext{$\kappa$-complete lattice}
for every infinite cardinal $\kappa$,
and in particular is a countably complete lattice.
Every countably complete lattice is a bounded lattice.