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 latticeis 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.