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 $\aleph_0$ complete lattice is the same as a countably complete lattice.)

Every complete lattice is a $\kappa$ complete lattice for every infinite cardinal $\kappa$ and in particular is a countably complete lattice. Every countably complete lattice is a bounded lattice.