Complete lattice

Remarks

It is enough to ask that every subset has a supremum: then $\bigwedge S$ is the supremum of the set of lower bounds of $S$. Taking $S = \varnothing$ shows that a complete lattice has a least element $\bigvee \varnothing$ and a greatest element $\bigwedge \varnothing$; in particular it is nonempty. Every complete lattice is a lattice, and every finite nonempty lattice is complete. Example: for a set $X$, the power set of $X$ ordered by inclusion is complete, with $\bigvee S = \bigcup S$ and $\bigwedge S = \bigcap S$, where $\bigcap \varnothing = X$. Non-example: $\Q$ with its usual order is a lattice but not complete, because $\set{q \in \Q : q^2 < 2}$ has no supremum in $\Q$.