0. The concept

Definition. A lattice $L$ is said to be complete if (i) every subset $S$ of $L$ has a least upper bound (denoted $\sup S$) and (ii) every subset of $L$ has a greatest lower bound (denoted $\inf S$).

Observation 1. A complete lattice has top and bottom elements, namely $0 = \sup \emptyset$ and $1 = \inf \emptyset$.

Observation 2. If (i) holds, then $L$ already satisfies (ii).