A lattice is an Algebraic structure defined as:

A unit element in a lattice $L$ is an element $1$ such that, for all $a \in L$, $a \preceq 1$. A null element in a lattice $L$ is an element $0$ such that, for all $a \in L$, $0 \preceq a$.

The lattice is complete if a Greatest lower bound and a Least upper bound exist for every subset $S$ of $L$ (all that is guaranteed by the definition of a lattice is that these bounds will exist for all finite subsets of L). If these exist, they are denoted as $\wedge S$, and $\vee S$, respectively.