\begin{definition}[Definition:Lattice/Definition 2]
Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure.
Then $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice''' {{iff}}:
:$\struct {S, \vee, \preceq}$ is a join semilattice
:$\struct {S, \wedge, \preceq}$ is a meet semilattice
That is, for all $a, b \in S$:
:$a \vee b$ is the supremum of $\set {a, b}$
:$a \wedge b$ is the infimum of $\set {a, b}$
\end{definition}