\begin{definition}[Definition:Lattice/Definition 3]
Let $\struct {S, \vee}$ and $\struct {S, \wedge}$ be semilattices on a set $S$.
Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$:
:$a \vee \paren {a \wedge b} = a$
:$a \wedge \paren {a \vee b} = a$
Let $\preceq$ be the ordering on $S$ defined by:
:$\forall a, b \in S: a \preceq b$ {{iff}} $a \vee b = b$
as on Semilattice Induces Ordering.
Then the ordered structure $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice'''.
Thus $\struct {S, \vee, \wedge, \preceq}$ is called a '''lattice''' {{iff}} the following axioms are satisfied and $\preceq$ is defined as above:
{{refactor|Extract the below into Axiom:Lattice Axioms}}
{{begin-axiom}}
{{axiom | n = \text L 0
        | lc= Closure
        | q = \forall a, b
        | m = a \vee b \in S
        | mm= a \wedge b \in S
}}
{{axiom | n = \text L 1
        | lc= Commutativity
        | q = \forall a, b
        | m = a \vee b = b \vee a
        | mm= a \wedge b = b \wedge a
}}
{{axiom | n = \text L 2
        | lc= Associativity
        | q = \forall a, b, c
        | m = a \vee \paren {b \vee c} = \paren {a \vee b} \vee c
        | mm= a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c
}}
{{axiom | n = \text L 3
        | lc= Idempotence
        | q = \forall a
        | m = a \vee a = a
        | mm= a \wedge a = a
}}
{{axiom | n = \text L 4
        | lc= Absorption
        | q = \forall a,b
        | m = a \vee \paren {a \wedge b} = a
        | mm= a \wedge \paren {a \vee b} = a
}}
{{end-axiom}}
\end{definition}