Structure with Commutative Idempotent Associative Operations satisfying Absorption Laws is Lattice

Theorem
Let $S$ be a set.

Let $\vee$ and $\wedge$ be binary operations which, when applied to $S$, are both:


 * closed operations
 * commutative operations
 * idempotent operations
 * associative operations.

Furthermore, let $\vee$ and $\wedge$ satisfy the absorption laws:


 * $\forall a, b \in S: a \vee \paren {a \wedge b} = a = a \wedge {a \vee b}$

Then there exists a unique lattice ordering $\preccurlyeq$ on $S$ such that:


 * $\forall a, b \in S$:
 * $x \vee y = \sup \set {a, b}$
 * $x \wedge y = \inf \set {a, b}$

That is:
 * $\struct {S, \vee, \wedge, \preccurlyeq}$ is a lattice.

Proof
We have that:


 * $\struct {S, \vee}$ is a commutative idempotent semigroup


 * $\struct {S, \wedge}$ is a commutative idempotent semigroup.

That is, $\struct {S, \vee}$ and $\struct {S, \wedge}$ are semilattices.

We are also given that $\vee$ and $\wedge$ satisfy the absorption laws:


 * $\forall a, b \in S: a \vee \paren {a \wedge b} = a = a \wedge {a \vee b}$

Finally we note that from Semilattice has Unique Ordering such that Operation is Supremum, there exists a unique ordering $\preccurlyeq$ on $S$ such that:
 * $a \vee b = \sup \set {a, b}$

where $\sup \set {a, b}$ is the supremum of $\set {x, y}$ $\preccurlyeq$.

Hence, by definition, the ordered structure:
 * $\struct {S, \vee, \wedge, \preccurlyeq}$

is a lattice.