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 by hypothesis 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 $\set {x, y}$ with respect to $\preccurlyeq$.

Hence, by definition, the ordered structure:

$\struct {S, \vee, \wedge, \preccurlyeq}$

is a lattice.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.23 \ \text {(b)}$