Brouwerian Lattice is Distributive

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Theorem

Let $\struct {S, \preceq}$ be a Brouwerian lattice.


Then $\struct {S, \preceq}$ is a distributive lattice


Proof

Let $x, y, z \in S$.

By Brouwerian Lattice iff Shift Mapping is Lower Adjoint:

$\forall x \in S, f: S \to S: \paren {\forall s \in S: \map f s = x \wedge s} \implies f$ is a lower adjoint

Define a mapping $f: S \to S$:

$\forall s \in S: \map f s = x \wedge s$

Then:

$f$ is a lower adjoint

By Lower Adjoint Preserves All Suprema:

$f$ preserves all suprema

By definition of preserves all suprema:

$f$ preserves the supremum of $\set {y, z}$

By definition of lattice:

$\set {y, z}$ admits a supremum

By preserves the supremum of set:

$\map \sup {\map {f^\to} {\set {y, z} } } = \map f {\sup \set {y, z} }$

Thus

\(\ds x \wedge \paren {y \vee z}\) \(=\) \(\ds \map f {y \vee z}\) Definition of $f$
\(\ds \) \(=\) \(\ds \map f {\sup \set {y, z} }\) Definition of Join
\(\ds \) \(=\) \(\ds \map \sup {\map {f^\to} {\set {y, z} } }\)
\(\ds \) \(=\) \(\ds \map \sup {\set {\map f y, \map f z} }\) Definition of Image of Subset under Mapping
\(\ds \) \(=\) \(\ds \sup \set {x \wedge y, x \wedge z}\) Definition of $f$
\(\ds \) \(=\) \(\ds \paren {x \wedge y} \vee \paren {x \wedge z}\) Definition of Join

Thus by definition:

$\struct {S, \preceq}$ is a distributive lattice

$\blacksquare$


Sources

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