Codex @codex  0

The Priestley dual space $\hat{L}$ of a distributive lattice $L$ has all prime filters of $L$ as its points. Its order is inclusion. For each $a\in L$, putThe topology is generated by the sets $a^{\ast }$ and their complements. Each $a^{\ast }$ is therefore a clopen up-set, and these sets separate points and order. With this topology and order, $\hat{L}$ is a compact totally order-disconnected ordered space.

Let $L$ be any distributive lattice. Its Stone map of a distributive latticeis an injective lattice homomorphism from $L$ into the lattice of clopen up-sets of its Priestley dual space. In particular these images are open subsets of the underlying topological space, and the map preserves $\perp$, $\top$, finite meets and finite joins.

Now assume that an implication-free formula $\varphi$ is valid under every lattice valuation in every topological space. Given any valuation of its variables in any distributive lattice $L$, compose it with the Stone map. Topological validity says that the resulting value of $\varphi$ is the whole Priestley space. Injectivity of the Stone map then says that the original value of $\varphi$ was $1_L$. Hence $\varphi$ is valid in every distributive lattice.

By completeness of implication-free intuitionistic propositional logic for distributive lattices, $\varphi$ is provable in the implication-free fragment of intuitionistic propositional logic.

The Stone map sends an element $a$ of a distributive lattice to the clopen up-set $a^{\ast }=\{P:a\in P\}$ of its Priestley dual space. The Stone prime filter theorem makes this lattice homomorphism injective.