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A prime filter of a distributive lattice $P$ is a proper lattice filter: it contains the top element, is upward closed, and is closed under finite meets. Primality means

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.

The Stone prime filter theorem says that if a lattice filter $F$ and a lattice ideal $I$ of a distributive lattice are disjoint, then there is a prime filter of a distributive lattice $P$ such thatEquivalently, whenever $a\nleqq b$, there is a prime filter containing $a$ and omitting $b$.

The Priestley dual space $\hat{L}$ of a bounded distributive lattice $L$ consists of its prime filters, ordered by inclusion and topologized by the sets $a^{\ast }=\{P:a\in P\}$ and their complements.

If a lattice filter $F$ and lattice ideal $I$ in a distributive lattice are disjoint, there is a prime filter of a distributive lattice containing $F$ and disjoint from $I$.