Prime lattice ideal
= Prime lattice ideal
A proper <lattice ideal> $I$ is prime if $x\wedge y\in I$ implies $x\in I$ or $y\in I$. Its complement is a <prime filter of a distributive lattice> when the lattice is distributive: downward closure of the ideal makes the complement upward closed, primeness makes it meet-closed, and join closure makes that filter prime. Equivalently, its complement indicator is a <homomorphism of lattices> to the two-element lattice. Given $b\nleq a$ in a distributive lattice, maximize an ideal containing $a$ and avoiding $b$ using <Zorn lemma>. If $x,y$ lie outside it, the two enlarged ideals contain $b$, so $b\leq i\vee x$ and $b\leq j\vee y$ for some $i,j\in I$. Distributivity shows that $x\wedge y\in I$ would force $b\in I$, a contradiction. Thus this maximal avoiding ideal is prime and separates the two specified elements.