nLab completely prime filter

Definition

Recall that a filter $F$ on a lattice $L$ is called prime if $\bot \notin F$ and, whenever $x \vee y \in F$, then $x \in F$ or $y \in F$. In other words, for every finite index set $I$, $x_k \in F$ for some $k$ whenever $\bigvee_{i\colon I} x_i \in F$.

We now generalise from finitary joins to arbitrary joins: A filter $F$ on a complete lattice $L$ is completely prime if, for any index set $I$ whatsoever, $x_k \in F$ for some $k$ whenever $\bigvee_{i\colon I} x_i \in F$. Equivalently, a completely prime filter is given by a simultaneous suplattice and lattice homomorphism from $L$ to the lattice $TV$ of truth values (which is classically the boolean domain $\mathbb{2}$).

In particular, if $L$ is a frame, then a completely prime filter of $L$ is given by a frame homomorphism from $L$ to $TV$. Thinking of $L$ as a locale, this is the same as a point of $L$.

 See also

• locale

• locale of real numbers