nLab countably 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 countable joins: A filter $F$ on a $\sigma$-complete lattice $L$ is countably prime or $\sigma$-prime if, for any countable index set $I$, $x_k \in F$ for some $k$ whenever $\bigvee_{i\colon I} x_i \in F$.

If $L$ is a $\sigma$-frame, then a countably prime filter of $L$ is given by a $\sigma$-frame homomorphism from $L$ to the initial $\sigma$-frame, which is the boolean domain in classical mathematics and in constructive mathematics that assumes the limited principle of omniscience. Thinking of $L$ as a $\sigma$-locale, this is the same as a point of $L$.

 Related concepts

• sigma-complete lattice

• sigma-frame

• sigma-locale

• locale of real numbers

• completely prime filter