completely prime filter in nLab

Contents

Definition

Recall that a filter FF on a lattice LL is called prime if ⊥∉F\bot \notin F and, whenever x∨y∈Fx \vee y \in F, then x∈Fx \in F or y∈Fy \in F. In other words, for every finite index set II, x k∈Fx_k \in F for some kk whenever ⋁ i:Ix i∈F\bigvee_{i\colon I} x_i \in F.

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

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

• locale

• locale of real numbers

• countably prime filter

 References

• Ming Ng, Adelic Geometry via Topos Theory, PhD thesis (pdf)