nLab
completely prime filter

Recall that a filter FF on a lattice LL is called prime if F\bot \notin F and, whenever xyFx \vee y \in F, then xFx \in F or yFy \in F. In other words, for every finite index set II, x kFx_k \in F for some kk whenever k:Ix iF\bigvee_{k\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 kFx_k \in F for some kk whenever k:Ix iF\bigvee_{k\colon I} x_i \in F. Equivalently, a completely prime filter is given by a simlutaneous 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.

Created on February 5, 2010 22:01:49 by Toby Bartels (173.60.119.197)