We now generalise from finitary joins to arbitrary joins: A filter on a complete lattice is completely prime if, for any index set whatsoever, for some whenever . Equivalently, a completely prime filter is given by a simlutaneous suplattice and lattice homomorphism from to the lattice of truth values (which is classically the boolean domain ).
In particular, if is a frame, then a completely prime filter of is given by a frame homomorphism from to . Thinking of as a locale, this is the same as a point of .
Created on February 5, 2010 22:01:49
by Toby Bartels