WEEK 42:

 

·             Stone’s Prime Filter Theorem

 

·             Representation Theorem for bounded distributive lattices

 

 

Homework:

 

1.* Exercise on filters and ideals as generalized elements.

(a) Let L be a finite lattice. Show that F is a filter of L if and only if F is a principal up-set, ­a, for some a in L, and state the dual fact for ideals.

(b) Let F =­a be a principal filter is some lattice L(not necessarily finite), show that F is prime if and only if a is join-irreducible in L.

(c) Let F =­a be a principal filter is some lattice L(not necessarily finite), show that F is maximal among proper filters if and only if a is an atom in L.

 

 

2.* Exercise about the lattice D consisting of the Cartesian product of  N upside-down with itself with a bottom added.

(a) Find the poset XD of all prime filters of the lattice D and prove that you aren’t missing any;

(b) For each a in D let Xa denote the set of prime filters of D containing a. Are there any elements of U(XD), the lattice of all up-sets of the poset XD, that are not in the image of the embedding from Stone's Representation Theorem that sends each a in D to Xa? Let O denote the closure under arbitrary unions of the collection { Xa | a in D}. Which sets are in O? Show that if a subcollection C of O covers XD (that is, the union of the sets in the collection is all of XD) then some finite subcollection C’ of C also covers XD.

(c) Let O’ denote the closure under arbitrary unions of the collection { Xa , X-Xa | a in D}. Which sets are in O’? Show that if a subcollection C of O’ covers XD (that is, the union of the sets in the collection is all of XD) then some finite subcollection C’ of C also covers XD.

 

3.* Let D be a bounded distributive lattice, and x an element of XD, the set of prime filters of D. Further let ( ^ ): D →  U( XD ) be the embedding from Stone's Representation Theorem.  Show that the principal up-set ↑x  is in the image of the map ( ^ ) if and only if the prime filter x is principal, i.e., x is of the form ↑a for some a є D.

 

4.* (a) Show that if a filter is maximal among the proper filters of a bounded distributive lattice, then it is prime.

    (b) Give an example to prove that the converse is not true in general.

    (c) Show that in a Boolean algebra every prime filter is maximal.

Conclude that for a Boolean algebra B, the set XB is an antichain and thus U( XB) = P( XB), the power set of XB.