WEEK 49:

 

·         Stone Prime Filter Theorem

 

·         Representation Theorem for bounded distributive lattices

 

Material for this week:    10.18, 10.20, 10.21

 

 

Homework 12:

 

1. Let P be a poset, and p,q є P. If  p v q exists then we have  ↑p ∩ ↑q= ↑(p v q).

 

2. Find all the prime filters of the lattice ^Å ( Z-  x Z- ) and prove that there aren't any others.

 

3. Let D be a bounded distributive lattice, and x an element of  XD, the set of prime filters of D. Further let φ : D →  U( XD ) be the embedding from Stone's Representation Theorem.  Show that the principal upset ↑x  is in the image of the map φ 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).