· Stone Prime Filter Theorem
· Representation Theorem for bounded distributive lattices
Material for this week: 10.18, 10.20, 10.21
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).