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 X_{D},
the set of prime filters of D. Further let φ : D → *U*( X_{D}
) 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 X_{B}
is an antichain and thus *U*( X_{B}) = *P*( X_{B}).