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 X_{D} of all prime filters of
the lattice D and prove that you aren’t missing any;

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

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

3.* Let
D be a bounded distributive lattice, and x an element of X_{D}, the set
of prime filters of D. Further let ( ^ ):
D → *U*( X_{D} ) 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 X_{B} is an antichain
and thus *U*(
X_{B}) = *P*(
X_{B}), the power set of X_{B}.