
Mathematics 567: Boolean Algebra
Instructor: Jonathan Smith, 496 Carver, 48172 (voice mail)
email: jdhsmithATmathDOTiastateDOTedu (substitute punctuation)
Office Hours: Mon. 1:10pm, 3:10 pm; Wed. 1:10pm, 3:10pm, 4:10pm (subject to change)
Grading: based on five graded homework assignments.
Click here for information about special accommodations.
Textbook: P.T. Johnstone, Stone Spaces, Cambridge University Press, ISBN 0521337798
Study Plan: reserve 1  2 hours for homework between each pair of classes. Successful performance in the class depends critically on completion of the homework assignments.
Syllabus: Selected material from chapters I  III, with possible additional topics of specific interest.
Summary of the key concepts
Homework Assignments
Final graded homework due 4/27:
Three questions from the following:

Let
( B , or , and , not, 0, 1 )
be a Boolean algebra, considered also as a Boolean ring
( B , + , · , 0, 1 ) .
Determine a polynomial f ( x , y ) such that
f ( x , y ) = x nand y
for all x , y in B .

Let B be a finite Boolean algebra. Show that the distributive lattice B^{J} of ideals of B is a Boolean algebra.

Let B be the power set of the set N of natural numbers. Show that the distributive lattice B^{J} of ideals of the Boolean algebra B is not a Boolean algebra.

Let G be a group, and let L be the lattice of normal subgroups of G, ordered by settheoretic inclusion.

Show that the lattice L .is complete.

Show that the lattice L .is algebraic.

Let L be the lattice of divisors of 72 , ordered by divisibility.

Show that L .is a locale.

Determine the topological space Pt L . of points of the locale L . [For full credit, you should specify both the set Pt L . and the set W( Pt L) of open sets in the topology.]

Determine the topological space Spec L for the distributive lattice L . [For full credit, you should specify both the set Spec L . and the set W( Spec L) of open sets in the topology.]
Fourth graded homework due 4/11:
Three questions from the following:

A bounded lattice L is atomic if for each nonzero element x of L, there is an atom a of L such that a < x . Give an example of a Boolean algebra that possesses an atom, but is not atomic. Justify your claims.

Show that a locale is spatial if and only if each element is the meet of a set of prime elements.

Let X be a topological space. Prove that the map
y : X > Pt W(X)
taking an element x of X to the complement of the
closure of { x } is continuous.

Let X be a topological space.

Show that X is the object set of a small category in which there is a (unique) morphism from x to y if and only if
x lies in the closure of the singleton {y} .

Show that a continuous map
f : X > Y
induces a functor from the category X to the category Y .

Let ( X , W(X) ) be a T_{0}space. Show that the following conditions are equivalent:

The topology W is the Alexandrov topology on the specialization order of the space X ;

The meet semilattice W is complete;

Every point of X has a smallest open neighborhood.
Third graded homework due 3/23:
Three questions from the following:
 Show that a lattice is a chain if and only if all its proper ideals are prime.
 Show that each distributive lattice is isomorphic to a sublattice of the lattice of subsets of some set.

 Show that a subset of a Boolean algebra is an ideal of the Boolean algebra if and only if it is an ideal of the corresponding Boolean ring.
 Show that a subset of a Boolean algebra is a prime ideal of the Boolean algebra if and only if it is a prime ideal of the corresponding Boolean ring.
 Give an example of a maximal ideal M in a lattice L such that M is not a prime ideal of L. Justify all your claims.
 Let M be a monoid. Recall that an Mset is a set X such that for each element m of M and x of X, a unique element x·m is defined, satisfying the unital law:
x·1 = x
and the mixed associative law:
(x·m)·n = x·(m n) .
A subset Y of X is called an Msubset if y in Y and m in M imply y·m in Y.
 Show that the set of Msubsets of X forms a Heyting algebra under the order relation of containment.
 If M is a group, show that this Heyting algebra is a Boolean algebra.
Second graded homework due 2/18:
Three questions from the following:

Show that the de Morgan law
( x + y ) ' = x ' · y '
holds in any Heyting algebra, while the dual statement may fail.

Show that the identity
( x · y ) ' ' = x ' ' · y ' '
holds in any Heyting algebra.

Show that in a Heyting algebra H, the following statements are equivalent:

For all x , y in H, the equation
( x · y ) ' = x ' + y '
holds;

For all x in H, the equation
x ' + x ' ' = 1
holds;

For all x in H, the equation
x = x ' ' implies that x
has a complement;

The subset
{ x  x = x ' ' }
of H forms a sublattice of H .

Recall the symmetric difference operation
x xor y = ( x · y' ) + ( x' · y )
in a Boolean algebra.

In a Boolean algebra, show that the operation
P ( x , y , z ) = x xor y xor z
is a Mal'tsev operation.

Express the operation P of (i) in terms of the meet, join, and implication.

If the result of (ii) is interpreted in a Heyting algebra, is it still a Mal'tsev operation? Justify your answer.

In a poset ( X, < ) , a subset U is said to be an upper set or upset if for each element x of X, the existence of an element u in U with u < x implies x in U . Prove that the set of all upsets of a poset ( X, < ) forms a topology on X .
First graded homework due 1/28:
Three questions from the following:
 Express { x  if P (x) then Q (x) else R (x) } in terms of { x  P (x) } , { x  Q (x) } , and { x  R (x) } .
 Define nand by
{ x  P (x) nand Q (x) } = [ { x  P (x) } · { x  Q (x) } ] ' .
Show that in a general Boolean algebra, each of the Boolean algebra operations + , · , and ' may be expressed entirely in terms of the nand operation x  y = ( x · y ) ' and the constant 0.
 In a ring R (with 1), each element x satisfies x^{3} = x . Is 3 x = 0 ?
 In a ring R (with 1), each element x satisfies x^{3} = x and 3 x = 0 . Prove that R is commutative.
 Is the assumption of the existence of a multiplicative identity 1 necessary in Exercise 4?
1/10 for 1/14: Read Sections 1.1  3, do the Exercise at the end of 1.3.

