 ## Mathematics 567: Boolean Algebra

Instructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail)

e-mail: jdhsmithATmathDOTiastateDOTedu (substitute punctuation)

Office Hours: Mon. 1:10pm, 3:10 pm; Wed. 1:10pm, 3:10pm, 4:10pm (subject to change)

Textbook: P.T. Johnstone, Stone Spaces, Cambridge University Press, ISBN 0-521-33779-8

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

Three questions from the following:
1. 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 .
2. Let  B  be a finite Boolean algebra. Show that the distributive lattice  BJ  of ideals of  B  is a Boolean algebra.
3. Let  B  be the power set of the set  N  of natural numbers. Show that the distributive lattice  BJ  of ideals of the Boolean algebra  B  is not a Boolean algebra.
4. Let  G  be a group, and let  L  be the lattice of normal subgroups of  G, ordered by set-theoretic inclusion.
1. Show that the lattice  L .is complete.
2. Show that the lattice  L .is algebraic.
5. Let  L  be the lattice of divisors of  72 , ordered by divisibility.
1. Show that  L .is a locale.
2. 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.]
3. 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.]

Three questions from the following:
1. A bounded lattice L is atomic if for each non-zero 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.
2. Show that a locale is spatial if and only if each element is the meet of a set of prime elements.
3. 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.
4. Let X be a topological space.
1. 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} .
2. Show that a continuous map
f : X --> Y
induces a functor from the category X to the category Y .
5. Let ( X , W(X) ) be a T0-space. Show that the following conditions are equivalent:
1. The topology W is the Alexandrov topology on the specialization order of the space X ;
2. The meet semilattice W is complete;
3. Every point of X has a smallest open neighborhood.

Three questions from the following:
1. Show that a lattice is a chain if and only if all its proper ideals are prime.
2. Show that each distributive lattice is isomorphic to a sublattice of the lattice of subsets of some set.
1. 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.
2. 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.
3. 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.
4. Let M be a monoid. Recall that an M-set 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·mn = x·(m n) .
A subset Y of X is called an M-subset if y in Y and m in M imply y·m in Y.
1. Show that the set of M-subsets of X forms a Heyting algebra under the order relation of containment.
2. If M is a group, show that this Heyting algebra is a Boolean algebra.

Three questions from the following:
1. Show that the de Morgan law
( x + y ) ' = x ' · y '
holds in any Heyting algebra, while the dual statement may fail.
2. Show that the identity
( x · y ) ' ' = x ' ' · y ' '
holds in any Heyting algebra.
3. Show that in a Heyting algebra H, the following statements are equivalent:
1. For all  x , y  in H, the equation ( x · y ) ' = x ' + y ' holds;
2. For all  x  in H, the equation  x ' + x ' ' = 1  holds;
3. For all  x  in H, the equation  x = x ' '  implies that  x  has a complement;
4. The subset  { x | x = x ' ' }  of H forms a sublattice of H .
4. Recall the symmetric difference operation
x xor y = ( x · y' ) + ( x' · y )
in a Boolean algebra.
1. In a Boolean algebra, show that the operation
P ( x , y , z ) = x xor y xor z
is a Mal'tsev operation.
2. Express the operation  P  of (i) in terms of the meet, join, and implication.
3. If the result of (ii) is interpreted in a Heyting algebra, is it still a Mal'tsev operation? Justify your answer.
5. 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 .