7. Problems

Problem N-1. (a) On a sketch of 2$^4$, indicate two elements that generate 2$^4$ as a Boolean algebra. (b) Choose a third element at the same level as your two generators, and express it in Boolean normal form in terms of the generators.

Problem N-2. In the examples of §, which are necessarily complete? Which are necessarily atomic?

Problem N-3. Prove Propositions 1 through 4 of § regarding Boolean rings.

Problem N-4. Decide which of the examples in § are atomic, which are atomless, and which (if any) are neither.

Problem N-5. Let $X$ be a countably infinite set.

(a) Show that Pow$(X)$ contains a chain isomorphic to the chain R of reals.

(b) Show that Pow$(X)$ contains an uncountable antichain of elements whose pairwise meets are finite subsets of $X$.

Problem N-6. (a) Show that if $p$ is an atom of a Boolean lattice $B$ and $x \in B$, then either $p \leq x$ or $x \leq p'$, but not both.

(b) Show that in a complete Boolean lattice $B$, any atom $p$ is completely join-prime (or sup-prime); in other words, $p \leq \sup S$ implies $p \leq s$ for some $s \in S$. (Suggestion: Somehow use $p'$.)

Problem N-7. Prove this representation theorem: Any atomic complete Boolean lattice is isomorphic to Pow$(X)$ for some set $X$. (A lattice is said to be atomic if every element is the $\sup$ of a set of atoms.)

(Notice that most of our representation theorems have used some subsets of a set; this representation theorem uses all subsets. Alternatively, this theorem can be regarded as a characterization of the lattices Pow$(X)$--they are the atomic complete Boolean lattices.)

Problem N-8. Let $L$ be a Boolean lattice with prime ideal space $\Pi (L)$. Each lattice property of $L$ should be reflected in some topological property of $\Pi (L)$. Here is one example: Show that $L$ is atomic if and only if the isolated points of $\Pi (L)$ form a topologically dense subset.

(An isolated point in a topological space is a point that is open, as a singleton. A subset is dense if its closure is the whole space.)

Problem N-9. Let $B$ be a Boolean lattice. Show that Open$(\Pi (B)) \cong$   Ideals$(B)$, where Open$()$ denotes the lattice of open sets of a topological space.

Problem N-10. As discussed in class, if $f:B \rightarrow C$ is a homomorphism of Boolean algebras, then there is a corresponding continuous map $\hat f: \Pi(C)\rightarrow \Pi(B)$, and if $h:X\rightarrow Y$ is a continuous map between Boolean spaces, then there is a corresponding homomorphism $\overline h :$   Clopen$(Y)\rightarrow$   Clopen$(X)$ of Boolean algebras.

Invent and state definitions for $\hat f$ and $\overline h$ (without writing the proof that they make sense) and then prove one of the two assertions in the following Proposition:

Proposition. $\overline {\hat f} = f$ and $\hat {\overline h} = h$, up to the identifications of Boolean algebras or Boolean spaces with their ``double duals¹.''

Problem N-11. Show that any two countable, atomless Boolean algebras are isomorphic.

(A Boolean algebra is atomless if (surprise!) it has no atoms. An example of a countable, atomless Boolean algebra is FBA$(\aleph _ 0)$, the free Boolean algebra on countably many generators, which can be constructed by first making FBA$(1) \subseteq$   FBA$(2) \subseteq$   FBA$(3) \subseteq \dots$ using Venn diagrams and then taking their union--all the subsets you get at all stages. Another example is Clopen$( 2 \times 2 \times 2 \times \dots )$, where $2$ means $\{0,1\}$ as a discrete topological space; this is the same as the lattice of all subsets of $2 \times 2 \times \dots$ that are describable by referring only to finitely many coordinates, for example, ``the subset consisting of all sequences whose second and fourth entries are either 1 and 0 or 0 and 1''.)