7. Problems

Problem J-1. Let $L$ be the lattice of all finite subsets of an infinite set $X$. Characterize Ideals$(L)$ up to isomorphism, preferably as a more familiar lattice.

Problem J-2. Prove Fact (e) in §. If the lattice is distributive, relate this to meet-irreducibility.

Problem J-3. Show that the following conditions are equivalent for an ideal $\pi$ in a lattice $L$:

1. $\pi$ is a prime ideal;

2. $L \! \setminus \! \pi$ is a dual ideal;

3. $L \! \setminus \! \pi$ is a prime dual ideal; i.e., if $D = L \setminus \! \pi$, $D$ is a dual ideal such that $x \vee y \in D$ implies $x \in D$ or $y \in D$.

Problem J-4. (a) Explain how homomorphisms of a lattice $L$ onto 2 correspond to prime ideals of $L$.

(b) For $L =$   2$\times$   3, there are several homomorphisms of $L$ onto 2. For each, indicate its kernel by darkening coverings on a copy of $L$.

Problem J-5. Let $X$ be a countably infinite set. In the Boolean algebra Pow$_ {fin}(X)$ of all finite and cofinite subsets of $X$, find a prime ideal that is not principal.

Problem J-6. Let ${\cal C}$ be the ring of all continuous functions $f : [0,1] \rightarrow$   R, where R is the field of reals and $[0,1] \subseteq$   R. For $x \in [0,1]$, let $M_x = \{f \in {\cal C} : f(x) = 0\}$.

(a) Show that $M_x$ is a maximal ideal, for each $x$.

(b) Show that every maximal ideal of ${\cal C}$ is of the form $M_x$.

(c) The support of a function $f$ is $\{x: f(x) \neq 0\}$, the set of points where the function does not vanish. Show that the supports of continuous functions on $[0,1]$ form a base for the topology of $[0,1]$. In other words, every open subset of $[0,1]$ is a union of supports.

For a commutative ring $R$ with 1, let ${\cal M}$ be its set of maximal ideals. The support of an element $r \in R$ is $\{M \in {\cal M} : r \not \in M\}$. The maximal ideal space of $R$ is ${\cal M}$ topologized by using the supports of elements as a base.

(d) Show that the maximal ideal space of ${\cal C}$ is homeomorphic to $[0,1]$. Thus $[0,1]$ can be reconstructed from ${\cal C}$, up to homeomorphism.

(e) Outline how it could be proved that ${\cal C}$ is not isomorphic to the ring ${\cal D}$ of real-valued continuous functions on the unit circle. (It is not necessary to prove the steps.)

Problem J-7. Let $L$ be a lattice and let $I _ \gamma, \gamma \in \Gamma$ be a family of ideals of $L$, possibly infinitely many. Describe (with proof) the elements of the ideal generated by all $I _ \gamma$, i.e., the smallest ideal containing all the ideals $I _ \gamma$. Your description should generalize §(c).