7. Problems

Problem AA-1. Prove Proposition .

Problem AA-2. Represent the 1-unary algebra $\langle {\cal A}; f \rangle$ explicitly as a subdirect product of SI algebras, where ${\cal A}$ has diagram $\parbox{1in}{\includegraphics{text/Adir/1-unary.eps}}$

Problem AA-3. Let $L$ be a distributive lattice and let $a \in L$. Define $\phi _ {\wedge a}: L \rightarrow L$ by $\phi _ {\wedge a} (x) = x \wedge a$ and likewise $\phi _ {\vee a}$ by $\phi _ {\vee a} (x) = x \vee a$. As you know, these are lattice homomorphisms.

(a) Show that $\ker \phi _ {\wedge a} \cap \ker \phi _ {\vee a} = 0$. (Make a one-line proof based on the absorption law for lattices.)

(b) What embedding does (a) give?

(c) Show that the only SI distributive lattice is    2$$. (Thus this fact is very elementary. The subdirection representation theorem then says that every distributive lattice is a subdirect product of copies of    2$$, a deeper fact that depends on the Axiom of Choice.)

Problem AA-4. Say how to represent the group $F _ {Q _ 8} (2)$ as a subdirect product of subdirectly irreducible groups, using as few factors as possible, by referring to the diagram of its normal subgroups.

Problem AA-5. (a) Which finite abelian groups are SI? (Use any facts you know about finite abelian groups and their subgroup diagrams. An SI abelian group has a smallest proper subgroup.)

(b) Find all SI abelian groups, finite and infinite. (They can be described as subgroups of the circle group--the multiplicative group of all complex numbers of absolute value 1.)

Problem AA-6. (a) Show that an SI 1-unary algebra has no ``fork'', i.e., distinct elements $a,b,c$ with $c = f(a) = f(b)$.

(Method: Let $\langle a \rangle$ denote the subalgebra generated by $a$, and similarly for $b$. For a subalgebra $S$ of ${\cal A}$ let $\theta _ S$ mean the congruence relation obtained by collapsing $S$ to a point. Show that $\theta _ {\langle a \rangle} \cap \theta _ {\langle b \rangle} \cap$   con$(a,b) = 0$ if $a,b$ give a fork. You may use the fact that con$(a,b)$ is obtained by first identifying $f ^ i (a)$ with $f ^ i (b)$ for each $i$ and then seeing what equivalence relation that generates.)

(b) Using (a), try to find all finite SI 1-unary algebras whose diagram is connected.

(A useful observation: In an $n$-cycle, you get exactly the same congruences as for the abelian group Z$_ n$, so the congruence lattice of an $n$-cycle is isomorphic to Subgroup$($Z$_ n)$.)

Problem AA-7. Show that the finite SI 1-unary algebras are

(i) The algebra consisting of two fixed points,

(ii) the ``cyclic'' 1-unary algebras ${\cal C} _ {p ^ k}$ of prime power order (with $k \geq 1$),

(iii) the algebras ${\cal D} _ k, f$ where ${\cal D} _ k = \{0,\dots, k\}$ and $f(0)=0$, $f(i)= i-1$ for $i>0$.

(iv) the two-component algebras where one component is a fixed point and the other is of kind (ii).

(In (ii), it is handy to make this observation, which you may justify very briefly: The congruence relations on an $n$-cycle regarded as a 1-unary algebra are exactly the same as those on the cycle regarded as the group or ring Z$_ n$. In all parts, you may justify briefly why these are SI; it is most important to explain why any finite SI must be of one of these forms.)

Problem AA-8. Consider the ring ${\cal A} =$   Z$_ 2 \oplus$   Z$_ 2 \oplus \dots$, the ``direct sum'' of countably many copies of the ring Z$_ 2$, or in other words, the subring of Z$_ 2 \times$   Z$_ 2 \times \dots$ consisting of the sequences that have only finitely many nonzero entries.

(a) Index the direct sum using $\omega = \{0,1,2\dots \}$. Show that the ideals of ${\cal A}$ correspond to subsets of $\omega$.

(b) Show that if ${\cal A} \equiv {\cal B} \times {\cal C}$, then at least one of ${\cal B}$ and ${\cal C}$ is isomorphic to ${\cal A}$. (Method: ${\cal A}$ would be the internal direct sum of corresponding ideals $I,J$, so that $I \cap J = (0)$ and $I + J = A$.)

(c) Show that ${\cal A}$ is not the direct product of directly indecomposable algebras. (Use a cardinality argument.)

Problem AA-9. (a) Show that direct-product decompositions of a commutative ring with 1 into two factors correspond to idempotents (elements $e$ with $e ^ 2 = e$).

(b) Let $R$ be the ring of all $\omega$-indexed sequences of zeros and ones that are ``eventually constant'', with sequences added and multiplied using the operations of $Z _ 2$ as a ring. Find all direct-product decompositions of $R$.

(c) In (b), does $R$ have a direct decomposition into directly indecomposable factors? (Why or why not?)

(d) What about the Boolean algebra Pow$_ {fin} (X)$ for countably infinite $X$?

Problem AA-10. Suppose that ${\cal A}$ is a finite algebra. An interesting question is whether Var$({\cal A})$ contains finite SI algebras larger than ${\cal A}$, or even contains an infinite SI algebra. If ${\cal A}$ is a lattice, for example, there are no larger SI's; if ${\cal A}$ is a nonabelian $p$-group, the answer is that there are arbitrarily large finite SI's and also infinite ones. An easy case:

(a) Show that Shallon's algebra is SI, and in fact is simple. (Method: Think about con$(r,s)$ for different possible distinct elements $r,s$.)

More generally, Let ${\cal A} _ n$ be the graph algebra based on a graph like Shallon's but with $n$ nodes, so that ${\cal A} _ n$ has $n+1$ elements and Shallon's algebra is ${\cal A} _ 3$. Show that ${\cal A} _ n$ is SI.

(b) Show that ${\cal A} _ n \in$   Var$({\cal A} _ 3)$. (Suggestion: Write ${\cal A} _ 3 = \{a _ 1, a _ 2, a _ 3,0\}$. Inside ${\cal A} _ 3 ^ n$, let $B$ be the subalgebra generated by elements whose entries are $a _ 1$'s (zero or more), then one $a _ 2$, and then the rest $a _ 3$'s. Let $\theta$ on $B$ be the equivalence relation obtained by identifying all elements of $B$ that have an entry of 0 and letting other blocks be singletons. Show that $\theta$ is a congruence relation on $B$. Then $B/\theta \cong \dots$.)

(c) Can you find an infinite SI in Var$({\cal A} _ 3)$?