8. Problems

(Some of these problems depend on additional material from lectures.)

Problem X-1. Describe (a) the free 1-unary algebra on $n$ generators;

(b) $F _ V (2)$, where $V$ is the variety of 1-unary algebras with $f ^ 3 (x) = f ^ 5 (x)$;

(c) the free 2-unary algebra on $1$ generator;

(d) $F _ S (3)$, where $S = \langle$   2$,\vee \rangle$, using the table method. (Here $S$ is a semilattice--a set with a single binary operation that is associative, commutative, and idempotent. A semilattice can also be defined as a set with a partial order such that any two elements have a least upper bound. Thus one way to obtain a semilattice is to take a lattice and ignore the meet operation, as has been done to make $S$.)

Problem X-2. Theorem. In a group $G$, every commutator $a ^ {-1} b ^ {-1} ab$ is a product of squares.

Proof #1. Let $S = \{$products of squares$\}$. Observe that $S$ is a normal subgroup. Moreover, $G/S$ satisfies $x ^ 2 = e$ and so is abelian. Then in $G/S$, $\overline a ^ {-1} \overline b ^ {-1} \overline a \overline b = \overline e$. This is the same as saying that in $G$, $a ^ {-1} b ^ {-1} ab \in S$.

This proof was indirect. A more direct proof would be to exhibit a law $x ^ {-1} y ^ {-1} xy = (\dots ) ^ 2 (\dots ) ^ 2 \dots (\dots ) ^ 2$ true in all groups, where each $(\dots )$ contains some expression in $x,y$.

(a) Before attempting to give such a proof, explain why there must exist a direct proof of this form.

(b) Somehow or other, find the direct proof.

Problem X-3. For Murskii's algebra $M$, suppose you want to compute $F _ M (2)$, using the table method. (a) Show what generating rows you would use. (b) Compute new rows in some reasonable order, labeling each row with the expression in the generators that produced it, until you generate a row that is already there. What law have you found? (c) If your law was in one variable, continue further until you get a law involving two variables. (d) Actually, $F _ M (2)$ has 11 elements. How many multiplications of rows would be involved in computing the whole free algebra and verifying that you are done?

Problem X-4. Two proofs of the existence of the free algebra $F _ V (n)$ are described in §6 above. They sound very different. Nevertheless, they are essentially the same. The problem: Explain why, by analyzing how the two elements of the first one are really present in the second.

Problem X-5. (a) Suppose that an algebra $F$ has a given set of generators $g _ 1,\dots, g _ n$. Show that if $F$ has the universal mapping property for maps into itself, then $F$ is free in some variety $V$. (Thus being free is in effect an absolute property of an algebra, without having to name a variety containing it.)

(b) An achievement of recent years was the solution of the restricted Burnside problem: For any $k$ and $n$, there is a largest finite group with $n$ generators that obeys $x ^ k = 1$. (There could also be infinite groups fitting this description; it's just that there is a largest finite one.) Is this largest finite group necessarily free? (Discuss.)

Problem X-6. Let $V$ be the variety of idempotent semigroups: 1-binary algebras whose operation is associative and obeys the law $x ^ 2 = x$.

By experimenting with expressions, make a conjecture as to whether $F _ V (3)$ is finite or infinite. Explain briefly how you arrived at your answer.

Problem X-7. The term algebra $T _ {\tau}(n)$ is described in §3 above; in §6 it is used in the second proof of the existence of free algebras in a variety.

For the variety $V$ of 1-unary algebras obeying the law $f ^ 3 (x) = f ^ 5 (x)$ and for $n=1$, explicitly describe $T _ \tau (1)$ and all $\theta \in$   Con$(T _ \tau (1))$ giving a quotient in $V$. (Here $\tau = \langle 1 \rangle$.)

Problem X-8. Consider the ``constructions'' H$,$S$,$P on classes of algebras.

(a) Say which containment relations between pairs of constructions must hold, e.g., SH$({\cal K}) \subseteq$   HS$({\cal K})$. (All the valid relations have easy proofs, but it is not required to write them down. Interpret H$,$S$,$P up to isomorphism.)

(b) For one such potential relation that does not hold, find a counterexample, with brief proof.

Problem X-9. Let $F = F _ {Q _ 8}(2)$. Refer to Figure . (a) Find a normal subgroup $N$ of $F$ such that $F/N \cong$   Z$_ 2 \times$   Z$_ 4$. (b) Find a normal subgroup $N$ such that $F/N \cong D _ 8$. Find a normal subgroup $N$ such that $F/N \cong Q _ 8$. Find the commutator subgroup $F'$ of $F$. (Determine the order of each subgroup. Recall the Correspondence Theorem, which says that the subgroups of $F$ that contain $N$ form the same diagram as the subgroups of $F/N$; the same is true if just normal subgroups are considered. From the previous problem you know that for abelian 2-groups (groups whose order is a power of 2), the group can be identified from the subgroup diagram. Recall that $F'$ is contained in every $N$ for which $F/N$ is abelian.)

Problem X-10. Figure shows homomorphisms of FML$(3)$ onto FDL$(3)$ and $M _ 3$, determined by mapping generators to generators.

On a copy of Figure , indicate ker$\; \alpha$ and ker$\; \beta$. (You will need to decide which elements go to which, but you need not write this information down. A congruence relation on a finite lattice is best diagrammed simply by darkening the coverings that are ``collapsed'', i.e., coverings between elements in the same block. Use different coloring or markings for the two congruence relations involved.)

Note. If there are surjections $A \rightarrow B$ and $A \rightarrow C$ whose kernels have intersection 0, then $A$ is embeddable in $B \times C$, as we'll discuss in class. Since this is the case in Figure , you have shown the interesting fact that FML$(3)$ is embeddable in the direct product of FDL$(3)$ and a single copy of $M _ 3$.

Figure: Two homomorphisms

Problem X-11. For the free algebra from the table shown in Figure :

(a) Whenever we subtract two rows we get a relation between generators, which is then a law, usually nontrivial. What relation between generators, and so what law, comes from the computation R8-R9 = 0 2 1 1 0 2 2 1 0 = R5, where R8 means row 8, etc.?

(b) Suppose we want to use the universal mapping property to map $F$ to $A$ with $g \mapsto 2$, $h \mapsto 1$. Which column of the table gives the projection that achieves this, and what is the homomorphism on $F$?