8. Problems

Problem V-1. Verify that any congruence relation on a group is simply the coset decomposition determined by some normal subgroup.

Problem V-2. For general algebras, prove (a) the first isomorphism theorem (Theorem ); (b) the correspondence theorem (Theorem ).

Problem V-3. (a) Any function $f:X\rightarrow Y$ on sets induces an equivalence relation on its domain $X$, where $x \sim x'$ means $f(x)=f(x')$. Show that for groups $G$ and $H$, if $\varphi: G \rightarrow H$ is a homomorphism then any single block of the equivalence relation it induces determines all the blocks. (This is why the ``kernel'' of $\varphi$ is defined to be a single block, the one containing $e$.)

(b) Give an example of two algebras and two homomorphisms $\varphi, \varphi'$ between them such that $\varphi$ and $\varphi'$ give different equivalence relations that do have at least one block in common. (This is why the ``kernel'' of $\varphi$ is defined to be the whole equivalence relation rather than a single block, for algebras in general.)

Problem V-4. Explain how the congruence lattice of ${\cal A}$ is a sublattice of the partition lattice of $A$ as a set.

Problem V-5. State and prove a version of Theorem that refers to two surjective homomorphisms $\varphi _ i: {\cal A} \rightarrow {\cal B} _ i$ ($i=1,2$), rather than to two congruence relations on ${\cal A}$.

Problem V-6. If $\varphi: {\cal A} \rightarrow {\cal B}$ is a surjective homomorphism, show that there is a lattice embedding of Con$({\cal B})$ into Con$({\cal A})$, with the image being an interval.

Problem V-7. Invent a correspondence theorem (like Theorem ) for a surjective homomorphism $\varphi: {\cal A} \rightarrow {\cal B}$ that relates subalgebras of ${\cal B}$ to certain subalgebras of ${\cal A}$. Somehow describe which ones. (No proof is required.)

Problem V-8. Show that the subdirect embedding theorem () holds for the intersection of a possibly infinite family of congruence relations.

Problem V-9. For the case ${\cal A} =$   Z, the ring of integers, give (a) an example of the subdirect embedding theorem in which the two congruence relations come from prime ideals, and (b) an example where neither comes from a prime ideal. In each case, say what the embedding does to each element.

Problem V-10. Prove that the congruence lattice of a chain of length $n$ (as an algebra with lattice operations) is the Boolean lattice 2$^ n$. (The length of a chain is the number of jumps, so a chain of length $n$ has $n+1$ elements.)

Problem V-11. Let ${\cal D}$ be a distributive lattice and consider any $d \in D$. Define maps $f _ {\vee d}: D\rightarrow D$ and $f _ {\wedge d}: D\rightarrow D$ by $f _ {\vee d}(x) = d \vee x$ and $f _ {\wedge d}(x) = d \wedge x$. (a) Show that $f _ {\vee d}$ and $f _ {\wedge d}$ are homomorphisms. (b) Show that the intersection of their kernels is 0 (i.e., equality). (c) Use Theorems and to show that ${\cal D} \hookrightarrow ( d ] \times [ d )$.

Problem V-12. Compute all the principal congruence relations in Figure . Indicate blocks by darkening coverings between two elements in the same block. You may omit examples already done in lecture.