Where I feel the uncorrected version could cause real misunderstanding, the page numbers are printed in bold face.
In the few cases where I have added a new exercise before the last one in a section, or the like, I show the new one as having the number of the preceding one with ½ added (e.g., see the notes on P.86 [81] and P.150 [p.141] below), to avoid the complication of how to refer to later ones; although in the revised version, the added item actually has the next number, and the later ones have +1 added to their numbers.
Here, now, are the errata and addenda:
P.iv (= back of title page, in published version only), explanation of the two lines in the middle of the page: These refer to the first edition, published for me in 1998, by the one-man publishing company which Henry Helson then ran.
P.7 [p.7], after the paragraph beginning The last two
sections, add another paragraph:
I have been teaching this material as a one-semester course,
which requires moving uncomfortably fast (especially since my university
declared that no new material should be presented in the
last week of the semester).
I think that in the future I will omit some nonessential sections
from the assigned material: at least 8.11, the latter
half of 9.10, and 10.13, and probably some of 4.18, 9.7, 10.7 and 10.9.
(I may vary the exact set from year to year.
Other inessential sections are 5.7 and 6.4, but since these don't
require the student to absorb new material, and are
interesting, I don't intend to drop them.)
P.13 [p.12], just before Exercise 2.2:3: Delete the sentence: I don't know the full answer to the following variant question: (I do now.)
P.16 [p.15], second sentence of condition (c): Change satisfying (aX)−(ae) to closed under the symbolic operations named in (aX)−(ae) .
P.21 [p.19], last sentence of first paragraph of section 2.7: substitution and evaluation maps should just be evaluation maps . (One can think of these as "substitution maps" when one focuses on how the outputs vary with the term, for fixed inputs from the group or other algebra, rather than the other way around; but they are the same functions.)
P.29 [P.26], 3rd-from-last line: ∈T} should be ∈Ri}.
P.37 [p.34], Exercise 3.3:6: In the first line, after the words operation p, add in n variables. On the next two lines, replace α, β, γ with α1, ..., αn, and likewise replace f(α), f(β), f(γ) with f(α1), ..., f(αn).
P.48 [p.45], 3rd line of Exercise 4.2:1: for {xi yi−1} read {xi yi−1 | i ∈ I} .
P.74 [p.70], Exercise 4.8:3: in the 2nd line, change
free groups and the abelianization of a group to
the free group on a set X and the abelianization of a
group G; and add, at the end of the exercise:
(If objects with the indicated universal properties exist for
some sets X or groups G, you might try to
determine exactly which X and/or G this
is true for.)
P.78 [p.74], Exercise 4.9:9, line after the display: x1, ..., xn should be x1, ..., xm .
P.79 [p.74], after Exercise 4.9:9 add:
Exercise 4.9:10. If A and B are abelian
groups, show that their tensor product A ⊗ B can
be written as a homomorphic image of a subgroup of
A * B (their coproduct
as not-necessarily-abelian groups).
P.85 [p.80], third line above Exercise 4.10:9: in inclusion should be the inclusion .
P.86 [p.81], after Exercise 4.11:3 add:
Exercise 4.11:3½. Show that if S is a
finite monoid, then the natural map
|S| → |Sgp| is surjective.
Show, in fact, that (Sgp)md can
be described as the monoid obtained by imposing
on S the relations
i = e for all idempotent elements
i ∈ S, i.e., all elements satisfying
i2 = i.
P.89 line 9 [p.83 last line]: as ring should be as a ring.
P.108 [p.102], last sentence of proof of Lemma 4.17.3: For clarity, change the last part of that sentence, from the words this would give to the end of the proof, to this set would have that empty intersection as a member, contradicting the first sentence above.
P.113 [p.106] end of section 4.17; add:
(However, in the preceding discussion, where I referred to the
fact that every ring
homomorphism B(X) → ℝ
is continuous, this implicitly used the fact that the only endomorphism
of ℝ is the identity, which allows one to deduce that
all ring homomorphisms B(X) → ℝ
are ℝ-algebra homomorphisms.
The corresponding fact about endomorphisms is not true
for ℂ so the results have to be stated in terms
of ℂ-algebra homomorphisms rather than ring
homomorphisms.)
P.115 [p.108]: In the diagram (which shows the second univeral property of U), the top arrow should be labeled u, not f.
P.129 [p.120], Exercise 5.2:1: Change the parenthetical statement (This requires ...) to (A solution must show that the two constructions give structures of the indicated sorts, and that performed successively in either order, they return the original data.)
P.129, last paragraph [p.121, paragraph preceding Exercise 5.2:2], 5th line: change depend on 5.2:2-5.2:3, and 5.2:5 to depend on 5.2:2(i) and 5.2:3. 5.2:5 .
P.131 [p.122], beginning of paragraph before Exercise 5.2:5: for algebra read associative algebra.
P.133 [p.124], Exercise 5.2:10(iii): After the final word topologies add on a finite set X .
P.141 [p.132]: In Exercise 5.3:7, add to the parenthetical note
before (a) the sentence: What we call below
the induced topology
on a subset Y of X,
sometimes called the subspace topology
, is the topology whose
open sets are the intersections of Y with the open subsets
of X.
P.150 [p.141], statement of Proposition 5.5.4; between conditions (i) and (ii), insert: (ii½) The least ordinal properly containing an ordinal α is α ∪ {α}. (This was part of our motivation for the definition of ordinal, but it has to be verified that it follows from Definition 5.5.2. The verification, which has been added to the proof, is not difficult.)
P.151 [p.142], after end of proof insert:
The relation ⊆ on the members of an ordinal is the well-ordering that we have constructed these objects to exemplify, so one makes
Definition 5.5.4½. Given ordinals α and β, we shall write α ≤ β if α ⊆ β.
In view of Proposition 5.5.3(iii), every two ordinals are comparable under ≤.
Then change the first sentence of the next paragraph to Returning to the proofs of the above two propositions, note that they make strong use of the Axiom of Regularity.
Finally, insert after that paragraph:
Exercise 5.5:3½. Find a set α which satisfies the condition γ ∈ β ∈ α ⇒ γ ∈ α of Definition 5.5.5, but which is not an ordinal.
P.152 [p.142], Exercise 5.5:6: Change ; equivalently, if and only if, as a set, it is to and that this, in turn, is equivalent to the condition that as a set, it be .
P.154 [p.144], Exercise 5.5:8: make the existing exercise part (i), and add: (ii) Does one or both of the distributive identities (α + β) γ = α γ + β γ, γ (α + β) = γ α + γ β, hold for ordinals?
P.163 [p.154], before final sentence of top paragraph beginning Note that ... insert: (That we can construct a map h in this way follows from Corollary 5.3.6.)
P.165 [p.155]: In first line of Exercise 5.6:6, before partially ordered insert nonempty , and in statement (ii') of that exercise, before disjoint add pairwise .
P.175 [p.164], beginning of last sentence: Replace Since the definition of Boolean algebra was modeled on the structure of the power set of a set, with Since Boolean algebras are defined by a family of identities which imply all identities satisfied by the corresponding operations on the power set of a set, it follows that , then replace the phrase beginning on the last line of p.175, and since we know that Boolean rings are equivalent to Boolean algebras with and in view of the relationship between Boolean rings and Boolean algebras .
P.198 [p.185], Lemma 6.3.6: in both (ii) and (iii) change closed to cl-closed . In (iii), after is compact insert (see Definition 6.2.4)
P.203 (top) [p.189, bottom]: After in two different
orders add Rather, it says If you throw some new
elements into a set X, the closure of the resulting set
doesn't lose any elements of cl(X).
P.208 [p.195] first line of Example 6.5.9: After a set of mathematical objects add of a given sort (e.g., groups, positive integers, topological spaces) .
P.219 [p.205]: In the next-to-last item in the column labeled Structure, the formula |S|ij → |S|ik should be |S|jk × |S|ij → |S|ik .
P.223 [p.209]: The second sentence of the paragraph
following Exercise 7.2:1 should say
Thus, to give an element of order n in a group G is
equivalent to displaying an embedding
of Zn in G, and
more generally, an element
satisfying xn = e is equivalent to
a homomorphism Zn → G.
P.224, line 7 [p.210, line 4]: after equalizers and coequalizers add (see section 4.10, last paragraph of text, and two final exercises).
P.228 [p.214], arrow-diagram in 3rd-from-last paragraph, 4th-from-last line, with a and b above the arrows: The last object should be labeled Z, not Y.
P.235 [p.222] Exercise 7.4:3: The third word, one should, of course, be on .
P.249 [p.235], Exercise 7.6:2(ii), line 3: change both occurrences of finite to pairwise . (Respecting pairwise meets and joins is equivalent to respecting finite nonempty meets and joins. But since it is simplest to define structures of Boolean ring, lattice, semilattice, etc. by just specifying these operations in the pairwise case, as we have done, it is most natural to pose this exercise in terms of these.)
P.264 [p.249], Exercise 7.8:4: at the end of the first sentence, add: with composition defined as in C.
P.268 [p.252], Exercise 7.8:19: after all torsion groups, add (defined as in the first sentence of the preceding exercise) .
P.269 [p.253], 3rd line above Definition 7.8.6: change but that to but we noted in the paragraph preceding Exercise 4.10:9 that .
P.274 [p.258] Exercise 7.8:28: Change the final parenthetical paragraph (Suggestion: ... ) to (Hint: This situation is roughly analogous to that of Lemma 6.5.1(iii).) (The original suggestion was essentially to deduce it from that result. I don't think that quite works, but it can be obtained by the same sort of considerations.)
P.276 [p.260], 4th-from-last and 6th-from-last lines of section 7.8: in both places, change G to Gmd .
P.289 [p.273], at the beginning of the line after (7.10.14), insert (cf. Definition 7.5.3), and on the same page, in the line following Exercise 7.10:3, after In making Cat a category , insert in Definition 7.5.3 .
P.299 [p.281], 5th line above Exercise 8.2:2: After = Uω(K), add: (See last sentence of Definition 7.8.5.)
P.302 [p.284], Exercise 8.2:10: Delete the initial words Show how to.
P.308 [p.290]: On the bottom line of the diagram (8.3.7), all three occurrences of C should be D. (The occurrences of C on the top line of the diagram are correct.)
P.315 [p.296], Exercise 8.3:13: In the third line of part (i), after R a ring add not isomorphic to ℤ.
P.321, 2nd paragraph [p.302, 2nd paragraph after Exercise 8.4:5]: Add at end of final sentence: (of which the fact that the equation x2 = −1 has a solution in the 5-adics is an easy application)
P.326 [p.307], Exercise 8.5:1(i), line 4: after of the direct limit of the given system add: , if one exists,
P.327 [p.308], Exercise 8.5:2, add at end: , in the sense indicated in the second sentence of Exercise 8.5:1(i) .
P.327 [p.308], Exercise 8.5:2: Number the existing statement
as part (i), and add
(ii) Deduce from parts (i) and (iv) of Exercise 8.5:1,
and part (i) above, a necessary and sufficient condition on
an isotone map J → I (not necessarily
surjective) for the conclusion of (i) above to hold.
P.327 [p.308], Exercise 8.5:3: at the end of part (i), (i < i0) should be (i ≤ i0).
P.338 [p.318], diagram with vertical arrows:
The labels on the two upward arrows should be interchanged
lim→ should go on the left, and
lim← on the right.
(Note: On this errata page,
I am using lim← and
lim→ for limits and colimits because I don't
know a way to put the arrow under the "lim" in html.)
P.341 [p.322]: in the paragraph preceding Lemma 8.6.8 and the statement of that lemma, everywhere change cE, cf, etc., to pE, pf, etc.. In the last formula of the first paragraph of that lemma, delete the small circle. Add after that formula: , i.e., the morphism lim← D pE1 F → lim← D pE2 F which forms commuting squares with the morphisms pf F(D): pE1F(D) → pE2F(D) for all D ∈ Ob(C).
P.348 [p.328], immediately after end of proof of
Theorem 8.8.7, add the paragraph:
The above is not quite what our general principle that
left universal constructions respect left universal constructions
and right universal constructions respect right universal
constructions
might have led us to expect: if a covariant
functor V is representable, the object R representing
it is left universal, yet the above theorem shows that
V respects the right universal construction of limits.
We must conclude that the functor and its representing object have a
mirror relationship to one another, like that between
a pair of adjoint functors, so that while R is left
universal, V patterns with right universal constructions.
Similarly, in the case of a contravariant representable functor,
the representing object is right universal, while the
functor represented behaves nicely with respect to colimits, which are
left universal.
P.350 [p.330], after Exercise 8.8:4, add the following.
Exercise 8.8:4½. Regarding the set ω of natural
numbers as ordered in the usual way, and letting
ωcat denote the category obtained
from it, let us define a functor
B: ωcat × ωcat → FSet
as follows.
For
(m,n) ∈ Ob(ωcat × ωcat),
let B(m,n) =
{i ∈ ω | min(m,n) ≤ i ≤ max(m,n)},
and for (m,n) ≤ (m′,n′) under
componentwise comparison,
let B((m,n), (m′,n′)):
B(m,n) → B(m′,n′)
carry elements of
B(m,n) ∩ B(m′,n′)
to themselves, and and carry elements of
B(m,n) that do not lie in
B(m′,n′)
to the least member of the latter set, min(m′,n′).
Verify that B is a functor.
Letting
D = E = ωcat, show that
the colimit in the middle of (8.8.10) is defined,
but neither of the outer colimits-of-colimits is.
Can you modify this example to get a functor such that the
middle colimit and one of the outer colimits-of-colimits are
defined, but not the other?
P.353 [p.333], end of section 8.8, add:
Let us return to Theorem 8.8.7 for a brief observation.
Though that result is symmetric in that covariant representable
functors behave nicely with respect to limits, and
contravariant representable functors with respect to colimits, it is
asymmetric in that both turn those constructions into limits.
Why are limits privileged above colimits in this way?
I can only give a vague intuitive answer: representable
functors take values in Set, and Set is
far from left-right symmetric in its properties.
In particular, limits in Set correspond to well-behaved
fundamental sorts of mathematical concepts; e.g., a product set
X × Y corresponds to the idea
take an element of X and an element of Y
,
while the coproduct corresponds to the less straightforward
operation of taking
an element of X or an element of Y
.
If we were looking at the analogs of representable functors
on enriched categories in the sense of section 7.11,
their behavior could be expected to vary considerably
depending on where those categories take their hom-objects.
P.354 [p.335], at beginning of proof of Lemma 8.9.1, insert: Note that by Lemma 8.6.8, the constructions lim→ D: CD → C and lim← D: CD → C can be regarded as functors; so we are comparing the comparison morphism for the behavior of the former functor with respect to the latter limit with the comparison morphism for the behavior of the lattter functor with respect to the former limit.
P.362 [p.342], immediately before Lemma 8.10.1, add:
Recall from the last sentence of Definition 8.6.1
that a category C is said to have small limits
if all
functors from small categories into C have limits.
P.363 fourth line after (8.10.2) [p.343, top line]: Change (second sentence) to (third sentence).
P.373 [p.358], paragraph following proof of Corollary 8.11.6: in the 6th line, after which we found add , once we had Yonda's Lemma, and at the end of the paragraph change (Cf. also Exercises 7.9:4(ii), 8.2:10.) to (This was part of the case of Exercise 8.2:10 referring to Exercise 7.9:4(ii).)
P.381 [p.360], last multi-line paragraph before Definition 9.1.3: change in English. to in medieval times .
P.383 [p.364], add at end of Exercise 9.1:11: Can you get such an example where the category over which the limit is taken is finite?
P.388 [p.367], Exercise 9.2:1: after the zeroary operation ... the unary operation ... the ω-ary operation respectively, add names for those operations: s, t, u.
P.393 [p.372], add at end of Exercise 9.3:2: (This will require finding a cardinality bound on an Ω-algebra generated by an X-tuple of elements. Such a bound will involve the cardinality of | Ω |, the arities of the operations of Ω, and the cardinality of X.)
P.395 [p.375], Exercise 9.3:6(ii): Change Show that (i) to Show that the second assertion of (i) .
P.401 [p.381], add at the end of Exercise 9.4:1: (Hint: Lemma 9.3:3 concerns the number of generators needed to get an element of a subalgebra generated by a set, as does the suggested variant with γ = 2. But what does this variant imply about the number needed to get the expressions on the two sides of an identity?)
P.402 [p.382], Definition 9.4.6: At the end of the first sentence, change some set J of identities to some set of identities J ⊆ IΩ (cf. (9.4.4)). And in the first line of the second paragraph, after is a variety add of Ω-algebras .
p.405 [p.386], insert after last word of Exercise 9.4:4: and the conclusion of the proposition again fails .
P.408 [p.388], Exercise 9.4:7: In the first line, change
to (i)(b)<==>(c) and to (ii) to to (i) and (ii),
and in the third line, after respectively, add:
and with no homomorphism to the
field ℤ/2ℤ
replaced by no homomorphism to any field
,
P.409 [p.387], Lemma 9.4.17, first line: change of algebras to of Ω-algebras .
P.412, line 2 [p.389, last line of short paragraph beginning "Note that"]: before induced add one for each x ∈ X, which are .
P.413 [p.391], lines 2-3 after (9.5.4): set of universes should be class of universes .
P.414 [p.391], replace the sentence introducing Exercise 9.5:4
with the exercise and preceding and following sentence below:
We end this section with some exercises about derived operations
on single sets.
Exercise 9.5:3½. In Exercise 2.7:1, we looked at
the derived operations of the ternary majority vote function
M3 on the set {0,1}.
(i) If you did parts (a)-(d) of that
exercise, you found that some of the functions listed there
are indeed derived operations of M3.
Show, in each of the cases where you found an operation s
to be a derived operation of M3, that
M3 is likewise a derived operation of s.
(So in these cases, the set of derived operations
of M3 and the set of derived operations
of s are the same.)
(ii) Does every derived operation s
of M3, other than the projection operations,
have the property that M3 is also
a derived operation of s?
The questions asked in the next two exercises are, as far as I know, open.
P.415 [p.392], Exercise 9.5:5: Bjorn Poonen used to have this open question on his webpage. More recently, he posted the = N case on a discussion website. One of the responses on that site notes that the question was raised earlier, as Problem C, in the brief final paragraph of a note by John S. Lew, Polynomials in Two Variables Taking Distinct Integer Values at Lattice-Points, American Mathematical Monthly, 88 (1981) 344-346. MR1539684. I've revised the statement of the exercise to note these facts.
P.420 [pp.397-398]: In the second line of Proposition 9.6.4(ii): operations of Ω should be primitive operations of V, while in the second line of Theorem 9.5.5, V should be Ω .
P.421 [p.401], point (d): (ti, vi)i∈ ari(σ) should be ((ti, vi))i∈ ari(σ) . (It is an indexed family of ordered pairs.)
P.424 [p.404], Exercise 9.6:11(iii): Change can be obtained from some variety W by adjoining one zeroary operation and no identities to can be obtained (up to an equivalence which respects underlying sets) by adjoining one zeroary operation and no identities to some variety W .
P.429 [p.409], second line of second paragraph before Exercise 9.7:5: Change functions to polynomial functions .
P.430 [p.407], next-to-last line of paragraph beginning If G: After form a Lie algebra add over the real numbers .
P.431 [p.408], change the paragraph preceding
Exercise 9.7:6 to
It can be shown that every finite-dimensional Lie algebra
over ℝ is the Lie algebra of a Lie group G.
Moreover, two Lie groups G1
and G2 determine isomorphic Lie algebras
if and only if they are isomorphic near the identity
,
i.e., have neighborhoods of the identity which are isomorphic under the
the restrictions of the group operations to partial operations on
those sets; and this in turn can be shown to hold if and only if the
universal covering groups of G1 and
G2 are isomorphic as Lie groups.
P.435 [p.412], end of Exercise 9.8:4(ii): After the operation lim sup add: , the definition of which was recalled in point (a) preceding Exercise 2.7:4.
P.453 [p.429], after Exercise 9.10.8, add:
Exercise 9.10.8½. If V and W are
varieties of algebras, will the category
V × W be equivalent to a variety of algebras?
If this is not always true, can you find necessary and/or
sufficient conditions for it to hold?
P.460 [p.435] top line: before the entries add that is, .
P.460 [p.435], bottom: change the line following (10.1.8), up to the words the identity , to be the morphism sending the entries of the universal element r ∈ |SL(n, ℝ)| to the corresponding entries of .
P.461 [p.436], line 2: cogroupin should be cogroup in .
P.465 [p.440]. On line 3; i.e., the first line of Lemma 10.2.8(iii), both occurrences of V should be Ω-Alg . And in the second line of the proof of that Lemma, |A| = should be A = .
P.469 [p.444], second line of Definition 10.3.5(ii): before Definition 8.2.3 add of .
P.470 [p.445], 3rd line of 2nd paragraph of section 10.4: FC should be FV .
P.478 [p.453], Exercise 10.5:4: in parts (iii) and (iv), after n a fixed integer add: > 1.
P.479 [p.454], Exercise 10.5:6: Change the final phrase beginning i.e., to a new sentence: (So if C has small colimits, Theorem 10.4.3 tells us that F is the left adjoint to a representable functor in the sense we have been studying.)
P.492 [p.466], display preceding Exercise 10.7:1: I now feel that introducing the abbreviations η and ε for ηU,V and εG,U only complicated things. So in that display, delete η = and ε = ; immediately before the display drop the words let us write these , after the display drop the two lines of parenthetical comment, and in the statement of Exercise 10.7:1 replace the one occurrence of η (in part (i)) with ηU,V and the three occurrences of ε (in parts (i), (iii) and (iv)) with εG,U .
P.510 [p.484], Exercise 10.10:12: Change varieties of algebras to varieties of finitary algebras .
P.513 [p.487] 4th and 5th lines of Exercise 10.11:3: After Cf. Definition 7.8:12 add and the notation introduced in Exercise 7.8:30(iii) .
P.515 [p.490], 3rd line below commutative diagram: || T || should be || R ||.
Pp.520-521 [494-495]: Number the three displays in Corollary 10.12.13 as (10.12.14), (10.12.15), and (10.12.16) .
P.521 [p.495], immediately after Exercise 10.12:11 add Is there a result analogous to Corollary 10.12.3 where one attaches a pair of adjoint functors on the left of the arrows of (10.12.4)? Certainly! The roles of C and D in (10.12.4) are identical, hence that analogous statement is just a case of the corollary itself, after a relabeling. Then add at the beginning of the next paragraph the words On the other hand, .
P.523 [p.497]: In the 2nd line of Exercise 10.13:2, after by generators and relations insert the words a universal example of.
P.528 [p.501], Exercise 10.13:11(i)(b): Change The clone of operations to The clone of derived operations .
P.529 [p.502], line before Exercise 10.13:14: Change A question I also don't know the answer to is to Two specific questions I don't know the answer to are posed in .