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 (*a _{X}*)−(

`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 ` ∈*R _{i}*}.

`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 ` α_{1}, ...,
α_{n}`, and likewise replace `
*f*(α), *f*(β), *f*(γ)`
with ` *f*(α_{1}), ...,
*f*(α_{n})`.
`

`P.48 [p.45], 3rd line of Exercise 4.2:1:
for ` {*x _{i} y_{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

`P.78 [p.74], Exercise 4.9:9, line after the display: `
*x*_{1}, ..., *x _{n}*

`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*| → |*S*^{gp}| is surjective.
Show, in fact, that (*S*^{gp})_{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
*i*^{2} = *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

` P.133 [p.124], Exercise 5.2:10(iii): After the final
word ` topologies

`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 *Z _{n}* in

`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 ` *G*_{md}` .
`

`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 ` = *U*^{ω}(*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 *x*^{2} = −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

` 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), ` (

` P.338 [p.318], diagram with vertical arrows:
The labels on the two upward arrows should be interchanged `
lim

(Note: On this errata page, I am using

**P.341** [**p.322**]: in the paragraph preceding
Lemma 8.6.8 and the statement of
that lemma, everywhere change *c _{E}*,

`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.`

Verify that

Can you modify this example to get a functor such that the middle colimit and

`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

,
while the coproduct corresponds to the less straightforward
operation of taking
*X* and an element of *Y* an element of

.
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.*X* *or* an element of *Y* `
`

`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}:
**C ^{D}** →

`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)

` P.373 [p.358], paragraph following proof
of Corollary 8.11.6: in the 6th line, after ` which we found

` P.381 [p.360], last multi-line paragraph
before Definition 9.1.3: change ` in English

`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* ⊆ *I*_{Ω}
(*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

`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
*M*_{3} 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 *M*_{3}.
Show, in each of the cases where you found an operation *s*
to be a derived operation of *M*_{3}, that
*M*_{3} is likewise a derived operation of *s*.
(So in these cases, the set of derived operations
of *M*_{3} and the set of derived operations
of *s* are the same.)

(ii) Does every derived operation *s*
of *M*_{3}, other than the projection operations,
have the property that *M*_{3} 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

` P.421 [p.401], point (d): `
(

`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 *G*_{1}
and *G*_{2} 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 *G*_{1} and
*G*_{2} 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: `*cogroup*in`
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: *F*_{C}` should be `
*F*_{V}` .
`

**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 ` η_{U,V} `
and ` ε_{G,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 `
η_{U,V} ` and the three occurrences of `
ε ` (in parts (i), (iii) and (iv)) with `
ε_{G,U}` .
`

` P.510 [p.484], Exercise 10.10:12: Change `
varieties of 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: `
||

`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` .
`