Where I feel the uncorrected version could cause real misunderstanding,
the page numbers are printed in bold face.
`
`

`P.iv (= back of title page), 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
recently 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.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.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 *R*` should be` as a ring *R*`.
`

`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.131 [p.122], beginning of paragraph before Exercise 5.2:5:
for` algebra

`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 is not difficult.)
`

`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 = αγ+βγ,
γ(α+β)=γα+γβ,
hold for ordinals?`
`

`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.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 on 4th-from-last line of 3rd
paragraph, with*a*` and` *b*` above
the arrows: The last object should be
labeled` *Z*`, not` *Y*`.
`

`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.276 [p.260], 4th-from-last and 6th-from-last lines
of section 7.8: in both places, change` *G*`
to` *G*_{md}` .
`

`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 exercise
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.

**P.341** [**p.322**], end of first paragraph of
Lemma 8.6.8: change*c _{f} ∘ F*

`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 *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}:
**C ^{D}** →

(Note: On this errata page, I am using

In using the above reference to Lemma 8.6.8, note that that lemma, unfortunately, uses

`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 third 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: drop the words` in English

`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.402 [p.380], Definition 9.4.6, first line of second paragraph:
after` *is a variety*`
add` *of* Ω*-algebras*` .
`

`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
by:`

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
`

` P.420 [p.398]: In the second line of part (ii)
Proposition 9.5.4:` operations of Ω

`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], paragraph preceding Exercise 9.7:6: After`
It can be shown`, add`
that every finite-dimensional real Lie algebra is the Lie algebra
of a Lie group, and, on the other hand, that` .
`

`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.461, 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.478** [**p.453**], Exercise 10.5:4:
in parts (iii) and (iv), after*n* a fixed integer`
add:` > 1`.
`

`P.515 [p.490], 3rd line below commutative diagram:`
||*T*||` should be` ||*R*||`.
`

`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`.
`