P.13, next-to-last line: μ and ι should be μ, ι, and e.
P.29, 2nd line of last paragraph: for the preceding section read §2.2.
P.37: Add to end of parenthetical explanation before first diagram: The symbol ∃1 (or ∃!) is a common shorthand for "there exists a unique".
P.42, line before Exercise 3.3:11 add: (In part (ii) below, note that Z is a common notation in group theory for the infinite cyclic group. The similarity to blackboard-bold Z as a symbol for the integers is a coincidence: the latter is based on German Zahl, meaning "number", the group-theoretic symbol on zyklisch, meaning "cyclic". Although the additive group of integers is an infinite cyclic group, a group denoted Z can either be written additively, or multiplicatively, e.g. as {xi | i∈Z}. The finite cyclic group of order n is likewise denoted Zn.)
P.66, after Exercise 3.10.7 add:
Exercise 3.10.7½
(i) Find a normal form for the monoid
presented by two generators a, b, and the one
relation abba = baab.
(This is hard, but can be done.)
(ii) (Victor Maltcev)
Does there exist
a normal form or other useful description for the monoid
presented by a, b and the
one relation abbab = baabb ?
(I do not know the answer.)
P.66, sentence before the diagram: the product should be the direct product.
P.77, line 4: 1S ⊗ T should be 1S ⊗ t .
Pp.126-127. In the 2nd paragraph of section 4.6, both occurrences of α should be γ; and the same is true in the second paragraph of the next page. (But the other occurrences of α on these pages are correct.) Also, on the 3rd-from-last line of p.126, where I say indexed by α, clearer wording would be indexed by an ordinal α.
P.145, 2nd line after Definition 5.2.2: all subsets should be all nonempty subsets.
P.164, Example 5.5.8, end of 4th line: S = M = should just be S = .
P.178, about an inch from the bottom: There is a diagram consisting of a circle, a dot on its left side, and a small solid triangle. That triangle is supposed to be an arrowhead, indicating that the circle is an arrow beginning and ending at the dot; but somehow it got placed badly.
P.205, Exercise 6.7:9(v): Where I have is not true for the general case of a category and a full subcategory, a clearer wording would be does not remain true if Ring1 and CommRing1 are replaced by an arbitrary category and an arbitrary full subcategory thereof.
P.207, Exercise 6.7:11: Add
(iii) Can you find a set-theoretic criterion for a
morphism in RelSet to be a monomorphism or
an epimorphism in that category?
To be left or right invertible?
P.209, Exercise 6.7:17(i): Where I have Cemb taken to consist of all monomorphisms of C, a clearer wording would be Cemb(X,Y) the set of all monomorphisms in C(X,Y).
P.224, last line: IdC should be IdSet.
P.227, Exercise 6.9:14: Add at the end the words: (including the part described as "straightforward").
P.230, 2nd diagram: U should be V.
P.232, 6th line above display: bicontinuous should be continuous. (I had somehow gotten the idea that "bicontinuous" meant "continuous as a function on the direct product space, and not just in each variable separately"; but it actually means "continuous, and having a continuous inverse", which is not in general the case here.)
P.239, end of Exercise 7.2:5: before on all sets insert the words with respect to U.
P.278, beginning of line 2: Δ(C) should be Δ(C).
P.278, 2nd line of Exercise 7.8:1: left and adjoint should be left and right adjoint.
P.300, First line of last paragraph: because C = D should be because U = V and F = G.
P.306, Definition 8.1.4, first word of 6th line: operation should be operations.
P.310, Exercise 8.1:6, first line of part (iii): ascending chain should be ascending chain condition.
P.315, proof of Theorem 8.3.3: In the first line of the proof, and in the expressions (u,x) in the first line of each of the displays, replace u by *. In particular, the formula u(x) = (u,x) becomes u(x) = (*,x). This avoids using the same symbol for two things, the arbitrary element chosen in the first sentence, and the function u it is used to define.
P.324, before Corollary 8.4.9, insert:
REMARK 8.4.8½.
If we consider classes of algebras defined by sorts of
propositions more general than identities, involving logical
operators such as ∃, ⇒, and ∨ (for instance,
torsion-free groups, and integral domains, both mentioned above,
divisible groups, which were considered in
Exercise 6.7:5, and fields, cf. Exercise 2.3:3),
we find, in general, that one or more of
the statements of Proposition 8.4.8 fail.
This is why we stated in Chapter 1 that it was
"better" to define the concept of group using three
operations and the identities (1.2.1), than using just
one operation, and the more general conditions (1.2.2).
Of course, it is also worthwhile studying what results are
true of classes of algebras defined by other sorts of propositions.
But varieties form a broad, useful, and well-behaved class, which we
will focus on in this course.
The parts of the above proposition saying that certain
constructions have the same form in V as
in Ω-Alg, together with some earlier
results yield:
P.328, next-to-last line of Lemma 8.4.16: the applies should be that applies.
P.343, after the exercise at the top of the page, add
(The result that
η(L): L → B(E(L)) is
one-to-one is actually known
to hold for a much wider class of Lie algebras
than those that are free as k-modules; but there are also
examples for which it fails.
See P.M.Cohn, A remark on the Birkhoff-Witt theorem,
J. London Math. Soc., 38 (1963) 197-203.
MR 26#6233.)
P.350, Definition 8.9.1: In the last display in that definition, in the label of the first arrow, the final term, tn−1 should be tm−1. (I.e., the arrow is given by an m-tuple of n-ary operations, not by an n-tuple.)
P.353, Definition 8.9.5, final parenthetical paragraph: Replace all occurrences of γ0 by γ1, and both occurrences of ≤γ0 by <γ1. Likewise, on p.354, in the last two lines of Definition 8.9.7, all occurrences of γ0 should be replaced by γ1.
P.368, last line: after the symbol δij you might add: (i.e., 1 if i = j, 0 if i ≠ j).
P.373, first line of Lemma 9.2:8(iii): the symbols Clβ and Cl1 should be Clβ and Cl1. I.e., Cl should not be italic in these symbols; cf. p.354, first sentence of Definition 8.9:7.
P.376, Definition 9.3.4, near end of first line of (iii): ≤ γ should be < γ.
P.379, 5th line of paragraph before the multi-arrow diagram: of S should be of A.
P.422, end of last paragraph, add: Below, when the prefix "Ab" is added to the name of a variety whose type involves one binary operation, the result will denote the subvariety determined by the additional identity of commutativity for that operation.
Pp.427-455: In the Word and Phrase Index, the Symbol Index, and the Exercise Index, most items after around p.60 (and some items shortly before that page) are shown with page-number too large by 1; so in looking for such an item, try the page before the one listed. (This probably means that I forgot to re-run the software that creates those indices after making some nontrivial changes around p.60. On the other hand, the Table of Contents (pp.iv-vii), and the page-references given in pointy brackets for the items in the Bibliography (pp.456-461) appear to be correct.)
P.453: in the description of Exercise 7.4:5 Aut(...) should be End(...).
P.459, reference [80]. The author's first name should be Philip (with just one l).