errata & addenda for ALGEBRA (chapters 8-14) in HAF
- In 8.10.i, the isomorphism from any group onto a subgroup
of the symmetric group is known as Cayley's Theorem.
- Hint for 8.20.c: If x^2 = y^2, then (x-y)(x+y)=0, so
x is plus-or-minus y. A simple counting argument finishes
- 8.23. Part d is wrong altogether. If h were such a
homomorphism, it would satisfy 1 = h(1) = h(5 times 1/5)
= h(5) times h(1/5) = (h(1)+h(1)+h(1)+h(1)+h(1)) times h(1/5)
= 0 times h(1/5) = 0. Part c is wrong too; it should say
that if F is a field of characteristic 0, then there is
a unique injective homomorphism from Q into F. Found by [JT]
- Comment for 8.36: Examples will be given in 8.37 and 8.38.
- In 8.40, on page 198, second equation from the bottom: the
superscript plus should be a subscript plus. [JT]
- In 8.51, on page 204, 3rd line from the bottom: "hence p=q
is a semantic theorem" should be
"hence p=q is a syntactic theorem." Or, in greater detail:
Since p=q is a semantic theorem, it is satisfied by every
algebraic system of the given type. In particular, it
is satisfied by the algebraic system Xi. That is,
p and q are equal in Xi. Thus p and q belong to the
same equivalence class. Thus p=q is a syntactic
- In 8.53 on page 205, this probably would be easier for
beginners to read if all the parentheses were of the same size. [JT]
- Clarification for Chapter 9, especially 9.15.
I am learning some more about category theory now, and
Saunders Mac Lane has been helping. Soon I hope to
clean up the discrepancies between my "category theory
for analysts" and conventional category theory.
When I have more details, I'll post them here.
To start with, probably some names should be changed:
My "initial structure" probably should be called
a "weak structure" so as not to confuse it with
the "initial object" of conventional category
theory, and my "subobject" could be called a
"trace object" so as not to confuse it with the
(slightly different) "subobject" of conventional
category theory. It is possible to reformulate
the results of my Chapter 9 so that the description
of objects does not explicitly mention structures;
the objects yield the underlying sets via a
forgetful functor. My "weak structure" construction
probably can be reformulated as the result of
an adjoint functor. I'm still looking into this.
- Further remarks for 10.28: We assume that the
beginner has some informal familiarity with sines and
cosines, as developed in an elementary course on trigonometry.
However, that informal familiarity is based on geometric
diagrams, and the reader of the present work probably has
reached the stage where he/she would prefer not to rely
so much on geometric diagrams. A more algebraic treatment
of sines and cosines can be based on power series. See
the further remarks, below, for section 22.24.
- In 11.3.(i) on page 273, we need a right parenthesis. [JT]
- In 11.3(iii),
x \otimes (c \star y) should be
changed to x \star (c \otimes y). [SS]
- In 11.29 on page 285, near the bottom, "Form the external
direct sum" should be "Form the external direct
- Additional remarks for 11.41. A few more remarks might
be helpful for motivation, i.e., to explain why I've developed
integration theory in this fashion.
Most books which introduce integration over positive measures
concentrate on integrals of real-valued functions. They
tend to mix together two different sorts of ideas; I believe that
these ideas can be grasped more readily if they
are kept separate. The two sorts of ideas are:
Many introductions to measure and integration work on the
"semi-extended real line" -- i.e., all of the real numbers,
together with either +infinity or -infinity. I have to admit that
that particular case is not covered by my book. However, it is my
feeling that that case is unnecessarily complicated and awkward.
All the really important ideas are displayed, in a more
elegant fashion, in the two cases I've covered (listed above).
- How big is something? Use notions of ordering. This
yields the positive integral, described in 11.41(iii) and
studied at the end of Chapter 21. A natural setting is
the extended nonnegative reals, [0,+oo]. (Some mathematicians
prefer to work in (-oo,+oo] or in [-oo,+oo), but I feel that
that adds complications without adding any important ideas.)
- How far apart are things? Use notions of distance. This
yields the Lp spaces, studied in the middle of Chapter 22.
A natural setting is in Banach spaces. Very little is gained
by restricting attention to real-valued functions; to
emphasize the role of distance I prefer to consider Banach-space-valued
functions and develop the Bochner-Lebesgue spaces.
- Add remark to 11.42: The point of all this is that later we'll
take the limits of simple functions, and thereby extend the definition
of the integral. -- Perhaps it would be better to postpone this material
until later in the book; this section doesn't do much good by itself.
- Clarification for 11.50 parts c and d. For some reason
I thought that these were only true in vector lattices, not
in lattice groups. I was mistaken -- these are also true in
lattice groups. Proof can be found in Birkhoff's book on
- In 12.15, part (E) is wrong. Each of the x's on the
right side of the inequality should be replaced by Mx, where
M is the sum of the mu's.
- 12.17.c. The last sentence is wrong. For instance,
the function that is equal to 0 when x is less than 0, and
equal to x-squared when x is greater than 0, is neither affine
nor strictly convex. (I don't know what I was thinking when
I wrote this. Is there some true statement that resembles
this false one?)
- 12.28. The theorem is right but the last line of the hint
is wrong. To show
instead proceed as follows: Assume
mS(x)<r; it suffices to
we conclude that there is some positive number k such
that x/k belongs to S and k<r. Since x/k belongs
to S, it follows that g(x/k)£1.
Since g is positively homogeneous, we have
- Add remark for 12.33, 23.10, 29.37, and 29.38: A student of mine
read 12.33 and asked for an example of a Banach limit. I guess I didn't
make it clear enough in 12.33: There *aren't any* explicit examples of
sequential Banach limits! That's one of the main consequences of
29.37-29.38 (together with 14.n76-14.78 and 20.30). See also 23.10 for
related discussion. I would guess that under additional mild
assumptions, there probably aren't any nondegenerate examples of
nonsequential Banach limits either, but I haven't investigated that.
- Correction for 13.28.a. There is some confusion in
the literature between "Heyting algebra" and "complete Heyting
algebra," and I have regrettably perpetuated this confusion.
HAF asserts that every Heyting algebra can be represented
isomorphically by a topology, but that should say every
complete Heyting algebra can be represented isomorphically
by a topology. We already know that every topology is a
complete lattice; that fact was pointed out in Section 5.21.
For a simple example of a Heyting algebra that is not complete,
take the set of all rational numbers in [0,1], with its usual
- Add remark for 14.16 through 14.26.
We should emphasize to beginners that there is no standard
treatment of this material. Different texts on logic use
slightly different ingredients for their first order languages
and for their rules of inference and logical axioms.
Most introductory treatments in the literature are equivalent -- they
yield classical logic, with the same Completeness Principle
and Compactness Principle, etc. Different introductory treatments
differ only superficially in their presentation of this material.
Still, the superficial differences may appear substantial to the
- Add remark for 14.20. Most logic books that I've looked
at use the Mendelson/Hamilton approach to bindings, which
permits a variable to be bound more than once. So far, the only
logic books that I've found that do not permit double-bindings
are Rasiowa-Sikorski  and
Lu Zhongwan, Mathematical Logic for Computer Science,
World Scientific, Singapore, 1989.
Surely there must be a few more. When I find them, I'll list
them here, since this is a viewpoint that I want to promote.
- In 14.23, the paragraph-break in the middle of the first
sentence was unintentional. [JT]
- In 14.71, a typographical error messed up my favorite
aphorism in the whole book. The second paragraph's
last sentence should read:
"Even a mathematician must accept some things on faith
or learn to live with uncertainty."