errata & addenda for ALGEBRA (chapters 8-14)

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 the proof.
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 theorem.
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 sum Phi="
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:
- 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 L^p 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.
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).
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 lattice theory.
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 g£m_S, instead proceed as follows: Assume m_S(x)<r; it suffices to show g(x)£r. From m_S(x)<r 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 g(x)£k, and therefore g(x)£r.
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 ordering.
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 beginner.
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 [1963] 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."