Errata and addenda to published version of An Invitation to General Algebra and Universal Constructions.

Bracketed page-numbers refer to the online pre-publication version.  (My pdf viewer includes the front-matter in the page-count.  If yours does the same, then add +6 to the page-numbers shown for that version to get the numbers to feed into your viewer.  E.g., for the correction marked P.37 [p.34], use 34+6=40 as the number to feed in.)  In the few cases where no bracketed number is shown, the correction applies only to the published version. 

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 (aX)−(ae)  to  closed under the symbolic operations named in (aX)−(ae) .

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  α1, ..., αn,  and likewise replace  f(α), f(β), f(γ)  with  f1), ..., fn)

P.48 [p.45], 3rd line of Exercise 4.2:1:  for  {xiyi−1}  read  {xiyi−1 | i ∈ 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 some sets  X  or groups  G,  you might try to determine exactly which  X  and/or  G  this is true for.)

P.78 [p.74], Exercise 4.9:9, line after the display:  x1, ..., xn  should be  x1, ..., xm .

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| → |Sgp|  is surjective.  Show, in fact, that  (Sgp)md  can be described as the monoid obtained by imposing on  S  the relations  i = e  for all idempotent elements  iS,  i.e., all elements satisfying  i2 = 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  read  associative 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  Zn  in  G,  and more generally, an element satisfying  xn  = e  is equivalent to a homomorphism  Zn → G. 

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  Gmd .

P.299 [p.281], 5th line above Exercise 8.2:2:  After  = Uω(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  x2 = −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  add:  , if one exists,

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),  (i < i0)  should be  (i ≤ i0)

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  cf ∘ F  to  cf 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.
  Exercise 8.8:4½.  Regarding the set  ω  of natural numbers as ordered in the usual way, and letting  ωcat  denote the category obtained from it, let us define a functor  B: ωcat × ωcat → FSet  as follows.  For  (m,n) ∈ Ob(ωcat × ωcat),  let  B(m,n) = {i ∈ ω | min(m,n) ≤ i ≤ max(m,n)},  and for  (m,n) ≤ (m′,n′)  under componentwise comparison, let  B((m,n), (m′,n′)): B(m,n) → B(m′,n′)  carry elements of B(m,n) ∩ B(m′,n′)  to themselves, and and carry elements of B(m,n)  that do not lie in  B(m′,n′)  to the least member of the latter set,  min(m′,n′). 
   Verify that  B  is a functor.  Letting  D = E = ωcat,  show that the colimit in the middle of (8.8.10) is defined, but neither of the outer colimits-of-colimits is.
   Can you modify this example to get a functor such that the middle colimit and one of the outer colimits-of-colimits are defined, but not the other? 

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: CD → C  and  lim← D: CD → C  can be regarded as functors; so we are comparing the comparison morphism for the behavior of the former functor with respect to the latter limit with the comparison morphism for the behavior of the lattter functor with respect to the former limit. 
(Note: On this errata page, I am using
  lim and  lim  for limits and colimits because I don't know a way to put the arrow under the "lim" in html.) 
In using the above reference to Lemma 8.6.8, note that that lemma, unfortunately, uses
  cE  in a way unrelated to the comparison morphism  cB  of Lemma 8.9.1. When I next revise the text, I will change the notation in Lemma 8.6.8 to  pE,  since these are essentially projection maps. 

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)  to  (third sentence)

P.373 [p.358], paragraph following proof of Corollary 8.11.6:  in the 6th line, after  which we found  add , once we had Yonda's Lemma,  and at the end of the paragraph change  (Cf. also Exercises 7.9:4(ii), 8.2:10.)  to  (This was part of the case of Exercise 8.2:10 referring to Exercise 7.9:4(ii).) 

P.381 [p.360], last multi-line paragraph before Definition 9.1.3:  drop the words  in English.  (It's not clear where the distortion arose.  The Oxford English Dictionary says it was in 15th-century Latin, based on the Greek word from which we get "arithmetic", while Wikipedia says that "al-Khwarithmi" was the Classical Arabic pronunciation of the Persian name, then adds an alternative explanation that does involve the root of "arithmetic", but not the name al-Khuwarizmi.) 

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  should be  class 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  M3  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  M3.  Show, in each of the cases where you found an operation  s  to be a derived operation of  M3,  that  M3  is likewise a derived operation of  s.  (So in these cases, the set of derived operations of  M3  and the set of derived operations of  s  are the same.) 
(ii)  Does every derived operation  s  of  M3,  other than the projection operations, have the property that  M3  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  should be  primitive operations of  V,   while in the second line of Theorem 9.5.5,  V  should be  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:  cogroupin  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