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

Below, the page-number beginning each note refers to the published version, while the bracketed page-number following this refers to the online freely readable pre-publication version.  (Later online versions incorporating some or all of these errata and addenda, the most recent of which is this, have page-numbering close to, but not exactly the same as, the pre-publication version.  Incidentally, 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, or +7 for later versions, to get the numbers to feed into your viewer.  E.g., for the correction marked P.37 [p.34], use 34+6=40, respectively 34+7=41, 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. 

In the few cases where I have added a new exercise before the last one in a section, or the like, I show the new one as having the number of the preceding one with ½ added (e.g., see the notes on P.86 [81] and P.150 [p.141] below), to avoid the complication of how to refer to later ones; although in the revised version, the added item actually has the next number, and the later ones have +1 added to their numbers.

Here, now, are the errata and addenda:

P.iv (= back of title page, in published version only), 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 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.13 [p.12], just before Exercise 2.2:3:  Delete the sentence:  I don't know the full answer to the following variant question:  (I do now.) 

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.29 [P.26], 3rd-from-last line: T}  should be Ri}. 

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  {xi yi−1}  read  {xi yi−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  i ∈ S,  i.e., all elements satisfying  i2 = i. 

P.89 line 9 [p.83 last line]:  as ring  should be  as a ring

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) →   is continuous, this implicitly used the fact that the only endomorphism of    is the identity, which allows one to deduce that all ring homomorphisms  B(X) →  are -algebra homomorphisms.  The corresponding fact about endomorphisms is not true for    so the results have to be stated in terms of -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.129 [p.120], Exercise 5.2:1:  Change the parenthetical statement  (This requires ...)  to  (A solution must show that the two constructions give structures of the indicated sorts, and that performed successively in either order, they return the original data.)

P.129, last paragraph [p.121, paragraph preceding Exercise 5.2:2], 5th line:  change  depend on 5.2:2-5.2:3, and 5.2:5  to  depend on 5.2:2(i) and 5.2:3.  5.2:5 .

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

P.133 [p.124], Exercise 5.2:10(iii):  After the final word  topologies  add  on a finite set  X .

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, which has been added to the proof, is not difficult.) 

P.151 [p.142], after end of proof insert:

 The relation  ⊆  on the members of an ordinal is the well-ordering that we have constructed these objects to exemplify, so one makes

 Definition 5.5.4½.  Given ordinals  α  and β,  we shall write  α ≤ β  if  α ⊆ β.

 In view of Proposition 5.5.3(iii), every two ordinals are comparable under  ≤.

 Then change the first sentence of the next paragraph to  Returning to the proofs of the above two propositions, note that they make strong use of the Axiom of Regularity.

 Finally, insert after that paragraph:

 Exercise 5.5:3½.  Find a set  α  which satisfies the condition  γ ∈ β ∈ α ⇒ γ ∈ α  of Definition 5.5.5, but which is not an ordinal.

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 = α γ + β γ,   γ (α + β) = γ α + γ β,  hold for ordinals? 

P.163 [p.154], before final sentence of top paragraph beginning  Note that ...  insert:  (That we can construct a map  h  in this way follows from Corollary 5.3.6.)

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.175 [p.164], beginning of last sentence:  Replace  Since the definition of Boolean algebra was modeled on the structure of the power set of a set,  with  Since Boolean algebras are defined by a family of identities which imply all identities satisfied by the corresponding operations on the power set of a set, it follows that ,  then replace the phrase beginning on the last line of p.175,  and since we know that Boolean rings are equivalent to Boolean algebras  with  and in view of the relationship between Boolean rings and Boolean algebras .

P.198 [p.185], Lemma 6.3.6:  in both (ii) and (iii) change  closed  to  cl-closed . In (iii), after  is compact  insert  (see Definition 6.2.4) 

P.203 (top) [p.189, bottom]: After  in two different orders  add  Rather, it says If you throw some new elements into a set  X,  the closure of the resulting set doesn't lose any elements of  cl(X).

P.208 [p.195] first line of Example 6.5.9:  After  a set of mathematical objects  add  of a given sort (e.g., groups, positive integers, topological spaces) .

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

P.235 [p.222] Exercise 7.4:3: The third word,  one  should, of course, be  on .

P.249 [p.235], Exercise 7.6:2(ii), line 3: change both occurrences of  finite  to  pairwise .  (Respecting pairwise meets and joins is equivalent to respecting finite nonempty meets and joins.  But since it is simplest to define structures of Boolean ring, lattice, semilattice, etc. by just specifying these operations in the pairwise case, as we have done, it is most natural to pose this exercise in terms of these.) 

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.274 [p.258] Exercise 7.8:28:  Change the final parenthetical paragraph  (Suggestion: ... )  to  (Hint: This situation is roughly analogous to that of Lemma 6.5.1(iii).) (The original suggestion was essentially to deduce it from that result.  I don't think that quite works, but it can be obtained by the same sort of considerations.) 

P.276 [p.260], 4th-from-last and 6th-from-last lines of section 7.8:  in both places, change  G  to  Gmd .

P.289 [p.273], at the beginning of the line after (7.10.14), insert  (cf. Definition 7.5.3),  and on the same page, in the line following Exercise 7.10:3, after  In making  Cat  a category , insert  in Definition 7.5.3 . 

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 

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 statement 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  lim  should go on the left, and  lim  on the right.
(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.) 

P.341 [p.322]: in the paragraph preceding Lemma 8.6.8 and the statement of that lemma, everywhere change  cEcf,  etc., to  pEpf,  etc.. In the last formula of the first paragraph of that lemma, delete the small circle.  Add after that formula:  , i.e., the morphism  lim← D  pE1 F → lim← D  pE2 F  which forms commuting squares with the morphisms  pf F(D): pE1F(D)  → pE2F(D)  for all  D ∈ Ob(C). 

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. 

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 fourth 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:  change  in English.  to  in medieval times .

P.383 [p.364], add at end of Exercise 9.1:11:  Can you get such an example where the category over which the limit is taken is finite? 

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.395 [p.375], Exercise 9.3:6(ii): Change  Show that (i)  to  Show that the second assertion of (i) .

P.401 [p.381], add at the end of Exercise 9.4:1:  (Hint: Lemma 9.3:3 concerns the number of generators needed to get an element of a subalgebra generated by a set, as does the suggested variant with  γ = 2.  But what does this variant imply about the number needed to get the expressions on the two sides of an identity?) 

P.402 [p.382], Definition 9.4.6:  At the end of the first sentence, change  some set  J  of identities  to  some set of identities  J ⊆ IΩ  (cf. (9.4.4)).  And in the first line of the second paragraph,  after  is a variety  add  of Ω-algebras .

p.405 [p.386], insert after last word of Exercise 9.4:4:  and the conclusion of the proposition again fails .

P.408 [p.388], Exercise 9.4:7: In the first line, change   to (i)(b)<==>(c) and to (ii)  to  to (i) and (ii),  and in the third line, after  respectively,  add:  and with no homomorphism to the field  /2 replaced by no homomorphism to any field, 

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 with the exercise and preceding and following sentence below:
   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.  I've revised the statement of the exercise to note these facts. 

P.420 [pp.397-398]:  In the second line of Proposition 9.6.4(ii):  operations of  Ω  should be  primitive operations of  V,   while in the second line of Theorem 9.5.5,  V  should be  Ω .

P.421 [p.401], point (d):  (ti, vi)i∈ ari(σ)  should be  ((ti, vi))i∈ ari(σ) . (It is an indexed family of ordered pairs.)

P.424 [p.404], Exercise 9.6:11(iii):  Change  can be obtained from some variety  W  by adjoining one zeroary operation and no identities  to  can be obtained (up to an equivalence which respects underlying sets) by adjoining one zeroary operation and no identities to some variety  W .

P.429 [p.409], second line of second paragraph before Exercise 9.7:5:  Change  functions  to  polynomial functions .

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], change the paragraph preceding Exercise 9.7:6 to  It can be shown that every finite-dimensional Lie algebra over is the Lie algebra of a Lie group  G.   Moreover, two Lie groups  G1  and  G2  determine isomorphic Lie algebras if and only if they are isomorphic near the identity, i.e., have neighborhoods of the identity which are isomorphic under the the restrictions of the group operations to partial operations on those sets; and this in turn can be shown to hold if and only if the universal covering groups of  G1  and  G2  are isomorphic as Lie groups.

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.460 [p.435] top line: before  the entries add  that is, .

P.460 [p.435], bottom: change the line following (10.1.8), up to the words  the identity , to  be the morphism sending the entries of the universal element  r ∈ |SL(n)|  to the corresponding entries of .

P.461 [p.436], 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.469 [p.444], second line of Definition 10.3.5(ii): before  Definition 8.2.3  add  of .

P.470 [p.445], 3rd line of 2nd paragraph of section 10.4:  FC should be  FV .

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

P.479 [p.454], Exercise 10.5:6: Change the final phrase beginning  i.e.,  to a new sentence:  (So if  C  has small colimits, Theorem 10.4.3 tells us that  F  is the left adjoint to a representable functor in the sense we have been studying.)

P.492 [p.466], display preceding Exercise 10.7:1:  I now feel that introducing the abbreviations  η  and  ε  for  ηU,V  and  εG,U  only complicated things.  So in that display, delete   η =  and  ε = ; immediately before the display drop the words  let us write these , after the display drop the two lines of parenthetical comment, and in the statement of Exercise 10.7:1 replace the one occurrence of η  (in part (i)) with  ηU,V  and the three occurrences of  ε  (in parts (i), (iii) and (iv)) with  εG,U .

P.510 [p.484], Exercise 10.10:12: Change  varieties of algebras  to  varieties of finitary algebras .

P.513 [p.487] 4th and 5th lines of Exercise 10.11:3:  After  Cf. Definition 7.8:12  add  and the notation introduced in Exercise 7.8:30(iii) .

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

Pp.520-521 [494-495]: Number the three displays in Corollary 10.12.13 as  (10.12.14),  (10.12.15), and (10.12.16) .

P.521 [p.495], immediately after Exercise 10.12:11 add  Is there a result analogous to Corollary 10.12.3 where one attaches a pair of adjoint functors on the left of the arrows of (10.12.4)?  Certainly!  The roles of  C  and  D  in (10.12.4) are identical, hence that analogous statement is just a case of the corollary itself, after a relabeling.  Then add at the beginning of the next paragraph the words  On the other hand, .

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

P.528 [p.501], Exercise 10.13:11(i)(b):  Change  The clone of operations  to  The clone of derived operations .

P.529 [p.502], line before Exercise 10.13:14:  Change  A question I also don't know the answer to is  to  Two specific questions I don't know the answer to are posed in .