page III.1, line 16: "there exists a family ๐‘… of finitary relations on ๐ด such that ๐ถ = ๐‘…โ–ต, the family of finitary ๐ด-operations that each preserve every relation in ๐‘…." -- The word "each" says nothing here. Delete. page III.2 and III.4 "clones of polynomial operations of an algebra; redundant and slightly inaccurate. Delete the ref to page 4. page III.5, Proof, line 3: "using Example 4 on page 135 Theorem 9.2," --- it seems to me that this clause ought to contain an "and." page III.7, exercise 5: "if n is such that f_C(n) is finite,.." Does this mean anything beyond "If f_C(n) is finite"? page III.26, four lines before Def 9.30: "a variety in which all congruences permute": how about "a variety for which each pair of congruences permutes."? Could also be, "all congruences permute with one another." page III.26, seventh line after Def. 9.29, opening paren after comma. Delete the comma. page III.30: our index listing for "residual bound" points to page 30. This would seeem to be an error. Find a better spot. page III.26-7; Theorems 9.31 and 9.32. It would be good to say that the proof will be completed on page III.29. Then page III.29 should have a couple of \endproofs. page III.31, Corollary 9.41. I would like the statement of this corollary to end with an \endproof. page III.32, sixth line after Theorem 9.42: "Since there are only two congruences 0๐ธ and 1๐ธ and [๐œ™, ๐œ“] โŠ† ๐œ™โˆฉ๐œ“." This reads like a dependent clause, not a whole sentence. page III.33, line 5: "is" should be "are." page III.34, lines-1-3: "Since A = ...and the latter factor is neutral but also Abelian (since it is the homomorphic image of the Abelian algebra ๐€)." There are one dependent clause and one parenthetical remark. I see it as lacking a principal clause. page III.34, Proof, line 9: "expanded all these operations" -- should be "expanded by all these operations" page III.37, display: I wish there were more space around "to" page III.49, Exercise 9.56.4. what is x+y ? page III.50, line -3: "An operation such as โŠ• in the last example, which generates all operations,.." (Is this true?) page III.56, line 3. Notation for an ordered pair. I somehow thought it was . Ditto page 54 and page 108. P.S. I poked around in alvin, and saw a lot of variation on this, so I guess anything goes. page III.59, Theorem 9.79: reference this in Chapter 10. page III.61, Exercise 9.80.10: "A finite algebra ๐€ is called paraprimal if and only if V(๐€) is congruence permutable ... and ,,,." This definition already occurred, on page III.31. Page 31 should be referenced both here and in the index (which is now inaccurate or at least incomplete). Also on page 62 there is a reference to page 50 page III.69, line 7: "for some prime ๐‘, we have {๐‘ฅ โˆฃ ๐‘๐‘ฅ = 0} (where 0 is the additive identity .... ) has at least two elements and is not all of ๐€." This sentence would be clearer if it said something like, "for some prime p, we have that the set {๐‘ฅ โˆฃ ๐‘๐‘ฅ = 0} has at least two ..." page III.95, line 5: certain sums of \iota(i) approach zero. This looks wrong until you realize that \iota(i) depends on k. Therefore it would be preferable to write \iota(i,k). IMO this change would be of benefit all through pages 95-6. page III.103, Theorem 9.110: "where the random variable X(A)= |Aut(A)| for each A \in \E^\sigma." But X(A) isn't a random variable. X is the random variable here; X(A) is one of its values. (I suppose this is analogous to saying that sin(x) is a continuous function, while in fact sin(x) is merely a value of a function. We all know what it means.) page III.103, line -3: delete the two "whether"s. page III.104, line 1: "indicator" -- this is the first appearance of this word in Volume III, and most likely first in all three volumes. I would delete it from where it is now, and then after the displayed definition, I might add a parenthetic remark, "(Similarly defined random variables are often called 'indicator variables.)'" Or I would simply delete the word indicator. page III.105, fifth display. "E(C,E_k) = ..." I am pretty sure this should be "E(X,E_k)." (Evidence: (a) Equation (*) is about X. (b) I couldn't find a capital C anywhere around here.) page III.105, next line: "Distribute the 1/k^{mk} across the sum. We break this sum into three pieces.." --- I would prefer, "We break the resulting sum into three pieces." page III.108, line -2, horizontal spacing after Cf. page III.109, Example 9.117: "This is equivalent to: for some Abelian group โŸจ๐ด, +, โˆ’, 0โŸฉ that ๐‘(๐‘ฅ, ๐‘ฆ, ๐‘ง) = ๐‘ฅ โˆ’ ๐‘ฆ +๐‘ง ." The following rendition may convey the idea more smoothly: "This is equivalent to: there exists an Abelian group such that p(x,y,z) = x - y + z for all x,y,z\in A." page III.109, Exercise 9.119.2.I think the specification of Z_n needs to include: "where 3\leq n <\omega and + is ordinary addition modulo n." page III.110, Exercise 8, line 6. Delete comma after "122". page III.110, Exercise 11, line 1. This could include "in Volume II." page III.114, Lemma 9.130, line 2. This line is a typographic mess. But I guess we have to just let Tex have its way. page III.118, Exercise 2. It would be nice to have \index entries for "Boolean group," and perhaps also \index{variety(ies)!of Boolean groups}. And similarly for semilattices with 0 [or with 1]. page III.119, Exercise 3: this is a two-star problem, and I do not doubt that it merits this accolade. Maybe there should be a couple of pointers to the literature. page III.120, line -12: "We call C essentially k-ary." Don't we really mean to say, "We call such a C essentially k-ary." ? page III.121, paragraph before Theorem 9.138: "The following may be seen as a theorem in abstract clone theory." That seems to be true, IF you have a definition of "idempotent clone" that works for abstract clones. I have such a definition, but I can't recall having mentioned it in Chapter 10 (mea culpa), and I don't know if it is in Chapter 9. page III.122, line 7: "The variety of all algebras with a single ๐‘˜-ary operation satisfying Equation (โ™ฆ) has been called the variety of '๐‘˜-dimensional diagonal algebras.'โ€ Compare this statement with Exercise 9.131.6, which tells us that diagonal algebras satisfy Equation (โ™ฆ) and ๐‘“(๐‘ฅ, . . . , ๐‘ฅ) โ‰ˆ ๐‘ฅ. From these remarks we easily derive that Equation (โ™ฆ) implies idempotence. Which appears to be false. Either the definition in line 7 is inaccurate (forgot idempotence) or the parenthetic remark in Exercise 6 is wrong. (OK, I have made the assumption that "diagonal algebra" is the same thing as "k-dimensional diagonal algebra for some k." If this isn't true, then all bets are off.) page III.134, end of first paragraph of second construction:" the free algebra ๐…๐’ฑ (๐œ”) always has this property." Might be better to say, "...is always generic for V." Chapter 10, Spring 2023 find place in Chap 10 where we talk abput semidegen when top cong is compact. Then refer back to def on "pinched lattice," page II.224. not sure of page number: Luis Sequeira, "near-unanimity is decomposable". AU 60 etc. Just a brief note to say this. page III.156, after Exercise 24: First equation. Don't we need a similar equation, in which the addition occurs in the second co-ordinate? (etc.) page III.181, Exercise 10.63.15: "Let be" -- should be "Let Phi be" page III.181, Exercise 10.63.18:"There is an infinite semigroup A whose reflection in W [group theory] is the one-element group. (The exercise is to find such a semigroup.) Right now this seems like an easy exercise: just let A be an infinite semigroup with constant multiplication. Right now (June 2023) I think the question should involve the forgetting functor from groups to monoids. Why else would we have awarded the exercise two stars? find place in Chap 10 where we talk about semidegenerate when top cong is compact. Then refer back to def of "pinched lattice," page II.224. --- August 17, 2023. Today I don't know what this means. page III.207 ff -- refer back to theorem 7.37, page II.222 page III.222, Theorem 10.124. Make a note to look at Theorem in Chapter 7. page III.240, description of Figure 10.5. Remark that Theorem 6.21 will not completed until Chapter 11. page III.251, Exercise 6, fifth line: "D \from\lambda\to T..." page III.251, line -6, "f". There is a lot wrong with the hints for this exercise. The equation involving "f" is not relevant, since f doesn't appear in the exercise as such. The rest of the hint is sketchy, and the claimed description of the clone is seriously false. page III.265, lines 1-2: "the commutative hypervariety" -- should be "the commutativity hypervariety." (This word occurs in the title of Example 2, on the previous page.) page III.269, Exercise 10.164.4(iii): "autofunctor" -- I think this should be "endofunctor." [Although neither term is defined in the book.] page III.274, line 12. "there exits tuples," should be "there exist tuples" page III.275, Definition 11.6: "A\models [1_A,1_A]=0." Strictly speaking, "\models" applies only to first-order (elementary) sentences in the relevant first-order language. This language does not include commutator. This error also occurs elsewhere in this section. (We know what you mean.) page III.276, lines 11-12: "there is an (k+1)-ary term ..." page III.277, Corollary 11.11: \endproof missing page III.277, Corollary 11.11: "what is A(\beta) ?" (This is said on the previous page, but ...) page III.280, line -7: "the equation (\Delta_n)-- shouldn't this be "equations"? page III.287, line 5: "Clearly \gamma is a union \Delta-classes." -- Should be "union of \Delta-classes." page III.289, diagram: The two bowties should be big join signs so the leftmost item of the figure should be [\bigvee \alpha_i,\beta] = \bigvee [\alpha_i,\beta]. page III.298, middle: Nathan Faulkner missing from index page III.300, line 13: change "is congruence semidistributive" to "is congruence meet semidistributive". page III.305, Lemma 11.41, Proof, ".. multilinearity of each f \in \tau.." This f is just a function symbol. As such, it can't be multilinear. page III.306, line 14. "... each belonging to (\alpha*,\beta*) and each uses .." == should be "..belonging ... and using .." page III.308, line -12: "We take G to consists of.." page III.308, line -11: "Note the twisting the second column." --- should be "Note the twisting of the second column." --- page III.313, Corollary 11.49: "Let \alpha and \beta be congruence on an algebra A .." -- Should be "Let \alpha and \beta be congruences.." page III.332, line 2! --- What? 7/28/23: at this point I can't see anything wrong here. WT page III.333, Theorem 11.84: add these two index commands: \index{variety(ies)!$n$-step solvable} \index{variety(ies)!$n$-step nilpotent}} page III.339, line -4: "in on" page III.358, display, line 2: "w.(u+v) = w.v + w.v" -- should be "w.(u+v) = w.u + w.v" page III.335, Lemma 11.89: the statement has polynomial operations mixed in with terms. [I think that ultimately we are permitting this. See bottom of page II.7.] page III.336, at Corollary 11.90, add this command: \index{variety(ies)!congruence-uniform} pages III.339-340. Single sentence overlapping these two pages. Problematic sentence. One sort of knows what it means, however. page III.341, subsection Abelian Varieties. BTW I couldn't find a definition of Abelian varieties. One could be added somewhere. [Maybe we can all guess what it is. One reason to have an expressly stated definition is that then we have a place for the index to point to.] page III.343, Exercise 11.104.2: add this: \index{variety(ies)!of entropic quasigroups} page III.337, third line after Equation (11.6.5): "R-modules" -- should be "R(V)-modules." page III.355, Exercise 11.116.4: "i^2 = 1." Shouldn't it be "i^2 = -1" ? page III.356, Condition (c): "See Corollary 9.41 on page 31 of Volume II" -- The location is actually Volume III. page III.356, line -9: "These varieties from a descending chain" -- Should be "form" page III.357, just after Theorem 11.117: "Bilinear algebras ... were introduced ... in Volume I." I would like to see something like \index{algebra(s)!bilinear} on this page. (And perhaps elsewhere as well.) page III.359, near middle, "the suggestion of John von Neumann," for one way to represent each n as an initial segment of \omega. This convention has been in place since the start of Volume I -- see item 13.b in the Preliminaries to Volume I (page I.7). In fact the convention is expressly stated one more place in Volume II or III, but I can't find it. page III.362, just before Contention: "this automatic algebra is richer than T_n." ???? definition of "richer"?? page III.383, line 15: "...every...group is the direct of its Sylow subgroups." -- Should be "is the direct product of.." page III.383, line 20: "showing that the above theorem is not a consequence of Theorem 11.119." Actually if A and B are theorems, then B is a consequence of A. One could say, "not an easy consequence." page III.384, Exercise 11.140.6: "Let A be a finite algebra in congruence modular variety.." -- Should be "in a congruence modular variety.." page III.416, entry for Mal'tsev: it says "see Mal'tsev (1963)" page III.424 -- George Bergman should have an index entry. Not sure where WTaylor, papre on regular permutable varieties and their s.i. algebras. Give it one sentence. page III.vi, List of Figures, Figure 11.8. "associate" should be "associated" page III.10.145*, Theorem 10.13. "Equations 10.1.2 and 10.1.3" --- should have parens page III.167, line 6: "the term condition allow us to ..." Right here through Theorem 1.15, there are no \endproofs. page III.174, theorem 10.58. Proof, line 6, "... and a function f:X-->D." The D is a set, should not be boldface letter. page III.179, Exercise 10.63.4: "use the second description of ๐‚(๐’ฑ) that was given in Example 3 of ยง10.1, page 134.)" Shouldn't this be the third description? page III.181, Exercise 13: "Let \Var V (resp. \var W) be..." -- should probably be "Let \Var V(resp.\ \var W)" (or use ~) page III.181, Exercise 15: "Let be ..." -- should be "Let \Phi be.." page III.193, Definition 10.83(ii): too much space. I don;t know where this is coming from, but there must be a way to enter correct source code. page III.194, "Ultraproducts," line 4: "(See \S8.2.)" should be "(See \S8.2 of Volume II.)" page III.199, fourth line of \S10.6: "al. 2011)" == should be "al.~2011)" page III.303, nine lines after display of (i),(ii),(iii): there is a sort of fuzzy star, used as an exponent on A here. I find this typography hard to see and hard to read. In fact "dagger" (see e.g. page III.187) is just as bad. page III.202, Corollary 10.93 and Definition 10.95. Here the Definition should probably come first, since it contains a def needed for the Cor. page III.202, just before Corollary 10.93 it says, "in Equations (10.6.1)โ€“10.6.7". The closing parenthesis is in the wrong location. Note -- this error appears to be widespread. For example, it occurs on page II.95, lines 2-3. page III.198, Theorem 10.111(v). Let's point to a reference to linear equations in this volume. page III.214, line 2, subscript V should be script page III.212, line -16; ",of finite signature," needs space before o page III.211, line -4; "Garcia and Taylor" -- should have a year (1984) page III.249, line 7: "See alsoWalter D. Neumann 1974andJohn" (spaces) page III.249, line 8: "and Joel Berman 1977 A" should be: "and Joel Berman 1977. A" (period inserted) page III.252, Exercise 12. Second line. The square-bracket expression there denotes an interval, and should be preceded by I. page III.302, line 4, middle: "of Volume II.)" -- should be "of Volume II)." page III.302, line 8: "Since ๐‘“๐‘–(๐‘ฅ, ๐‘ฆ, ๐‘ฅ) โ‰ˆ ๐‘”๐‘–(๐‘ฅ, ๐‘ฆ, ๐‘ฅ)." \Wavy equality doesn't express anything. Need to say "[equation] \models [equation]" page III.302, line 9: "the term condition allow us to...": should be "...condition allows us..." page III.303, line -11: "is natural bijection" -- should be "is a natural bijection" page III.303, line 18, condition (iii). "vectors of distinct variables." page III.308, line -11 (in proof of 11.44): "Note the twisting the second column" -- should be "note the twisting of the second column." page III.308, line -2: "inductively." -- I would say "recursively." page III.309, four lines above (I),(II),(III): "there is an direct-up edge." page III.312: "labeling" and "labelling" both occur on this page. I believe that up to this page we have used "labeling" most of the time. page III.313, Corollary 11.49: "Let \alpha and \beta be congruence on an algebra A." -- should be "congruences" page III.315, line 12: "If A affine." -- should be "if A is affine," page III.315, Theorem 11.56. This theorem does not appear to be connected to any \endproof mark. I suppose it's supposed to be obvious from the historical remarks just before Theorem 11.56. I would appreciate some help on it. page III.356, line 22: "according the Baker's Finite Basis Theorem" should be "according to Baker's Finite Basis Theorem" page III.357, top. Subsection heading: "Nonfinitely Based Algebras Belonging to Congruence Permutable Varieties" -- But there is no mention of permutability until the next subsection heading, (page 366). page III.409, refs, P. M. Cohn 1965, spacing is wrong. This error recurs here and there throughout the references. page III.424, index: Cayley, Arthur. get name to occur once only in the index page III.274, line 14: "there exits tuples" should be "there exist tuples" page III.xxx, bibliographic entry for Post (1920). ".. theory of elementary proposition." This feels wrong to wt; he surmises that it is really "propositions."