vol2 index: line 5, condtition, 27 terms, w-universal set of.. needs a hyphen index: critical aglebra, 353 index: Jackson, M. and Jackson N. seem to be the same person index: include "rectangular bands," page 250 --------------------------- page II.5, line 2: "the signature of lattices is {\meet,\join}" The signature is supposed to be a function, so we could say "the signature of lattices is {<0,\meet>,<1,\join>}" page II.5, line 2: "the signature of lattices is {\meet,\join}" (signature is defined to be a function) page II.6, last paragraph: rewrite this some day page II.7, first line after (6.1.2): boldface A page II.9, line -14: "A Mal'tsev condition, in fact defines a filter..." -- delete comma page II.23, line 2: "there is" should be "there are." page II.23, end of line 6, "of" -- delete this word page II.23, bottom, after (e) "Indeed, the tree depicting any slender term has all these properties." I think that what's wanted here is to say that conditions (a)-(e) constitute a definition of "slender." Then this remark is a tautology. Or wait, perhaps the remark in second para page 23 is intended as a def of "slender term." Get clear on this. page II.24, line -8, "invariant respect to every unary polynomial" -- should be "invariant with respect to..." page II.24, line 13: "enhanced" -- a defined word should be boldface page II.25: something slightly off about the diagrams on this page. The text mentions only a single complexity \ell, whereas the drawings show distinct complexities \ell_0, \ell_1, etc. page II.24, line 8: "operations symbols" page II.30, line 14: "occurring is" shoud be "occurring in" page II.26, end of line 10, "above." -- This word should be followed by a comma, not preceded. Also, it seemed to me that this should say "below" rather than "above." page II.24, line -8, "invariant respect to every unary polynomial" -- should be "invariant with respect to..." page II.24, line 8: "operations symbols" page II.24, line 13: "enhanced" -- a defined word should be boldface page II.25: something slightly off about the diagrams on this page. The text mentions only a single complexity \ell, whereas the drawings show distinct complexities \ell_0, \ell_1, etc. Am I missing something? page II.25, line -2. "sets" -- should this be "sets out"? page II.26, top: "Let F be the set of all terms x+a with a ranging over all constants in our algebra. Then \ell = 1, and F is infinite." Is this a counterexample to the first sentence on on this page. (It seems to me that, for finiteness, we need the next sentence also: "for convenience we also assume ... .") page II.26, line -5: "congruences lattices" should be "congruence lattices" page II.27, line 4; middle line of display (6.4.1): "for all n"-- this appears on the right side, but not on the left side. probably both sides should have the same treatment. In any case, it seems to me, when you are using recursion (or induction) over \omega, the "for all n" part is tacitly assumed, and doesn't need to be said. In other words: the inductive proof apparatus automatically covers all n (that's its job). page II.26, end of line 10, "above." -- This word should be followed by a comma, not preceded. Also, it seemed to me that this should say "below" rather than "above." page II.27, last line: "by associating a ternary term t_i(x,y,z) with each bracket, left and right... ." This description of C_2(b) is hard to parse, and I'm not sure it really contains all that is wanted. For elucidation I went back 1/13/23 to George's paper with Baker and Wang (2004); the description of C_2(b) in that earlier paper is (to my ear) much more comprehensible. (Nb the designation C_2(b) does not appear in BMW-2004.) Recommendation of WT: (a) add this: "for more details, consult (Baker-McNulty-Wang2004)." (b) for a larger revision, simply paste in the appropriate parts from BMW-2004. page II.28, two lines before line 2: "and it is observed that [\alpha,\beta] = \alpha\meet\beta .." -- I would like to see a comma before "and" -- right now it seems to be a run-on sentence. page II.28, top, perhaps rewrite M'v condition for a bracket expression (see elsewhere) to accord with other needed changes page II.28, just before Theorem 6.21. One could insert, "See M. Olsak, 'Maltsev conditions for general congruence meet- semidistributive algebras,' The Journal of Symbolic Logic Volume 86, Number 4, December 2021, 1432--1451. This article extends Theorem 6.21 with further equivalent conditions, including one Mal'tsev condition of a more familiar shape." page II.30. first line of (iib)==>(iv): "occurring is" should be "occurring in" page II.30, line 14: "occurring is" should be "occurring in" page II.30, line -11: "in assumed" should be "is assumed" page II.32, EXAMPLE 6.22. Several occurrences of \script V, up to the end of the EXAMPLE, should be changed to \script S (for "semilattices"). page II.34, line -4: "When D = I[a) is the principal filter generated by a, we denote this congruence by ... " I think that what is meant is really, "When D is I[a), the principal filter generated by a, we denote ..." page II.34, line 12: "less" should probably be "fewer" page II.34, four lines before \endproof: "these combined" should be "these combine" page II.40, THEOREM 6.32, part (iv): "... the following identities hold in .." This thought needs to be completed: hold where? page II.42. line 1: "weaken" really should be "remove" page II.45, Theorem 6.38: "Let A be..." really should be "If A is..." page II.32, proof of (i)=>(vi). How do we get a non-trivial module variety U that satisfies C_2(b)? page II.35, Proof of 6.25, line 5: "every congruences" should be "every congruence." page II.38, just before Thm. 6.30: "let K_{fin} be the finite members" should be "let K_{fin} be its finite members." page II.32, proof of (i)=>(vi). How do we get a non-trivial module variety U that satisfies C_2(b)? page II.35, Proof of 6.25, line 5: "every congruences" should be "every congruence." page II.38, just before Thm. 6.30: "let K_{fin} be the finite members" should be "let K_{fin} be its finite members." page II.40, THEOREM 6.32, part (iv): "... the following identities hold in .." This thought needs to be completed: hold where? page II.42, line -1: "this is sufficient" might be clearer if it said, "this collapse is sufficient for modularity." page II.42. line 1: "weaken" really should be "remove" page II.45, Theorem 6.38: "Let A be..." really should be "If A is..." page II.51, before Theorem 6.48: did Csakany also prove that regularity is Mal'tsev-definable? Have to check this. page II.55, end of page. large displayed implication. This could be double checked. One sentence could be added: "(There is one implication for each vector of values \la n,m,p,i,j \ra.)" page II.56. line -13: "Let f be the function..." -- should be "Let f be a function..." page II.58, line -7, requires a period. page II.59, before exercise 9. Reference to Exer 6.16.17. The "17" is hard-coded, and regrettably no longer accurate. Essentially the same hard-coding occurs on page II.58, line 8. page II.60, Exercise 19, line 5: "of type <2,3,1,0,1>" The term "type" is out of date. Refer instead to the "signature." The same comment applies to Exercise 21 on page II.62, and to ... page II.60, Exercise 19, line 8: "a,and" should be "a, and" page II.71, middle: issue around D\subseteq V. WT should study this. page II.71, Example 6.65, Equation (6.7.14) -- "or" not in the right place. page II.71, Example 6.66: "f(x,...,x) \Wavy x" -- The connective \Wavy does not make assertions. page II.77, line 3. End with period. page II.82, Example 1, line 5. The first period here should be a comma. page II.97, Figure 6.12: there are slight errors in the rendering of this figure. E.g. left-right symmetry fails at a couple of the vertices. (The figure does, however, accurately portray the finite lattice L_{14}.) Hardly worth the effort. page II.97, before Equation (6.9.2): change the comma on this line to semicolon or to colon or to dash. page II.101, bottom, statement of theorem 6.104: should include a year, page II.109, line 14. "see \S10.8, page 232" -- should be "Volume III, \S10.8, page 232" page II.111. line 10: "there be" should be "there to be" page II.122, Theorem 6.120. Proof: "Let \alpha, \beta, and \gamma be elements Con A with ..." -- Should be "elements of Con A." page II.126, after display: "Here \not x denotes the complement of x." ("s" is missing). page II.130, table for a binary ternary operation. Some explanation? Probably not necessary. page II.131: "For any 𝐀 ∈ 𝒱, a finite partial operation 𝑓 has an interpolating term operation of 𝐀 if and only if 𝑓 preserves all subuniverses of 𝐀𝑛." Some definitions might be wanted here? page II.133, Exercise 19, line 2: "we can viewed" -- change to "we can view" or "can be viewed." page II.133, line 3: "Abelian" should be "\boldface{Abelian}" page II.137, Proof, line 2: "proceed" should probably be "proceed via" page II.143, Exercise 24. Let 𝐀 = ⟨{0, 1}, π‘š, β€²βŸ© be the algebra where π‘š is the majority function and ...": should probably say, "where m is the ternary majority function and ..." page II.146, line 4: "and F(0) is" is better as "and F(0) to be" page II.146, five lines after Def 7.2: "varies" should be "differs" page II.149, line 3: "particular" -- delete word page II.151, is \models really "logical consequence".) page II.153, Exercise 8.56.1: "Use Theorem 8.44 to prove Quackenbush's .. Corollary 7.78 that ..." -- And at 7.78 we should add a notice about this. page II.155, line 11: "there is" should be "there are" page II.163, line -11: what is meant by "invariants of T"? page II.164, Exercise 7.12.1, line 6: "the any" page II.164, Exercise 7.12.8, "axioms." Could we add "(equational)"? page II.169, Exercises 25 and 26. "[n,m]" stands for an interval, hence should be written "I[m,n]" page II.172, line 10: "the only basic operations of V have rank less than 2" -- delete "only" page II.174, line 3, "We first build the algebra A_n" -- should be "an algebra A_n." page II.175, line 4: "but leave" should be "but leaving" page II.175, line 8: "one square of the table: in B_n..." (It seems to me that a single element. such as B_n, can only specify a row of the table or a column of the table (depending on how our table is laid out). It doesn't specify a single entry (square). Maybe o.k. page II.184, line 10: "countable many blocks" -- "countably"? "nonfinitely in the finite sense" [missing "based"] page II.187, line 2: "adjoining the constantly 0 tuple to 𝐆, where the value 0 to all products undefined in 𝐆.": The dependent clause "where the value 0 .." needs completion page II.187, CLAIM 5: Do we have a definition for a congruence relation of a partial algebra? Assuming we do, does this Claim say anything beyond "\tau\Compos\rho is a congruence on G, and |G/\tau\Compos\rho|=2(k^2-k)m'"? I'm just asking. Not really proposing an action item here. page II.190, Jonsson's speculation: bad \vspace (Compare with the Examples section on page II.191.) page II.192, Fact 2, Proof, line 13: "Evidently, \theta an equivalence relation." Should be "is an equivalence relation." page II.194, Exercise 6: "Prove than" should be "Prove that" page II.196, near bottom: "[A certain pointed group] is perhaps the most notable of a number of other counterexamples." Maybe it's ok, but it sure seemed contradictory at first. P/S. looked over this recently. I don't know what exactly was my complaint, but it does seem you have a long sentence that might benefit from splitting in two. page II.197, line 6: "What follows is, to all intents and purposes McKenzie’s original publishedoge II.233G argument," -- needs a comma after "purposes" page II.199, Key Lemma, line 7, "plug" -- Is this a technical term? page II.199-200 -- WT check this again some time. page II.200, line 2, "particular" -- word says nothing here; delete page II.201, just before (*): "finish the proof of the Theorem" -- supply Theorem number (It already occurs just below here; perhaps that is enough.) page II.202, second para. Extreme case of run-on sentence. Not really recommending to rewrite it at this stage. page II.202, eight lines above the \diamond-condition: "the associated flat graph algebra" -- change to "G_G" page II.203, line -16: "Also. it follows." -- use comma page II.213, line 10, "each edge ... produces ... " -- "will produce" page II.223, line -14: "entries" -- I would be happier with "components" (occurs elsewhere also) page II.225, just before the Pinched Lattice Theorem. "Type 𝑛 joins has a similar definition." -- either write "Type n joins" in quotes (which makes it singular) or change "has" to "have. page II.2xx, Exercise 10.150.3. It would be nice to have an index item that points to the definition of "undemanding." page II.234, please add: \index{variety(ies)!of bands} \index{variety(ies)!of rectangular bands} page II.229, after the diagram. "x \goesto Dxy and y \goesto Bxy." I don't find any support for \goesto here. (I sort of guess the first one means F(x)=Dxy, but that isn't written here.) page II.229, two lines further on: "Let 𝜎 be the signature that provides ..." This really should say, "let \sigma be _a_ signature that ... " page II.230, Exercise 1, line 3: "then infinite cardinal πœ… is..." -- should be "then the infinite cardinal πœ… is... page II.233, third line after display. "... equational theories that include the group axioms" -- better to say, "... "equational theories that page II.236, line -5: "i_0 is the index associated be s." Make this "associated to" or "associated with." page II.238, line 4: "some" -- change to "a" page II.238, line -5: "and in which" -- this should be "in which". The rest of this run-on sentence is hard (for me) to parse with any assurance. E.g. "these segments" most likely means v and v', but it's hard to be sure. page II.234, please add: \index{variety(ies)!of bands} \index{variety(ies)!of rectangular bands} page II.239, middle: "Let 𝐱 be the tuple that assigns the variable π‘₯ to every variable." OK every tuple is a function, so "assign" coud make sense here, but the domain of a tuple is always an initial segment of \omega. page II.240, Theorem 7.53 of Pigozzi. First sentence of proof. "The two statements ... are ... equivalent." --- All theorems are equivalent. Here it could say "easily interderivable," or something like that. Not sure where: wtaylor, paper on regular, permutable varieties and their s.i. algebras. Give it one sentence. page II.240, line 12: "problem." We generally are calling these exercises. page II.240, Theorem 7.53 of Pigozzi. First sentence of proof. "The two statements ... are ... equivalent." --- All theorems are equivalent. Here it could say "easily interderivable," or something like that. Not sure where: wtaylor, paper on regular, permutable varieties and their s.i. algebras. Give it one sentence. page II.243, line 17: "T is a map from Con A to" [a certain interval of theories]. Then in Contention 4 (line -8) we are told that T stands for an equational theory. later: ok, now I see, one of those T's is really a capital tau. (How else does one get, in math mode, a non-slanted non-bold capital T?) page II.244, line 9: "So using .. can be simplified." Needs a comma somewhere, or some other treatment. page II.244, middle: "done,so" page II.244, line -6: "gives value in A" -- should be "gives a value in A." page II.245, line 7: "occurs in either s and t." page II.245, line 13: "So no variables occur in p and q." Does this mean "no variable occurs in p or in q," or does it mean "there is no variable that occurs both in p and in q."? page II.245, eight lines before Contention 5. "the basic subterms ... produce ... " ???? page II.249, at end of text paragraph, "obvious automorphisms" -- why not boldface? (This seems to be a definition.) page II.255, line 3: "ordered pairs of terms in the variables x_0, ...x_{r-1}"--shouldn't this be "x_{n-1}"? page II.255, Fact 1. -- missing \endproof page II.256, line 18: ",." page II.257, Theorem 7.61, proof, line 1: "let is ..." page II.258, very bottom."If all the equations in Ξ” turns up on the second list,..." page II.259, line 6: "in with no variables occur" -- should be "in which..." page II.259, line 14: "recursive enumerable" should be "recursively" page II.260, "comprise" page II.260, line -12: "based on the complexity terms" -- should be "of terms" page II.260, line -5: "which becomes an equality in 𝐒" -- should be "which becomes an equality in 𝐒," (comma added) page II.262; three lines before Claim 1: "We say it is in normal forms." -- should be "in normal form." (no s) page II.262; two lines before Claim 1: "the that" page II.262; proof of Claim 1, line 7: "the induction hypothesis holds asserts that ... " page II.263; proof of Claim 2, line 6: "the t reduces" -- should be "t reduces" page II.263; proof of Claim 2, line 7: "𝑠 reduces of 𝑑 in one step" -- should be "reduces to" page II.263; line -8: "Now put u_i' to be the normal form" -- change to, "Now let u_i' be the normal form.." page II.265; line 13: "there is a term t_w that looks for example like.. where.." -- this does not read like an existential statement; rather, it reads like a definition (although the "for example" kind of clouds any authority it might have had as a definition). My guess is that one could just delete any mention of t_w in line 13. Actually t_w is not mentioned until in the statement of Thm 7.65, where it is well defined. page II.266, Corollary 7.66: I would like to see an \endproof. page II.270, "Case" line 8: "Last" -- should be "Lastly" ?? page II.274, line x: "The first, 𝐿 sends 𝑄" should be "The first, 𝐿, sends 𝑄" page II.276, line 4. Add refs about K_{\infty} in Chapter 10 page II.316, line 14: "larger then" page II.317, last statement on page: "We use ..." -- If I is another name for \omega, then we're in the clear. Otherwise not. page II.320, line 7: "Two structures A and B are elementarily, ... if and only if ... ." -- should be "elementarily equivalent" page II.322, second paragraph after Example 8.10, middle of page: "the inclusion map in an elementary embedding" -- should be "is an elementary embedding." page II.322, Theorem 8.11. \endproof? page II.325, middle of page, condition F: "..the lattice of all equational theories of a given fixed signature": -- delete "fixed", or delete "given", or delete them both. page II.325, Exercise 8.17.1, line 2: "A satisfies \phi is A if and only if b satisfies \phi in A." -- change "is" to "in." page II.326, line 2: "formula obtained for \phi," -- should be "obtained from \phi" page II.327, \S8.2, line 1: Give reference on Galois connections: pages I.51-52 page II.338, line 11: "symbols added the L'" -- should be "added to L'" page II.329, last para plus few lines on p. 330: imho it would be preferable to say, soon and directly: "Obviously Th 8.20 does not hold for all formulas." page II.339, line -18: "This set of sentence" -- should be "sentences." page II.339, seven lines before Theorem 8.38: "on the other hand" -- here this phrase should be followed by a comma. page II.339, two lines before Theorem 8.38: "set of sentence" should be "set of sentences"" page II.339, four lines after Theorem 8.38: "let \phi(x) be ... and a is..." page II.339, eight lines after Theorem 8.38: "Let πβˆ—1 be that substructure of ⟨𝐁1, 𝐴1⟩ generated by..." -- "that" should be "the" page II.347, two lines before Corollary 8.50. : "this" should be capitalized (in order to agree with the previous line). page II.351, line 9, "The structure A is finitely subdirectly reducible provided there is ... ": Is this the _definition_ of "finitely subdirectly reducible"? It should say so, and best if it had an index entry. Also boldface. page II.351, line -6: "the B*" should be "B*" page II.153, Exercise 8.56.1: "Use Theorem 8.44 to prove Quackenbush's .. Corollary 7.78 that ..." -- And at 7.78 we should add a notice about this. page II.356, line -5: "this last class includes of the subdirectly irreducible algebras in 𝒱." -- "includes all of"? or something else? page II.357, line -4: "the lattice of equational theories" -- of a single signature? page II.357, line -1: "Jonsson" should be "Jonsson's" page II.358, line 10: "were" -- I woud prefer "are" here page II.360, line 18: "components linked to the i-th of A0A0" -- probably should say, "linked to the i-th component of A0A0" page II.360, before the bullet points: C really ought to be C_i; however, since C is almost not mentioned again, and plays no big role, it may as well stay like this, page II.361, line 4: "is the only subdirectly irreducible lattice is V𝐋 βˆ– VF," -- should be "only s.i.lattice in V𝐋 βˆ– VF," (I other words,"is" should be "in.") page II.363, line 4: "operations symbols" should be "operation symbols." page II.368, line -6: "The length of a lattice is the least upper bound on length of any chain" -- I would say, "of the length of any chain." page II.373, Exercise 10,: "..lattices, which we assume are disjoint." -- should it not be something like "whose universes are assumed to be disjoint"? page II.378, line -3: Delete "large" and "(the universe)". page II.379, line 16: comma preceding "which" page II.385, fifth line under "Primal algebras". "the only properties the 2 that come into play" == should be "the only properties of 2." page II.389, line -14: "8.6.3," being an equation number, should be enclosed in parentheses. page II.395, top two lines: "such that ... and with ...." I would prefer "such that ... and such that ... ." page II.395, Exercise 4, line 3: "but H x G \iso H." This is wrong; it should say "\not\iso H." (An earlier writeup contained an error in stating the revised version. This one is correct 7/18/23.) page II.396, line 6: "amendable" should be "amenable." page II.396, line 14: "nice." Delete. (Our definition did not say anything about a _nice_ vector space.) page II.396, line -4: "Some model M of \Lambda." -- has \Lambda been defined? I couldn't find it. page II.397, line 1: K needs to be introduced. page II.397, first bullet, "in every L+ structure," -- Possibly delete. (Without this phrase, it says more or less the same thing. Or, one could keep a more detailed version as follows: "The set Ξ› includes sentences that assert, in every 𝔏+-structure A, that π‘ˆ^A denotes a nonempty set that is closed under all the operations denoted by operation symbols of 𝔏." page II.397, Claim 1, line 2: "so that xxxx that has no model." I think this should be "so that xxxx has no model." (Thus an extra "that" is to be deleted.) page II.399, line -9... "Let .... is a finite subset of ,,, ." This should say, "Let .... be a finite subset of ,,, ." page II.402, line -10: "each \eps_i as well as \gamma are equations." -- Change to "each \eps_i, as well as \gamma, is an equation." page II.403, right after "Quasivarieties." -- " ... where each πœ–π‘– as well as 𝛾 are equations." I generally feel this oughtto say ".. each πœ–π‘– ... is an equation." page II.404,Theorem 8.04. (Compare page II.398, lines 2 and 4.) For implication (as a connective in elem formulas), do we use an arrow with one horizontal line segment, or two parallel segments? --> or ==> ?? And then, near the end of page 404 there is a particularly long double arrow. (Well, I think this question/problem goes way beyond any project of correcting typos. Any guidance from you, George? page II.405, "preserved with respect to the formation of arbitrary direct products." I don't see that the word "arbitrary" adds anything here. page II.405, Cor. 8.106, second sentence. This is a run-on sentence. Easy to fix. page II.405, line -5: "giving notion of formal deduction.." -- should be "giving a notion ..." page II.405, ine -5: "proving corresponding completeness thm." -- should be "proving the corresponding .." page II.409, Exercise 9(i)."PsK is the relativized reduct of some elementary class." -- This probably ought to say, "... a relativized reduct of... ." Up to this point, the authors have established (mostly by example) that "the" is used for those cases where U is specified. page II.411, Def 8.111, line 7: "is smallest set ... such that ... ." -- should be, "is the smallest set ... ." page II.413, Def 8.114(4): "... complete list of all the .." -- we could delete "complete" or delete "all." page II.415, line -8: "we pick 𝑏𝑖 ∈ 𝐴𝑖 is the following manner" -- should be "in the following manner." page II.416, Corollary 8.116, "Let 𝐀𝑖 and 𝐁𝑖 be structures for the same language for every 𝑖 ∈ 𝐼." page II.416, Corollary 8.116. Here we encounter the symbol \curly<=. This symbol will appear again in Theorem 8.123 (Tarski's Elementary Chain Theorem), on page II.422. Also Thms 8.124, 8.125, 8.126, 8.127. I haven't been able to find this symbol elsewhere in Alvin. More specifically, I have not found a definition of the symbol (nor words for the concept). (Maybe they are in there--Alvin's a big place.) The closely related symbol \curly< was defined on page II.322. The traditional difference between the symbols < and <= is the reflexivity of <=. However, \curly< is already reflexive (definition on p. 322). Thus extending \curly< to \curly<= via reflexivity would not change anything. Recommendation: (i) If my analysis is correct, then remove \curly<= in favor of \curly<, throughout Alvin. (ii) In the alternate case, retain \curly<= and make necessary changes, including a definition and an index entry. P.S. one can reconstruct a likely definition for \curly<= from what is written at the start of the proof of Thm 8.123. It turns out to be the same as our def of \curly<, thus somewhat confirming clause (i) above. page II.417, just after the paragraph about the Fefferman-Vaught Theorem, with commentary about some Boolean algebraists. "We say a sequence ... is partitioned provided ..." This definition has one typo: missing a right paren, right before the semicolon. page II.418, line -2: "with such that" -- should be "such that" page II.419, line 8: "There are least s atoms" -- should be "at least" page II.4i9, line 4. I would not say that we are formalizing \theta_s in the language of boolean algebras. If that were the case, then Claim 1 tells me that there is a very specific number s, which presumably we could evaluate. Claim 1 would turn into somethihAg like "\theta_7 is true in every finite Boolean algebra." However, this doesn't happen because really s is an implicit function of the \psi's. I guess this is justifiable since Claim 1 isn't being sold as a theorem. page II.420, line 3. Same theme continues. page II.419, line 11: "There are at exactly..." -- should be "at least" or "exactly." Then same mistake, page II.420, line 13. page II.421, large display, last line: "relations symbol" should be "relation symbol." page II.421, line -6: "that must be" -- should be "there must be" page II.424, Theorem 8.125, "a substructure C" -- what is C supposed to be a substructure of? page II.424, Theorem 8.126, line 6: "A curly<= C\reduct_to L_1." shouldn't this C be boldface? page II.425, four and five lines above the display: "the new constants symbols" -- should be "constant symbols" page II.426, line 7: "Therefore T\models\not\theta and so that \not\theta\isin\Phi." If I understand things here, what this needs is to have "that" deleted. page II.426, Theorem 8.129 (Beth) \Sigma's signature is said to be defiined by appending {R} to L. But a signature (or language) has got to also include the arity of R. ("A signature is a function ...") page II.426, line -4: "with the help of the connectives to be \and, \or and \not, and the quantifiers..." If we delete "to be" then I will understand this sentence. page II.435, Exercise 6: there is a ... sentence \sigma and a formula... \psi ... -- should be "there are." page II.441, Exercise 8.141.1. "a signature which includes, among its .. symbols a binary symbol ..." -- commas not balanced. Perhaps use one more or one less. In the index: line 5, condtition, 27 terms, w-universal set of.. needs a hyphen critical algebra, 353 Jackson, M. and Jackson N. seem to be the same person include "rectangular bands," page 250 ---------------------------