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\begin{document}

\title{response to referee's report}
\maketitle

Both authors are extremely grateful to the referee for his 
careful and critical reading of our manuscript.
We made an effort to comply with his suggestions.

In this response, we describe how we dealt with the suggestions 
in the main review, and then we describe how we dealt with the suggestions
made on the PDF markup.

\bigskip
\bigskip

{\bf Referee's remarks, from file {\tt kearnes-kiss-review.pdf}}

\medskip

{\bf Big suggestion 1.} Add a list of open problems. Done!

\medskip

{\bf Big suggestion 2.} Add a subsection 
to Chapter 2 summarizing tame congruence theory.

\smallskip

After some consideration, we decided against this.
Chapter 2 functions to brief readers on topics that are actually
used in the manuscript. While tame congruence theory does motivate
much of the work here, it is not used anywhere.

It seemed better for us to point out where the results
of tame congruence theory influenced us than to summarize
the subject. Some remarks of this type were in the manuscript
before, but we added more of these remarks.


\medskip

{\bf Lengthy remarks.}

{\bf Chapter 1.}

\medskip

Section 1.3:
``Can a general definition or description be given of what is
meant by a commutator theory?''

\medskip

We don't know.
We refer to commutator theories that are known,
but we aren't ready to define the limits of what could be a commutator
theory. For example, could an algebra with a compatible
up and down directed partial order be the model
of an abelian algebra for some commutator theory?
No such theory exists, but we know of no reason 
blocking the development of 
a theory of this type.

We addressed this comment by slightly drawing back
from our earlier assertion that the existing commutator
theories \emph{are} approximations to the four we describe,
saying rather that the \emph{seem to be} 
approximations to the four we describe.

\medskip

{\bf Chapter 2.}

\medskip


Section 2.3: 
``if you have any sympathy for an unrepentant category theory-phobe, 
you could consider providing a little more detail on what
natural bijections, adjoints, and functors are.''
\medskip

The purpose of this subsection is to establish that
Birkhoff's Theorem holds for meet continuous lattices:
i.e., that a class of meet continuous lattices closed under
meet continuous homomorphic images, sublattices
and products is axiomatizable by meet continuous lattice identities.
(This is used later.)
The proof requires that the forgetful functor have a left
adjoint. We establish this and derive the needed conclusion.
(Note that the assumption of 
meet continuity is essential, and the statement is false
under the weaker assumption
of completeness.)

This fact about meet continuous lattices 
is a tangential but important point. 
We feel that we should
say enough to explain ourselves to those who might
be concerned about the point, but don't feel that it 
justifies a review of category theory.

\medskip

Section 2.5: Some details of Theorem 2.19 could be left to the reader.

\medskip


True, and this is what is usually done when proving the
basic properties of the commutator. But there is value in having the 
proof written out completely in at least one reference.
(Part of the reason for publishing is to complete the historical
record.) We left this as it was.

\medskip
{\bf Chapter 4.}

\medskip

Section 4.4: A case was overlooked in Lemma 4.21.
It has been fixed.

\medskip
{\bf Chapter 6.}

\medskip

Section 6.3: It is suggested that we add 
a lemma before Theorem 6.36 that establishes 
that there is a prime congruence above every congruence
below 1. This statement is not exactly true.

We are silent on whether the congruence 1 is prime,
but (in analogy with integer arithmetic or ring theory)
the convention should be that $1$ is not prime (nor semiprime).
With this assumption, an $\infty$-solvable algebra has no prime
congruences at all.

What is true is that there is a prime congruence
above any congruence
that is not $\infty$-solvably related to $1$.
(This is proved in Theorem 6.35.)
However (as the referee points out in the markup)
the proof is correctly written provided you interpret
the empty intersection of congruences to be the top
congruence. We added words to the proof to ensure that
readers do interpret it this way.

\medskip
{\bf Front/end matter.}

\medskip

page 194: Fixed the spelling of Gr\"atzer \& Schmidt in the bibliography.

page 199: Add ``quasi-variety'' and ``quasi-order'' to the index?
Currently we index notation and terminology that is defined in the text 
and identifiers for people or theorems. We have now added the definition of 
``quasi-variety'' (and indexed it), but not  ``quasi-order''.

\bigskip
\bigskip
\vfill
\pagebreak

{\bf Detailed corrections, addressing remarks in
{\tt kearnes-kiss-markup.pdf}}

{\bf Chapter 1.}

\medskip

{\bf Referee's remarks.}

\medskip

page iii: addition to keywords. Done!

\smallskip

page iii: question about abstract. The question is relevant to the spirit
of the monograph, but the abstract does not seem to be the place to discuss
this. We added a problem at the end related to one of these remarks.

\smallskip

page 2: uniformize hyphenation. Done! (All quasi words are hyphenated now.)

\smallskip

page 4: replace ``a'' with  ``the''. Done! 

\smallskip

page 9: provide a reference to McKenzie's development
of nonmodular commutator theory based on the 
term condition. 
Could not find a reference, so reworded the sentence to avoid
any erroneous attribution.

\medskip

{\bf Additional changes.}

\medskip

page 6, footnote 7: corrected spelling of ``occurrences''

\smallskip

page 7, first line of second paragraph: altered wording to remind
the reader that algebras in a congruence permutable variety have a 
Maltsev term.

\smallskip

page 8, line -4: altered the wording to emphasize that Con(A) may be
considered to be an interval of itself.

\smallskip

page 10, 3rd paragraph: clarified the meaning of ``we''.

\smallskip

page 13, last paragraph: altered wording to indicate that
the open problems posed have been collected at the end.

\bigskip

{\bf Chapter 2.}
\medskip

{\bf Referee's remarks.}
\medskip

page 14, line -8: suggestion to make universes nonempty. OK.

\smallskip

page 20, Thm 2.1: remove extra ``a''. Done!

\smallskip

page 21, line 16: remove extra ``by''. Done!

\smallskip

page 21, line -10: was the ``I(L)'' notation introduced? Yes, on page 17.
Added this to the index to make it easier to find.

\smallskip

page 22, line 13: remove extra ``of''. Done!

\smallskip

page 24, line 10: add s to ``variable''. Done.

\smallskip

page 25, line 11: change citation 2.15 to 2.14. Done!

\smallskip

page 25, line -4: A lattice filter in a finite lattice is the same as
a principal filter. Right, this could be said either way.
We left it as lattice filter.

\smallskip

page 28, Thm 2.19: some parts could be left to the reader.
True, but we wanted this proof to be written out completely.

\smallskip

page 30, line -9: insert ``of''. Done!

\smallskip

page 32, line -9: change $t_i$ to $w_i$. Done!

\smallskip

page 32, line -8: insert ``of''. Done!

\smallskip

page 33, line -7: $\mathbf L$ need not satisfy quasi-identity
(2.6) for this statement to hold. True! Modified the wording 
to make this clear.

\smallskip

page 34, line -13: must $w_i$ be below $t$? Yes. Added this comment.

\smallskip

page 34, line -3: perhaps mention the origin of these lattice words.
We don't know the origin of these words.
They appear in Hobby-McKenzie 1988 without attribution, 
in Czedli 1981 without attribution, and in Jonsson-Rival 1979
without attribution.
%\color{red} Emil?? Do you want to do anything with this,
%or just leave things as they are?\color{black}

\medskip

{\bf Additional changes.}
\medskip

page 19, 3rd line after definition 2.4:  changed ``iff'' to ``if and only if''.

\smallskip

page 31, line 13:  changed ``identity'' to ``quasi-identity''.

\bigskip

{\bf Chapter 3.}
\medskip

{\bf Referee's remarks.}
\medskip

page 37, lines -8 and -7: change $\m a$ in superscripts 
to $\m a({\vr M})$. Done!

\smallskip

page 38, line -1: change ``rectangulation'' to ``rectangularity''. 
We eliminated all use of ``rectangularity''.

\smallskip

page 43, line 14: change RS to SR. Done!

\smallskip

page 44, line 8: correct the argument that $\tau_n\subseteq \tau_{n+1}$. Done!

\smallskip

page 49, line 1: change ``the'' to ``that''. Done!

\smallskip

page 50--51:  conflicting uses of $n$; make one of the $m$. Done!

\smallskip

page 53, lines -11 and -10: remove extra )'s (twice). Done!

\smallskip

page 55, Thm 3.20: correct the variable patterns and one subscript. Done!

\smallskip

page 57, line -6: Explain that $N = \{1,\ldots,n\}$. Done!

\smallskip

Page 60: this figure should not be alone on the page. Done!

\medskip

{\bf Additional changes.}
\medskip

Changed ``rectangularity'' to ``rectangulation'' throughout.
(Nine instances in Chapter 3 and one instance in Chapter 5.)

\smallskip

page 37, lines -8 and -7: corrected the font for M
(3 times).

\smallskip

page 51, paragraph beginning ``Item (3)'':  changed ``iff'' to 
``if and only if''.

\smallskip

page 59, line -1: changed ``an abelian'' to ``a nontrivial abelian''.

\bigskip

{\bf Chapter 4.}
\medskip

{\bf Referee's remarks.}
\medskip

page 61, Problem 4.2: Change ``If'' to ``Is''. Done.

\smallskip

page 63, second line of Section 4.1: add ``$\m l$''. Done!

\smallskip

page 63, third paragraph of Section 4.1: Change ``second'' to ``third''. Done!

\smallskip

page 64, first paragraph of proof of Lm 4.5: change ``continous'' to 
``continuous''. Done!

\smallskip

page 70, middle display: change $f_k$ to $f_j$. Done!

\smallskip

page 72, 2nd paragraph: has the notation $\beta\circ_k\gamma$ been defined?
Yes, in a footnote on page 6. Added a footnote here reminding the reader.

\smallskip

page 74, first line of proof of Lm 4.11: Change comma to period. Done!

\smallskip

page 75, 3rd paragraph of proof of Thm 4.12: Tell the reader where $D_1$
was introduced. Done!

\smallskip

page 83, first paragraph: Labelling the lattice is suggested. Done!

\smallskip

page 83, equation (4.6): Changed $=$ to $\approx$. Done! 

\smallskip

pages 84-86: Fix Lemma 4.21.
Modified the discussion about graph lattices to 
address the remarks on these pages.

\medskip

{\bf Additional changes.}
\medskip

page 61, line 2: ``is any nontrivial way'' to ``in any nontrivial way''.

\smallskip

page 62, first 2 sentences: modified the punctuation.

\smallskip

page 64, line 1: corrected spelling of ``occurring''.

\bigskip

{\bf Chapter 5.}
\medskip

{\bf Referee's remarks.}
\medskip

Definition 5.1: How does this differ from McKenzie's definition?
The paper containing McKenzie's definition is cited in the introduction
on page 10. His definition uses tame congruence theory, so we don't
want to include it.

\smallskip

page 89, line 6: changed ``as indicated in line (9)'' to 
``as indicated by (5.3)''.

\smallskip

page 95, 3rd paragraph of proof of Claim 5.12: remove ``it''. Done!

\smallskip

page 99, line 2: change ``corollary'' to ``theorem''. Done!

\smallskip

page 104, big matrix: remove extra )'s. Done!

\smallskip

page 106, 3rd paragraph of proof: change ``SInce'' to ``Since''. Done!

\smallskip

page 106, later, same paragraph: change ``$r_r$'' to ``$r_i$''. Done!

\smallskip

page 106, later, same paragraph: change ``impliy'' to ``imply''. Done!

\smallskip

page 106, later, next (=4rth) paragraph: change ``$a\sim_B$'' to 
``$a\sim_B b$''. Done!

\smallskip

page 106, later, next (=4rth) paragraph: add 
``Item~$(2)$ is a special case of $(5)$'' to the end of the proof. Done!

\smallskip

page 108, second paragraph of proof: remove extra ``the''. Done!

\smallskip

page 112, 3rd and 4rth paragraph of proof: the notation for the arguments
in these two paragraphs is not in harmony. Fixed.

\smallskip

page 113, line 5: adjust the spacing. \color{red} Emil??\color{black}

\smallskip

page 114, line 8: $E_1$ and $E_2$ are not defined correctly. Fixed!

\smallskip

page 115, line 12: $p*q$ 
is not a member of G, but is a $\gamma$-class of a member.
Fixed! 
%\color{red} Emil?? Please check this proof also.\color{black}

\medskip

{\bf Additional changes.}
\medskip

page 106, line -3: changed ``homomomorphic'' to ``homomorphic''.

\smallskip

page 113, line -2: displayed this line to avoid a bad break.

\bigskip

page 114, line 16: changed ``$\m g$'' to ``$\m g\times \m g$''.

\bigskip

page 115, line -8: changed $\overline{p}$''
to ``$\overline{\wec{p}}$'' (same for $\overline{q}$
and $\overline{r}$).


\bigskip

{\bf Chapter 6.}
\medskip

{\bf Referee's remarks.}
\medskip

page 120, line -4: Explain why ``difference'' is used in ``difference term''.
We do not know a factual explanation that we could supply here.
%\color{red} Emil?? Do we want to do this? I don't know what to write.
%\color{black}

\smallskip

page 130, line -13: remove extra ``the''. Done!

\smallskip

page 131, line -2: remove ``a''. Done!

\smallskip

page 132, line 22: change join to meet. Done!

\smallskip

page 136, line 2: add ``a''. Done!

\smallskip

page 138, line 2: remove extra ``the''. Done.

\smallskip

page 138, line 12: change $<$ to $\leq$. Done!

\smallskip

page 138, line -7: change $y$ to $z$. Done!

\smallskip

page 139, line 2: change ``prove'' to ``proves''. Done!

\smallskip

page 139, lines 4 and 5: change ``interval'' to ``intervals''. Done!

\smallskip

page 139, line -1: change ``theorem'' to ``lemma''. Done!

\smallskip

page 141, last paragraph: no need to refer to earlier claims
to establish that $\alpha$ and $\widehat{\beta}$ 
are above $\delta$. Fixed!

\smallskip

page 142, line 5: add space to separate ``is'' from ``a''. Done!

\smallskip

page 142, line 6: no need to separate this statement out as a claim. Changed!

\smallskip

page 142, line -4: need to 
adjust wording in 2 places. Done (in the way suggested)!

\smallskip

page 143, line -7: this intersection could be empty. Right! That doesn't
affect the argument. Added a comment pointing this out.

\smallskip

page 144, line 14: change ``$\pi$ avoids $\sigma$'' to 
``$\sigma$ avoids $\pi$''. Done!

\medskip

{\bf Additional changes.}
\medskip

page 122, line 4: mentioned that Lipparini's result can be derived from
Corollary~6.3.

\smallskip

page 122, line 7: changes ``succesor'' to ``successor''.

\smallskip

page 135, paragraph above Problem 6.21: changed
``speaks only about''
to 
``refers only to''.

\smallskip

page 143: added the hypothesis that $\theta$ is proper to 
the statement of Theorem 6.36.

\smallskip

page 144: added a remark before Cor 6.37 saying that the corollary
shows how to define the earlier concepts of the chapter in terms
of prime congruences.

\bigskip

{\bf Chapter 7.}
\medskip

{\bf Referee's remarks.}
\medskip

page 146, line -1: remind readers of ``closed''. Done!

\smallskip

page 149, middle of page: simplify subscript $I[\mu,\nu]_S$ to $I_S$. Done!

\smallskip

page 151, lines 1 and 2: simplify subscript (3 times). Done!

\smallskip

page 151, line 6: refer to figure by label. Done!

\smallskip

page 152, lines 11 and 13: could simplify subscripts from 
U/(V-U)/(N-V) to P/Q/R. Decided not to do this, because 
P/Q/R = U/(V-U)/(N-V) in this paragraph but 
P/Q/R means something different in the next paragraph.

\smallskip

page 156, paragraph before Lemma 7.11: is this the first
use of nonsolvability? It is the first use in this chapter.
Prior to the statement of the lemma we indicate that
the $\blacktriangleleft$ relation was studied
in Chapter 6, and this symbol is indexed, so the reader
should have no trouble refreshing his memory about this relation.
%I don't see what is achieved by pointing this out, though. 
%\color{red} Emil??\color{black}

\smallskip

page 156, Lemma 7.11(3): is $\nu$ arbitrary? Yes -- made this clear.

\smallskip

page 156, line -11: adjust the spacing. \color{red} Emil??\color{black}

\smallskip

page 157, Lemma 7.12: fix primes on (d)' (4 times). Done!

\smallskip

page 158, 2nd line of subsection 7.2: add ``a''. Done!

\bigskip

{\bf Chapter 8.}
\medskip

{\bf Referee's remarks.}
\medskip

page 162, first paragraph: Change ``be'' to ``by''. Done!

\smallskip

page 166: a hypothesis of Theorem 8.2 is not stated. Fixed!

\smallskip

page 168, last displayed line: order is wrong. Fixed!

\smallskip

page 169, 3 lines after Fig 8.4: change superscript from $2n-5$ to $2n-4$.
Done!

\smallskip

page 175, line 6: change ``an'' to ``a''.

\smallskip

page 175, line 7: change ${\vr V}$ to ${\rm{CON}}({\vr V})$. Done!

\smallskip

page 180: Compare Thm 8.2 to Thm 9.11 of Hobby-McKenzie?
Added this to the introduction of the chapter.

\smallskip

page 184, Cor 8.15: Add ``idempotent''? Done!

\medskip

{\bf Additional changes.}
\medskip

Added references to some omitting types theorems of Hobby-McKenzie
in the introduction of the chapter.

\bigskip

{\bf Chapter 9.}
\medskip

{\bf Referee's remarks.}
\medskip

Lemma 9.1: Correct spelling of ``isomorphic''. Done! 

\smallskip

Lemma 9.2: (Are there similarities between this and anything in 
Hobby-McKenzie?) This is hard to say. Lemma 10.2 of Hobby-McKenzie
is the closest result. Our hypothesis 9.2(1) holds in 
Lemma 10.2 of Hobby-McKenzie, but our hypothesis 9.2(2) need not hold
there. The construction of large SI's in Hobby-McKenzie 
uses a different subalgebra of a power. It also
depends on affine structure on a minimal, which we have no access to.
We don't know what comparison we can make that would be accurate and useful.

\smallskip

Mention that Theorem 9.5 generalizes Theorem 10.4 of Hobby-McKenzie: Done!

\smallskip

Change of $(*)$' to $(*)^{\prime}$ throughout chapter: Done!

\medskip

{\bf Additional changes.}
\medskip

First sentence of chapter: ``A variety is {\bf residually large} 
if it has a proper class of subdirectly irreducible
members up to isomorphism,'' (added ``up to isomorphism'').

\bigskip

{\bf Problems.}

\medskip

Added a page collecting the problems in the text and 
adding a few more.

\bigskip

{\bf Bibliography.}
\medskip

Updated entries [51] and [60].

\end{document}
