Hi,

I decided to referee this myself. Great paper and very nicely written. I
have only a few minor corrections and some observations.

Minor corrections:

page 5, top line: you might replace "so" with "and"

page 10, au line # 30: change "then so if" to "then so is"

ref [14]: the ~ isn't showing. I use $\sim$ to get this.

the email addresses at the bottom are off (this may be an AU error).


Observations: (These are not required)

In my paper with Day on Polin's variety we characterize which congruence
identities imply CM and in my paper with Czedli we show this is effective.
This is only tangentially related to your paper.

I have a slight variant of the Maltsev condition for SD-join given in HM
and KK that is more symmetric. In Theorem 8.14 of Shape replace 3(ii) and
3(iii) with

(a) d_i(xyy) = d_{i+1}(xyy),        i \equiv 0 or 1 mod 3
(b) d_i(xyx) = d_{i+1}(xyx),        i \equiv 0 or 2 mod 3
(a) d_i(xxy) = d_{i+1}(xxy),        i \equiv 1 or 2 mod 3

If \Sigma is this condition, then \Sigma^(k) implies d_k(xyz) = x, and so
by your result SD-join implies a congruence identity.

Again none of these observations is required to go into your paper.

I will put this through the AU manager so you will get instructions from
them.

  Ralph


page 2: another possible example is \Sigma = the identities of an nu term,
in which case \Sigma' is inconsistent, while \Sigma = the identities of a
weak nu term gives \Sigma' = \Sigma.

More generally if \Sigma has only a single operation symbol F(x_1, ...,x_n)
and its equations are linear, you have the equivalence relation on B(F),
the subsets of S = {1,...,n} as defined in the Shape monograph. Let S^(0)
= S and S^(k+1) be the intersection of all U \subseteq S such that U is
equivalent to some V containing S^(k). I think S' = empty set is
equivalent to \Sigma implying modularity and S(k) empty to a nontrivial
congruence identity.

Dear Keith and Agnes,

     I chased down the reference.  Here is the paper:


     George F. McNulty, ``Undecidable properties of finite sets of
     equations", Journal of Symbolic Logic, 41 (1976) 589--604.

The theorem to look at is

   Theorem 2.5

     Let   H  be a collection of finite sets of equations in a
     strong signature.
     If
      (i)    H is not empty,

      (ii)   H is closed under logical equivalence,
            (if  \Delta\in H and \Gamma is a finite set of
             equations of the same signature so that  \Delta
             and \Gamma have the same models, then \Gamma\in H) and

      (iii)  for each \Gamma\in H there is a term  t  in which
             both  x  and  y  occur such that  t=x is a logical
             consequence of  \Gamma,

    then  H  is not a recursive set.


A strong signature is one that has some operation symbol of
rank at least two.

I am pretty sure something like this is in my dissertation as
well (that would be 1972).  I am in my office right now and
can't find my copy---so it must be at home...

I hope this is what you were after.

Best Regards

G