On page 80 (next-to-last paragraph) I proclaimed, without proof, that the lattice \(\mathbf L_{\operatorname{Clo}^A}\) is both algebraic and coalgebraic. While it is true that \(\mathbf L_{\operatorname{Clo}^A}\) is algebraic for any set \(A\) (Corollary 4.4), it is coalgebraic only for finite \(A\). Ralph McKenzie kindly sketched a counterexample for me. Here is the construction if you are interested.
page 134, Exercise 4.80.4. This exercise is a total loss. I begin by making the
false assertion that we proved in 3.28 that every member of \(Cr_n\) has
an identity. We did no such thing. While it is true that every
finite ring in this variety has an identity (this follows from
Theorem 6.3 for instance), there are infinite members with no
identity. Here is a construction, which is a useful addition to your bag
of tricks.
Let \(\mathbf F\) be a finite field. Set \(A=\left\{\,x \in
F^{\omega} : (\exists n)\,(\forall m>n)\;x_m=0\,\right\}\). It is
easy to see that \(\mathbf A\) is a subring of \({\mathbf F}^\omega\). But
\(\mathbf A\) has no identity. To show this, let \(e\) be any element
of \(A\). Fix \(n\) so that for all \(m>n\), \(e_m=0\). Choose an
element \(a\in A\) such that \(a_{n+1}\neq 0\). Then \((e\cdot a)_{n+1} = 0\),
so surely \(e\cdot a \neq a\). Thus \(e\) is not an identity of \(\mathbf A\).
On page 287, I managed to make three errors in the second paragraph of the proof of Theorem 8.54. First, the sentence \(\gamma(f)\) is not correct as written. One must distribute the universal quantifiers over the disjunction. Second, the set \(\Sigma \cup \{\gamma(f) : f \in F\}\) does have a model, namely a trivial algebra, so we need to add an additional sentence asserting nontriviality. Finally, the argument involves a logical fallacy that I wouldn't accept from my undergraduate students! It should go as follows: If the set of sentences has a model, then the variety \(\mathcal W\) contains a nontrivial set, and hence is passive. Contradiction. Then we can proceed to apply compactness. Here is a rewritten passage you can clip and save.
Page 289. Just to be clear, the last paragraph on the page applies only to locally finite varieties. Furthermore, it is not known whether an arbitrary variety possessing a Taylor term will have a term \(t\) satisfying the two equations at the bottom of the page.
p(l) refers to page p, line l (or l lines from the bottom, if negative).