# Serious Errors

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.

# Minor Errors

p(l) refers to page p, line l (or l lines from the bottom, if negative).

25(-11)
“$$c=y$$” should be “$$c=z$$”.
27(3)
The identity in the book is incorrect. ($$\mathbf L$$ does not satisfy that identity.) The correct identity is: $$y\wedge(x\vee (y\wedge z)) \approx y\wedge (x \vee (z \wedge (x\vee y)))$$.
35(13)
"$$\bigvee \mathcal X \subseteq Y$$" should be "$$\bigvee \mathcal X \subseteq C(Y)$$"
61(-2)
The lattice of subgroups of $$\mathbb{Z}_{p^k}$$ is a chain of length $$k$$, not $$k+1$$.
76(15ff)
Numerous appearances of $$\mathbf B_n$$ should be $$\mathbb B_n$$.
81(-1) and 82(5)
on both lines $$\operatorname{Clo}_m A$$ should be $$\operatorname{Clo}_m F$$.
85(19)
$$\mathbf{Clo}_1(\mathbf A)$$ contains 6 of the 27 elements of $$A^{A^1}$$.
90(21)
$$\hat{\mathbf a}_j = (a_{ij},\ldots,a_{nj})$$
101(18)
“$$\mathbf F_{\mathbf A}(X_n)$$ “ should be “$$\mathbf F_{\mathcal A}(X_n)$$”.
109(-1)
“$$9x\approx 0$$” should be “$$3x \approx 0$$”
110(1)
“$$x^n \approx x$$” should be “$$x^n \approx e$$”.
145(-12)
“$$\mathbf a^2$$” should be “$$\mathbf b^1$$”
222(-15ff)
$$\mathbf F$$ is actually a homomorphic image of $$x\mathbb Z[x]$$. Also, instead of $$g_1g_2 = x$$ it should be $$g_1h_1+g_2h_2+\cdots g_kh_k = x$$. But it all still works out.
254(5)
“Lemma 8.13” should be “Theorem 8.13”.
280(24)
“possible” should be “practical.”
281(-4)
“binary” should be “unary.”