ERRATA: MODES, ROMANOWSKA & SMITH

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1 ERRATA: MODES, ROMANOWSKA & SMITH Line : There is a function f : X Ω P(X) with xf = {x} for x in X and ωf = for ω in Ω. By the universality property (1.4.1) for (X Ω), the mapping f can be uniquely Line 39 10: ω ωτ Line : Lemma Line : is satisfied in all rectangular bands Line 53 10: holds in all normal bands Line 63 7: Hence B = p + W. Line 64 1: operations P and Line : then c else a Line 97 5: j j n (not j r ) Line 97 4: Delete first sentence. Line 97 3: k k n Line : 0 x < 1 Line 106 7: A s i I A i i I ker π i = Â and each π i surjects. Line 106 4: i I θ i = Â and i I, A i = A θ i Line : i I θ i = Â and i I, A i = A θ i Line : iff PK = u = v. Line : (c) ((u, v) Σ and (v, w) Σ) ((v, w) Σ); Line : Define J : B P Line : and (v, w) Σ) Line : φ ki,k Line : A φ l A l Line : h k : XΩ φ A k Line 166 1: a / U. Line : Consequently A/U = Line : DPK Line 171 4: & m i Line : {X P < (M) Line : P < (M) Line 174 5: Exercise 3.7R

2 2 ERRATA: MODES Line 175 5: h i : i I Lines to +4: Replace Example as follows: Example In the free semigroup X + (cf. Section 1.4) over X = {x, y}, one has x 2 x 2 y and y 2 xy 3, but it is not true that x 2 y 2 x 2 yxy 3. It follows that this algebra is not naturally quasiordered. Line : fibre (B i α, Ω) Line 218 7: a subdirect product of (I, Ω) and the envelopes (E i, Ω) for i I, Line 218 6: s i I (E i, Ω) (I, Ω) Line 218 3: It follows that µ ker π = Â. Line 218 2: all the (A, Ω) µ i and (I, Ω). Line : of the algebras (I, Ω) and (C i, Ω) for i I, Line : (A, Ω) s (C i, Ω) (I, Ω). Line : Lemma Line : It follows that µ ker π = Â. Line 219 9: the (A, Ω) µ j and ker π. Line 219 8: Lemma Line : all the (Ci, Ω) and (I, Ω). Line : the (Ci, Ω) or (I, Ω). Line , 10: Corollary Line : Theorem Lines to 8: Replace text as follows: and Let i I P E j := (E i i j) P j := (A i i j). Obviously Pj E is a subalgebra of (B, Ω) and a functorial sum of the E i. The union P j is a subalgebra of (Pj E, Ω). Let µ(j) E be the relation on Pj E defined by (a k ψ k,r,b l ψ l,s ) µ(j) E : a k ψ k,j = a k ψ k,r ψ r,j = b l ψ l,s ψ s,j = b l ψ l,j

3 ERRATA: MODES 3 for k r, l s and r, s j. It is easy to see that it coincides with the relation δ of Proposition Hence it is a congruence relation of (Pj E, Ω), and (Pj E, Ω) µ(j)e = Ej. Now let µ(j) be the relation µ(j) E restricted to P j, i.e. (a k, b l ) µ(j) : a k ψ k,j = b l ψ l,j. Then by the Second Isomorphism Theorem 1.2.4, µ(j) is a congruence of P j. It obviously preserves (A j, Ω), and (P j, Ω) µ(j) = Ej [Exercise 4.5I]. Note also that (a k ψ k,r, b l ψ l,s ) µ(j) E if and only if (a k, b l ) µ(j). We will show that (E j, Ω) is an envelope of (A j, Ω). Let λ µ(j) be a congruence on (P j, Ω) preserving (A j, Ω). For k, l j, let (a k, b l ) λ. Since (a k ψ k,j, a k ) µ(j) E and (b l ψ l,j, b l ) µ(j) E, it follows that (a k ψ k,j, b l ψ l,j ) µ(j) E. Hence (a k, b l ) µ(j). This implies that λ µ(j), and consequently that λ = µ(j) is a maximal congruence preserving (A j, Ω). Line : concrete strongly irregular varieties Line 264 3: Line : for x = a 1, Line : R : (A, ) (End(A, ), +); Line 290 5: ( = xp q pq + y p + q pq + z q ) (p + q pq) p + q pq Line 290 2: pq (p q ) + q = 1 (p q ) Line : as a congruence of (XC, I ), Line : subalgebra of (XC XC, I ) Line 300 5: = xy(w 2 w 1 ). Line 303 7: φ : S X X X Line : (d) Show that if there is a faithful action of a semigroup (S, ) on S satisfying x yzs t = xyt xzt s xys uvs t = xut yvt s, then (S, ) is commutative. Line 368 6: (Cf. Exercise 6.4O.) and

4 4 ERRATA: MODES Line : quasi-identity (6.2.8) Line 377 6: violates (6.2.8) Line 377 3: condition (6.2.8) Line 385 3: characterize certain Lallement Line : let I be the quasivariety of Ω-semilattices. Line : (E j, Ω) Line : i j Line 386 1: a 1... a ωσ 1 c i a ωσ+1... a n ω Line : holds for each m-ary Lines to 9: Replace text as follows: We conclude this section with a version of Theorem concerning the case where I is any irregular variety of modes. By P lonka s Theorem 4.3.2, the variety I is defined by a set of regular identities and an identity x y = x. Let (A, Ω) be a Lallement sum as in the introduction to Lemma Now I-algebras all have a full algebraic quasi-order. Thus for each j I, the sets P j coincide with A and E j = A µ(j) for a maximal congruence µ j of (A, Ω) preserving (A j, Ω). Define a new relation µ j on A as follows: (b, c) µ j : a A j, a b = a c. Then a proof similar to that of Lemma shows that the relation µ j is the largest congruence on (A, Ω) preserving (A j, Ω), and the envelope (E j, Ω) is cancellative. This gives part of the proof of the version of Theorem for the case where I is an irregular variety of modes. The last part of the proof of that version goes like the proof of Theorem for the case where I is the quasivariety of Ω-semilattices. We do not know if Theorem holds also in the general case where I is any quasivariety of naturally quasi-ordered Ω-modes. Note that in this situation, we could not use the congruence µ as defined before Lemma (or above). Indeed, the elements and a 1... a ωσ 1 ba ωσ+1... a n ω a 1... a ωσ 1 ca ωσ+1... a n ω in the second paragraph of the proof of Lemma would not necessarily both be in A j. A similar situation arises in the last paragraph of the proof of Lemma 7.4.1, and in the last paragraph of the proof of Theorem Line 390 8: Replace with:

5 ERRATA: MODES 5 Corollary Let I be an irregular variety of modes. Line : zero bands form an irregular variety, Corollary Lines to + 13: Replace Example as follows: Example Consider the differential groupoid A given in the example following Proposition It is a Lallement sum of two subgroupoids A 0 and A 1 with the envelopes E 0 = {0 µ(0), 1 µ(0), 0 µ(0) } and E 1 = {0 µ(1) = 1 µ(1), 0 µ(1) } over the two element left-zero band {0, 1} by the sum homomorphisms 0 φ 1,0 = 0 µ(0) and 0φ 0,1 = 1φ 0,1 = 0 µ(1) and 0 φ 0,1 = 0 µ(1). Note that none of A 0, E 0 and E 1 is cancellative. Note also that and 0φ 0,1 = 1φ 0,1, but 0φ 0,0 = 0 1 = 1φ 0,0. It follows that it is not possible to extend the sum homomorphisms to functorial homomorphisms between E i, or to embed A into a functorial sum of these envelopes. Line 385 6: affine spaces over D Line : for j < k and Line 415 2: ω is an m-ary operation Line : exists Lines to + 11: over an Ω-semilattice or an algebra in an irregular variety embeds as a subreduct into a functorial sum of affine spaces. We provide two theorems. One concerns sums over Ω-semilattices, while Line 428 3: Theorem Line : = cψ j,k... yψ j,k... eψ j,k ω Lines to 10: and a variety Q of Ω-semilattices or of irregular Ω-modes. Line 433 8: By Theorem and the remarks that follow it, the mode Line : L(p x, λ, γ) = Lines , + 10: over its semilattice replica. Examples Line 552 8: Note that the submodules of the module (E, +, R) also form Line : in AS, for Line : A r Line : variety of V -modals defined by a linear set of identities. Line : F = A(, T ) and G = X (, T )

6 6 ERRATA: MODES Line : A A(A, T ) fα f F ff B A(B, T ) α X X (X, T ) gα g G gg Y X (Y, T ) α Line : tion of the variety of distributive lattices, using Line 586 9: [1969] Symmetric Spaces I: General Theory, Benjamin, New York, New York. Line : Line 615 6: differential groupoid 269