Interpolation and Beth’s property in propositional many-valued logics: A semantic investigation

Annals of Pure and Applied Logic 141 (2006) 148–179 www.elsevier.com/locate/apal

Interpolation and Beth’s property in propositional many-valued logics: A semantic investigation✩ Franco Montagna Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini 44, 53100 Siena, Italy Received 20 January 2005; received in revised form 7 November 2005; accepted 7 November 2005 Available online 7 December 2005 Communicated by S.N. Artemov

Abstract In this paper we give a rather detailed algebraic investigation of interpolation and Beth’s property in propositional manyvalued logics extending H´ajek’s Basic Logic BL [P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer, 1998], and we connect such properties with amalgamation and strong amalgamation in the corresponding varieties of algebras. It turns out that, while the most interesting extensions of BL in the language of BL have deductive interpolation, very few of them have Beth’s property or Craig interpolation. Thus in the last part of the paper we look for conservative extensions of BL having such properties. c 2005 Elsevier B.V. All rights reserved.
Keywords: Many-valued logic; BL-algebras; Interpolation

1. Introduction Our interest in interpolation in many-valued logic was attracted by some discussions with specialists of manyvalued logic, with either a logical or an algebraic background like Matthias Baaz, Dov Gabbay, Andrew Glass, Daniele Mundici and Hiroakira Ono. From these discussions, two seemingly incompatible results emerged. The first one (cf. [4]) says that Łukasiewicz logic has no interpolation, whilst the second one (a combination of [29, 30], where amalgamation is shown for MV-algebras, and [31], where it is shown that in many cases amalgamation implies interpolation) says that not only Łukasiewicz logic has interpolation, but it satisfies also a strong version of Robinson’s joint consistency property. Needless to say, both results are correct, but they refer to different kinds of interpolation. Roughly speaking, a propositional logic L has interpolation if whenever B can be derived from A in L, the information from A which is relevant to derive B only contains propositional variables common to A and B. This property, however, can be formulated in several inequivalent ways. Here we mention three of them: (a) Interpolation with respect to the consequence relation (deductive interpolation): if A  L B, then there is a formula C whose propositional variables are common to A and B such that A  L C and C  L B, where  L denotes the consequence relation in L. ✩ To the beloved memory of my father Gianni Montagna.

E-mail address: [email protected]. c 2005 Elsevier B.V. All rights reserved. 0168-0072/$ - see front matter
doi:10.1016/j.apal.2005.11.001

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(b) Interpolation with respect to implication (Craig’s interpolation for short): if L  A → B, then there is a formula C whose propositional variables are common to A and B such that L  A → C and L  C → B. (c) Uniform interpolation (with respect to implication): for every formula A( p, q ) (where p and q denote sequences of mutually distinct variables), there are formulas ∃ q A( p, q ) and ∀ q A( p, q ) (in the variables p only, and depending only on A and on p, denoted by and called right interpolant and left interpolant respectively) such that for every formula B( p, r) such that the sequences p, q and r have no common variables, one has: (c1) If L  A( p, q ) → B( p, r), then L  A( p, q) → ∃ q A( p, q ) and L  ∃ q A( p, q ) → B( p, r) (c2) If L  B( p, r) → A( p, q ), then L  B( p, r) → ∀ q A( p, q ) and L  ∀ q A( p, q ) → A( p, q ). We will see that these formulations are not equivalent in many-valued logic. A property related to interpolation is Beth’s definability property. Also this property can be formulated in (at least) two ways. We start from the deductive Beth property. We say that a formula A( a , p) ( p a propositional variable, a a string of propositional variables) implicitly defines p in L iff A( a , p), A( a , q)  p ↔ q. We say that A( a, p) explicitly defines p in L if there is a formula D( a ) in the variables a only such that A( a , p)  p ↔ D( a ). We say that A( a , p)
weakly explicitly defines p in L if there are formulas D1 ( a ), . . . , Dn ( a) in the variables a only such that A( a , p)  ni=1 ( p ↔ Di ( a )). We say that a logic L has the deductive Beth property if whenever A( a , p) defines implicitly p in L, then A( a , p) explicitly defines p in L. We say that L has the weak Beth property if whenever A( a , p) defines p implicitly in L, it weakly explicitly defines p in L. The relationship between the deductive Beth property and the weak Beth property is similar to the relation between the existential property in intuitionistic logic ( ∃x A implies  A(x/t) for some term t) and Herbrand’s theorem for classical logic. As said before, Beth’s property has an alternative formulation in terms of implication. More precisely, we say that L has the implicative Beth property iff whenever L  (A( a , p)& A( a , q)) → ( p ↔ q), then there is a formula D( a) such that L  A( a , p) → ( p ↔ D( a )). For the logics we are interested in, the implicative Beth property follows easily from Craig interpolation: if L  (A( a , p)& A( a , q)) → ( p ↔ q), then L  (A( a , p)& p) → (A( a , q)) → q). If D( a ) is an interpolant of A( a , p)& p and A( a , q) → q, then it is readily seen that L  A( a , p) → ( p ↔ D( a )). To the contrary, we will see that in many-valued logic deductive interpolation does not imply the deductive Beth property. The properties mentioned above are strictly related to various kinds of amalgamation properties. We will investigate two of them (amalgamation and strong amalgamation) in Sections 3 and 4. In [25], it is shown that for superintuitionistic logics both deductive and implicative interpolation are equivalent to both amalgamation and strong amalgamation. Moreover, Beth’s property is weaker than interpolation in these logics. However, superintuitionistic logics have the deduction theorem and many-valued logics do not in general, therefore the latter logics behave quite differently. The situation of modal logics, investigated in full detail in [27], is closer to that of many-valued logics, because several modal logics do not have the deduction theorem. In [27], cf. also [17], deep relations are shown between interpolation, Beth’s property and amalgamation properties in modal logics and in their corresponding varieties. For instance, deductive interpolation is equivalent to amalgamation, implicative interpolation is equivalent to superamalgamation, and strong amalgamation is equivalent to amalgamation plus Beth’s property. While the first equivalence (deductive interpolation-amalgamation) extends to many-valued logic, at the moment we do not know the situation of the other equivalences. We plan to investigate it in the future. We also mention the work by Ono [31], where it is shown that for algebraizable logics the amalgamation property and the congruence extension property imply the deductive interpolation for the corresponding logic. Finally, we quote [18], where it is shown that in varieties of commutative residuated lattices (hence in all varieties we are interested in), amalgamation is equivalent to deductive interpolation for the corresponding logic. Moreover for non-commutative substructural logics, implicative interpolation does not imply deductive interpolation. In this paper we offer a rather systematic algebraic investigation of interpolation and Beth’s property in manyvalued logics extending H´ajek’s basic logic BL, possibly with additional operators. The logic BL was introduced by H´ajek in [21]. This logic can be presented both as a common fragment of the three most important many-valued logics, Łukasiewicz logic, G¨odel logic, and product logic, as well as the logic of all continuous t-norms and their residuals, cf. [9]. We recall that a continuous t-norm is a continuous commutative, associative and weakly increasing binary function · on [0, 1] such that x · 1 = x for all x ∈ [0, 1]. Continuous t-norms have been proposed as a very general semantics for fuzzy sets and for many-valued logics. Thus BL really deserves the name of basic many-valued

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logic. Our choice to treat extensions of BL rather than other weaker many-valued logics has another motivation: the extensions of BL are easier to investigate by semantic means rather than by proof theory, and the present author is a specialist of the algebraic semantics of many-valued logic rather than of proof theory. In order to describe the motivations of this paper, we introduce a quotation from [17]: The mathematician will probably be quite happy with an exhaustive classification of systems with and without interpolation and will take a special pride and delight in the various methods and counterexamples used. The computer scientist, however, may take a different view. He may not want a ‘laundry-list’ of variations on meaningless technical results arising from ‘obvious’ lack of expressive power. He may want to know the reasons behind the results. Part of the present paper is devoted to ‘a (unfortunately not completely exhaustive, but still rather detailed) classification of systems with and without deductive interpolation’ and to an exhaustive classification of extensions of BL with and without Beth’s property and with and without Craig interpolation. When doing this, we establish some ‘meaningless technical results’ showing that both Beth’s property and Craig interpolation fail in most extensions of BL. Finally, we try to make a first step towards a better understanding of ‘the reasons behind the results’. That is, we try to understand what kind of expressive power is missing in order to get Craig interpolation or Beth’s property. For the most interesting logics for which either Craig interpolation or Beth’s property fail, we will look for a conservative extension having the properties above. Our plan is only partially successful, for the following reasons: first of all, we have not been able to determine whether all of the expressive power added is really necessary, or a weaker language would be sufficient to get interpolation. Second, the extensions of BL satisfying Craig interpolation are only partially conservative over BL. More precisely, for every n we find an extension BL(n) of BL which has Craig interpolation and which is conservative over BL with respect to formulas in n variables only. Summing up, in our opinion the present paper constitutes a good starting point towards an investigation of interpolation in many-valued logic, but such research is far from concluded. For instance, in the first section of [17], the reader may find a lot of perspectives of research concerning interpolation (e.g. Lyndon interpolation, connections with quantifier elimination, complexity of algorithms for finding interpolants, etc.) which are not discussed in the present paper. The paper is organized as follows: Section 2 contains an exposition of some fundamental properties of the logic BL and of its algebraic semantics, the BL-algebras. Then in Section 3 we begin our algebraic investigation of interpolation. Thanks to the above mentioned results of Ono [31] and of Galatos–Ono [18], deductive interpolation reduces to the amalgamation property for the corresponding variety. Thus we investigate the amalgamation property in the most important subvarieties of the variety BL of BL-algebras, like BL itself, the variety W of Wajsberg algebras, the variety P of product algebras, and the variety G of G¨odel algebras. We show that all these varieties, as well as their extensions by Baaz projection ∆ have the amalgamation property. However, there are uncountably many subvarieties of BL without the amalgamation property: among them, in view of a general result by Larisa Maksimova (cf. [25,26] and [17]), the varieties of n-valued G¨odel algebras with n > 3, and the join of W and any of G or P (or both). To the contrary, the join of G and P has the amalgamation property. In Section 4, we investigate the strong amalgamation property. This property is related to the deductive Beth property for the corresponding logic. We prove that the only non-trivial subvarieties of BL with strong amalgamation are G, the variety of Boolean algebras and the variety of 3-valued G¨odel algebras. However, we show that the class of linearly ordered divisible Wajsberg algebras, as well as the class of linearly ordered product algebras with nth roots, have strong amalgamation. Finally, we introduce a class of BL-algebras with division operators, called divisible BL-algebras, whose linearly ordered members have strong amalgamation. In Section 5, we derive from results by Ono (cf. [31]) and Galatos–Ono [18] that BL, Łukasiewicz logic, product logic, and G¨odel logic, as well as their extensions with ∆, have the deductive interpolation property, whilst several many-valued logics extending BL do not. Then we investigate the deductive Beth property. It follows from [24] (cf. also [17]), that any schematic extension of G in the language of BL has the deductive Beth property. In this paper we prove that no other extensions of BL have the deductive Beth property. In other words, an extension of BL has the deductive Beth property iff it extends G. So, none of Łukasiewicz logic, product logic and BL has the deductive Beth property. However, we prove that divisible Łukasiewicz logic, product logic with n t h roots and divisible BL logic have the weak Beth property. We do not know if they also have the deductive Beth property, but we prove that their extensions with ∆ have. Thus there is a conservative extension of BL which has the deductive Beth property. Finally, in Section 6 we investigate Craig’s interpolation and the implicative Beth property. First of all, we

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improve a result by Baaz and Veith [4] by proving that the only consistent extensions of BL with Craig interpolation are G¨odel logic, classical logic and the 3-valued G¨odel logic. Then, we show that the extensions of BL in the language of BL having the implicative Beth property are precisely the extensions of G¨odel logic. In the light of these negative results, after observing that deductive interpolation in any extension of BL with ∆ gives a modal version of Craig’s interpolation, we look for partially conservative extensions of BL having both the implicative Beth property and the Craig interpolation property. (Conservative extensions of Łukasiewicz logic and of product logic with the uniform interpolation property can be found in [4].) For every n, we produce an extension BL(n) of BL which has uniform interpolation and the implicative Beth property, which is conservative over BL with respect to formulas in n variables, that is: if A is a formula in the language of BL with no more than n propositional variables, then B L(n)  A iff B L  A. 2. Preliminaries The logics considered in the present paper are extensions of H´ajek’s Basic Logic BL (cf. [21]) (or in some cases of its falsum-free fragment), possibly with additional operators. The formulas of BL are built from propositional variables and from the constant ⊥ by means of two binary connectives & and →. BL is axiomatized as follows: (A1) (A2) (A3) (A4) (A5) (A6) (A7) (A8)

(A → B) → ((B → C) → (A → C)) (A&B) → A (A&B) → (B& A) (A&(A → B)) → (B&(B → A)) (A → (B → C)) → ((A&B) → C) ((A&B) → C) → (A → (B → C)) ((A → B) → C) → (((B → A) → C) → C) ⊥ → A.

The only rule of BL is Modus Ponens: A

A→B . B

(MP)

We will write ¬A for A → ⊥, A ↔ B for (A → B)&(B → A), A ∧ B for A&(A → B), and A ∨ B for ((A → B) → B) ∧ ((B → A) → A). In order to introduce the equivalent algebraic semantics for BL (in the sense of [7]), namely the BL-algebras, we start from the following definitions. Definition 2.1 (cf. [6]). A hoop is a structure A, ·, ⇒, 1 such that A, ·, 1 is a commutative monoid, and ⇒ is a binary operation such that x ⇒ x = 1,

x ⇒ (y ⇒ z) = (x · y) ⇒ z

and

x · (x ⇒ y) = y · (y ⇒ x).

In any hoop, the operation ⇒ induces a partial order ≤ defined by x ≤ y iff x ⇒ y = 1. Moreover, hoops are precisely the partially ordered commutative integral residuated monoids (pocrims) in which the meet operation  is definable by x  y = x · (x ⇒ y). Finally, hoops satisfy the following divisibility condition: If x ≤ y, then there is an element z such that z · y = x.

(div)

Definition 2.2. A hoop is said to be basic iff it satisfies the identity (x ⇒ y) ⇒ z ≤ ((y ⇒ x) ⇒ z) ⇒ z.

(lin)

A Wajsberg hoop is a hoop satisfying: (x ⇒ y) ⇒ y = (y ⇒ x) ⇒ x.

(W)

A cancellative hoop is a hoop satisfying: x ⇒ (x · y) = y.

(canc)

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A bounded hoop is a hoop with an additional constant 0 satisfying the equation 0 ≤ x. A BL algebra is a bounded basic hoop. A Wajsberg algebra is a bounded Wajsberg hoop. A product algebra (cf. [21]) is a BL-algebra satisfying the equations x ∼ x = 0 and ∼∼ z ≤ ((x · z) ⇒ (y · z)) ⇒ (x ⇒ y), where ∼ x is an abbreviation for x ⇒ 0. A G¨odel algebra (cf. [21]) is a BL-algebra satisfying the equation x 2 = x. The varieties of BL-algebras, of Wajsberg algebras, of G¨odel algebras, of product algebras, of basic hoops, of Wajsberg hoops and of cancellative hoops will be denoted by BL, W, G, P, BH, WH and CH respectively. In any BL-algebra, as well as in any basic hoop, the join operation  is definable by x  y = ((x ⇒ y) ⇒ y)  ((y ⇒ x) ⇒ x). Moreover, the identity (x ⇒ y)  (y ⇒ x) = 1

(prl)

holds in any BL-algebra and in any basic hoop. Thus in particular, BL-algebras are commutative, integral and bounded residuated lattices (cf. [33] or [22]) satisfying conditions (div) and (prl). BL-algebras can be also characterized as those bounded hoops which are isomorphic to a subdirect product of linearly ordered bounded hoops. We also recall that a cancellative hoop is a Wajsberg hoop, and a Wajsberg hoop is basic, cf. [16] and [1]. Moreover, a linearly ordered Wajsberg hoop is either cancellative or the reduct of a Wajsberg algebra, cf. [16]. Finally, we recall that Wajsberg algebras are term equivalent to Chang MV-algebras: any Wajsberg algebra is a MV-algebra with respect to ∼ and to the operation ⊕ defined by x ⊕ y =∼ x ⇒ y. Conversely, any MV-algebra is a Wajsberg algebra with respect to the operations · and ⇒ defined by x · y =∼ (∼ x⊕ ∼ y) and by x ⇒ y =∼ x ⊕ y, cf. [21] or [10]. Definition 2.3. Let I, ≤ be a totally ordered set with minimum  i 0 . For all i ∈ I , let Ai be a hoop such that for i = j , Ai ∩ A j = {1}, and  assume that Ai0 is bounded. Then i∈I Ai (the ordinal sum of the family (Ai )i∈I ) is the structure whose base set is i∈I Ai , whose bottom is the minimum of Ai0 , whose top is 1, and whose operations are  A  x ⇒ i y if x, y ∈ Ai x⇒y= y if ∃i > j (x ∈ Ai and y ∈ A j )   1 if ∃i < j (x ∈ Ai \ {1} and y ∈ A j )  A  x · i y x·y= x   y

if x, y ∈ Ai if ∃i < j (x ∈ Ai , y ∈ A j \ {1}) if ∃i < j (y ∈ Ai , x ∈ A j \ {1}).

In [3] the following is proved: Theorem 2.4. Every linearly ordered BL-algebra A is the ordinal sum of an indexed family Wi : i ∈ I  of linearly ordered Wajsberg hoops, where I is a linearly ordered set with minimum i 0 , and Wi0 is bounded. In the following, the Wajsberg hoops Wi in Theorem 2.4 will be called the Wajsberg components of A. Using the fact that the Wi are closed to prove (cf. [3]) that with reference to Theorem 2.4, the  under hoop operations, it is easy subalgebras of A = i∈I Wi are those of the form B = i∈I Ui , where for i ∈ I , Ui is a subhoop of Wi (possibly trivial if i = i 0 ), and Ui0 is a Wajsberg subalgebra of Wi0 . In the following we set [0, 1]W = [0, 1], ·W , ⇒W , 0, 1, [0, 1]G = [0, 1], ·G , ⇒G , 0, 1 and [0, 1]Π = [0, 1], ·Π , ⇒Π , 0, 1, where ·Π is an ordinary product in [0, 1], and the remaining operators are defined as follows: x ⇒W y = min{1 − x + y, 1}, x ·W y = max{x + y − 1, 0},   1 if x ≤ y 1 if x ≤ y x ⇒G y = x ⇒Π y = y otherwise. y otherwise x

x ·G y = min{x, y}

Moreover, let (ω)[0, 1]W denote the ordinal sum of ω copies of [0, 1]W . Then: Theorem 2.5. The following conditions hold:

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(i) (ii) (iii) (iv)

153

W is generated as a quasivariety by [0, 1]W [10,21]. G is generated as a quasivariety by [0, 1]G [21]. P is generated as a quasivariety by [0, 1]Π [21]. BL is generated as a quasivariety by (ω)[0, 1]W [1].

BL-algebras constitute the equivalent algebraic semantics for BL in the sense of [7]. This property can be explained as follows. An evaluation of BL into a BL-algebra A is a map e from the set of BL-formulas into A such that e(⊥) = 0, and for any two BL-formulas A and B, one has: e(A&B) = e(A) · e(B) and e(A → B) = e(A) ⇒ e(B). For every formula A, let T A denote the term obtained from A by replacing ⊥ by 0, any propositional variable pi by the individual variable x i , & by ·, and → by ⇒. Conversely, given a term t, let F t be the formula obtained from t by the inverse substitution, that is, replacing x i by pi , 0 by ⊥, · by &, and ⇒ by →. Let |= B L denote the consequence relation in the equational logic of BL-algebras, and let  B L denote the consequence relation in BL. Then we have: Let Γ be a set of BL-formulas, and let A be a BL-formula. Let Γ T = {T B = 1 : B ∈ Γ }. Then, Γ  B L A iff Γ T |= B L T A = 1. Let Σ be a set of equations, and let η be any equation in the equational logic of BL-algebras. For every equation σ : t = s, let (σ ) F denote the BL-formula F t ↔ F s . Moreover, let Σ F = {σ F : σ ∈ Σ }. Then, Σ |= B L η iff Σ F B L ηF . Since the equational logic of any variety is strongly complete with respect to the variety itself, we can deduce: Theorem 2.6. BL is strongly complete with respect to the class of BL-algebras [21], that is, for every set Γ of BLformulas and for every BL-formula A, one has: Γ  B L A iff for every BL-algebra A and for every evaluation e in A, if e(B) = 1 for all B ∈ Γ , then e(A) = 1. Moreover, if Γ is finite, then by Theorem 2.5(iv) (cf. also [1]) we also have: Γ  B L A iff for every evaluation e in (ω)[0, 1]W , if e(B) = 1 for all B ∈ Γ , then e(A) = 1. This relationship between BL and BL-algebras can be extended both to the subvarieties of BL and to varieties of BL-algebras with additional operators. Let V be a variety of BL-algebras, possibly with further operators. For every n-ary function symbol f not in the language of BL-algebras, we introduce a new n-ary connective c f . Let L be the language of BL plus all connectives of the form c f for some operation f of V. To every term t of the language of V, we associate a formula F t as usual, replacing every additional function symbol f by c f . Then we introduce the logic L V associated with V as follows: • • • •

The formulas of L V are those of L. All axioms and all the rules of BL are axioms (rules) of L V . For every defining equation t = u of V the formula Ft ↔ Fu is an axiom of L V . For every n-ary function symbol f not in the language of BL algebras, the rule A 1 ↔ B1 , . . . , A n ↔ Bn c f (A1 , . . . , An ) ↔ c f (B1 , . . . , Bn ) is a rule of L V .

We will call L V the logic associated with V. It is readily seen that V is an equivalent algebraic semantics for L V in the sense of [7]. Due to this fact, we feel free to work in varieties rather than in many-valued logics. The notation L V to denote the logic associated with V, as well as the notation c f to denote the connective associated with f , will be used for general varieties and for general function symbols, but not for special varieties and function symbols: e.g., we will denote by BL, by Ł (Łukasiewicz logic), by G (G¨odel logic or Dummett logic, cf. [14]) and by Π (product logic) the logics associated with BL, W, G, and P respectively. In this paper we will mainly refer to the logics extending BL, as well as to some extensions of them by division operators, which will be defined later, and to their extensions by the operator ∆, which we are going to define. Definition 2.7. Let V be a variety of BL-algebras, possibly with additional operators. Then Vδ is the variety consisting of all structures for the language of V enriched by a new unary symbol δ and satisfying the defining equations of V plus the following ones: (1) δ(1) = 1.

154

(2) (3) (4) (5) (6) (7)

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δ(x ⇒ y) ≤ δ(x) ⇒ δ(y). δ(x) ∼ δ(x) = 1. δ(x) ≤ x. δ(δ(x)) = δ(x). δ(x  y) = δ(x)  δ(y). For every n-ary function symbol f not in the language of BL-algebras, n

δ(x i ⇔ yi ) ≤ f (x 1 , . . . , x n ) ⇔ f (y1 , . . . , yn ).

i=1

(In the following,




and



denote finite joins and finite meets respectively.)

It is readily seen that in any subdirectly irreducible algebra of Vδ , one has: δ(1) = 1, and δ(x) = 0 if x = 1. The logical operator associated with δ is denoted by ∆. Moreover, if L is an extension of BL associated with a variety V of BL-algebras, then the extension of L by ∆, that is, the logic associated with Vδ is denoted by L ∆ . The interest of extensions by δ is both logical and algebraic. On the logical side, we have that for any formulas A and B we have: A L∆ B

iff

L ∆  ∆(A) → B.

This can be obtained by induction on the length of the derivation of B from A in L ∆ , using the logical translation of the axioms (1) . . . (7) for δ. On the algebraic side, we have that Vδ is always a discriminator variety (cf. [8] for the definition). A discriminator is given by d(x, y, z) = (∼ δ(x ⇔ y) · x)  (δ(x ⇔ y) · z). Thus in every subdirectly irreducible algebra of Vδ every universal formula can be expressed by an equation [28]. Definition 2.8. Let A be a hoop, possibly with additional operators. A congruence filter of A is the congruence class of 1 with respect to some congruence of A. A congruence filter F is said to be prime iff whenever a  b ∈ F, then either a ∈ F or b ∈ F. If A is just a (bounded or unbounded) hoop, then its congruence filters are precisely its implicative filters (filters for short), that is, the subsets F of A such that 1 ∈ F, and whenever x and x ⇒ y ∈ F, then y ∈ F. It is easy to prove that if V is a variety of BL-algebras, possibly with additional operators, then the congruence filters of algebras in Vδ are precisely the implicative filters closed under δ. Indeed, if F is such a filter and x i θ F yi for i = 1, . . . , n, then from the identities (1). . . (7) for δ (and from (7) in particular), we obtain that for any n-ary function symbol f we have f (x 1 , . . . , x n )θ F f (y1 , . . . , yn ). The lattice of congruences and the lattice of congruence filters of a hoop with operators are isomorphic under the isomorphism which associates with a congruence filter F the congruence θ F defined by xθ F y iff x ⇒ y ∈ F and y ⇒ x ∈ F. The inverse isomorphism associates with every congruence θ the set Fθ = {x : xθ 1}. We will denote by A/F the quotient of A modulo θ F , and for a ∈ A, we will denote by a/F the equivalence class of a modulo θ F . It is relatively easy to prove that if A is a BL-algebra or a basic hoop (possibly with additional operators), then a congruence filter F is prime iff A/F is linearly ordered. This is because 1 = (a ⇒ b)  (b ⇒ a) ∈ F, therefore if F is prime then either a ⇒ b ∈ F, or b ⇒ a ∈ F. Conversely, if A/F is linearly ordered, then the linearity of the order implies that either (a  b) ⇒ a ∈ F, or (a  b) ⇒ b ∈ F, and this immediately implies that F is prime. We conclude this section reviewing some categorical equivalences between varieties of BL-algebras or of hoops and lattice-ordered abelian groups. Definition 2.9. A lattice ordered abelian group (cf. [5] or [20]) is an algebra G = G, +, −, 0, ,  such that G, +, −, 0 is an abelian group, G, ,  is a lattice, and for all x, y, z ∈ G, one has x + (y  z) = (x + y)  (x + z). A strong unit of a lattice ordered abelian group G is an element u ∈ G such that for all g ∈ G there is n ∈ N such that g ≤ u · · + u.  + · n times

The variety of lattice ordered abelian groups will be denoted by LAG.

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Definition 2.10. The radical of a Wajsberg algebra A (denoted by Rad(A)) is the intersection of its maximal filters. A Wajsberg algebra A is said to be perfect if for all x ∈ A, either x ∈ Rad(A), or ∼ x ∈ Rad(A). The functor Γ (cf. [29]) from the category of lattice ordered abelian groups with strong unit into the category of Wajsberg algebras is defined as follows: if G is a lattice-ordered abelian group and u is a strong unit of G, then Γ (G, u) denotes the algebra A whose universe is {x ∈ G : 0 ≤ x ≤ u}, and whose operations are x · y = (x + y − u)  0 and x ⇒ y = ((u − x) + y)  u. Moreover if α is a morphism of lattice ordered abelian groups with strong unit from (G, u) into (H, w), then Γ (h) denotes its restriction to Γ (G, u). The functor Υ from the category of lattice ordered abelian groups into the category of cancellative hoops is defined as follows: if G is a lattice ordered abelian group, then Υ (G) is the algebra whose universe is the negative cone {x ∈ G : x ≤ 0} of G and whose operations are the restriction of + to the negative cone of G and the operation ⇒ defined by x ⇒ y = (y − x)  0. Moreover if α is a morphism of lattice ordered abelian groups from G into H, then Υ (α) is the restriction of α to Υ (G). Finally, we define a functor Λ from the category of perfect Wajsberg algebras into the category of cancellative hoops as follows: given a perfect Wajsberg algebra A, Λ(A) is the algebra C whose universe is Rad(A) and whose operations · and ⇒ are the restrictions to C of the monoid operation and its residuum (note that any filter is closed under such operations). Moreover, given a morphism α of perfect Wajsberg algebras from W into U, Λ(α) denotes the restriction of α to Rad(W). It is well-known that Γ , Υ and Λ are equivalences of categories. The results are due to [29,16] and [12] respectively. We will denote their inverses by Γ −1 , Υ −1 and Λ−1 respectively. 3. Amalgamation As we said in the introduction, deductive interpolation and amalgamation are strictly related. This explains our choice to start our investigation from the amalgamation property. Definition 3.1. Let K be a class of algebras of the same type. We say that K has the amalgamation property (a.p. for short) iff whenever A, B and C are in K and i and j are monomorphisms from A into B and from A into C respectively, there are D ∈ K and monomorphisms h and k from B into D and from C into D such that the compositions h ◦ i and k ◦ j coincide. In this case, (D, h, k) is said to be an amalgam of (A, B, C, i, j ). In the following, if A, B and C are algebras of the same type, we will write A = B ∩ C to mean that A is a subalgebra of both B and C and the domain of A is the intersection of the domains of B and C. The following lemma is almost obvious. Lemma 3.2. Let K be a class of algebras of the same type closed under isomorphic images. Suppose that for every A, B and C in K such that A = B ∩ C, there are an algebra D ∈ K and embeddings h and k of B and C respectively into D such that the restrictions of h and k to A coincide. Then K has the a.p. Notation. (a) If A, B, C, h and k satisfy the conditions of Lemma 3.2, we will simply say that (D, h, k) is an amalgam of (A, B, C). (b) Given a class K of hoops, possibly with additional operations and constants, we denote by Klin the class of linearly ordered members of K. We recall that a class K of algebras has the congruence extension property (CEP) iff whenever A, B ∈ K and A is a subalgebra of B, then for every congruence θ of A there is a congruence θ  of B such that θ = θ  ∩ A2 . Let us say that a variety K of hoops, possibly with additional operators and constants, has the prime filter extension property (PFEP) iff for every A, B in K, if A is a subalgebra of B, then for every prime congruence filter F of A, there is a prime congruence filter G of B such that F = G ∩ A. Lemma 3.3. Let K be a variety or a quasivariety of BL-algebras possibly with additional operators, such that: (i) K has the PFEP. (ii) For any A ∈ K, for any congruence filter F of A and for any a ∈ A such that a ∈ / F, there is a prime congruence filter P of A such that a ∈ / P and F ⊆ P.

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(iii) Klin has the a.p. Then K has the a.p. Proof. Throughout this proof, we write filter for congruence filter. Let A, B, C in K such that A = B ∩ C. For any / Pb . Let Mb = A ∩ Pb . Then Mb is a prime filter of A, and by the b ∈ B \ {1}, take a prime filter Pb of B such that b ∈ PFEP there is a prime filter Q b of C such that Q b ∩ A = Mb . Let Bb = B/Pb , Ab = A/Mb , and Cb = C/Q b . Then Ab , Bb and Cb are in Klin, as Mb , Pb and Q b are prime filters. / Q c . Moreover, Mc = Q c ∩ A is a prime filter of Similarly, if c ∈ C \ {1} there is a prime filter Q c of C such that c ∈ A, and we can extend it to a prime filter Pc of B such that Pc ∩ A = Mc . Let Ac = A/Mc , Bc = B/Pc , Cc = C/Q c , and note that Ac , Bc and Cc are in Klin . Now for d ∈ B ∪ C, Ad embeds into both Bd and Cd via the embeddings i d and jd defined by i d (x/Md ) = x/Pd and jd (x/Md ) = x/Q d (it is readily seen that i d and jd are well-defined). After replacing Ad , Bd and Cd with isomorphic copies, we can suppose that Ad = Bd ∩ Cd . By (iii), there is an amalgam (Dd , h d , kd ) of (Ad , Bd , Cd ) such that Dd ∈ Klin. f

Moreover, for b ∈ B \ {1}, b/Pb = 1, and for c ∈ C \ {1}, c/Q c = 1. Thus the mappings f : x → (x/Pd : d ∈   g B ∪ C) and y → (y/Q d : d ∈ B ∪ C) are embeddings of B and C respectively into d∈B∪C Bd and d∈B∪C Cd respectively.Moreover, f andg coincide on all the elements of A.  Finally, d∈B∪C Bd and d∈B∪C Cd embed into d∈B∪C Dd∈ K via the embeddings h and k defined by h(x)d = h d (x/Pd ), and k(x)d = kd (x/Q d ) respectively. Let D = d∈B∪C Dd . Then (D, h ◦ f, k ◦ g) is an amalgam of (A, B, C).
Remark. A closer look at the proof of Lemma 3.3 shows that condition (iii) can be replaced by the following one: If A, B, C ∈ Klin and A = B ∩C, then there is an amalgam (D, h, k) of (A, B, C) such that D ∈ K (not necessarily D ∈ Klin). It is well-known that any variety of (bounded or unbounded) hoops has the CEP: any congruence θ of a hoop A is associated with a filter F of A, and if A is a subalgebra of B and G is the filter of B generated by F, then G ∩ A = {x ∈ A : ∃a1 , . . . , an ∈ F : a1 · . . . · an ≤ x} = F. Thus if θ  is the congruence of B associated with G, one has θ = θ  ∩ A2 . In a similar way, we see that any variety of B L δ -algebras possibly with additional operations has the CEP. Given a subalgebra A of B and a filter F of A, if G is the congruence filter of B generated by F, then G ∩ A = {x ∈ A : ∃a1 , . . . , an ∈ F : δ(a1 · . . . · an ) ≤ x} = F. Lemma 3.4. Any variety V of basic hoops or of BL-algebras, as well as any variety of B L δ -algebras, possibly with additional operators, satisfies conditions (i) and (ii) of Lemma 3.3. Proof. (i) Suppose first that V is a variety of BL-algebras or of basic hoops. Let A, B in V be such that A is a subalgebra of B, and let F be a prime filter of A. The set K of all filters G of B such that G ∩ A = F is non-empty (since V has the CEP), and it is closed under unions of chains. By Zorn’s Lemma, K has a maximal element P. We claim that P is prime. Let for X ⊆ A, Fg(X) denote the filter generated by X. Suppose by contradiction that for some a, b ∈ B we have a  b ∈ P, a ∈ / P and b ∈ / P. Then by the maximality of P there are c, d ∈ A \ F such that c ∈ Fg(P ∪ {a}), and d ∈ Fg(P ∪ {b}). Thus c  d ∈ Fg(P ∪ {a}) ∩ Fg(P ∪ {b}). Now Fg(P ∪ {a}) ∩ Fg(P ∪ {b}) = Fg(P ∪ {a  b}). Indeed, a  b ∈ Fg(P ∪ {a}) ∩ Fg(P ∪ {b}), therefore Fg(P ∪ {a  b} ⊆ Fg(P ∪ {a}) ∩ Fg(P ∪ {b}). On the other hand, if x ∈ Fg(P ∪ {a}) ∩ Fg(P ∪ {b}), then there are p, q ∈ P and m, n ∈ N such that p · a n ≤ x and q · bm ≤ x. If k = max{m, n} and r = p  q, then r · (a  b)k = (r · a k )  (r · bk ) ≤ x. So x ∈ Fg(P ∪ {a  b}), and the other inclusion is proved. Summing up, c  d ∈ Fg(P ∪ {a  b}), and since a  b ∈ P, it follows that c  d ∈ P ∩ A = F. Since F is prime, either c ∈ F or d ∈ F, and a contradiction is reached. If V is a variety of B L δ -algebras with operators, we repeat the same argument, thus obtaining a congruence filter P which is maximal among those whose intersection with A is F. As in the previous case, it is sufficient to prove that for all a, b ∈ B, one has Fg(P ∪ {a}) ∩ Fg(P ∪ {b}) = Fg(P ∪ {a  b}). For the non-trivial inclusion, if x ∈ Fg(P ∪ {a}) ∩ Fg(P ∪ {b}), then there are p, q ∈ P such that δ( p · a) ≤ x and δ(q · b) ≤ x. If r = p  q, then δ(r · (a  b)) = δ(r · a)  δ(r · b) ≤ x. So x ∈ Fg(P ∪ {a  b}), and the other inclusion is proved.

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(ii) That condition (ii) holds in any BL-algebra and in any B L δ -algebra is proved in [21], and the proof for basic hoops or for B L δ -algebras with operators is quite similar. The idea is the following: given a congruence filter F and an element a ∈ / F, using Zorn’s Lemma one obtains a congruence filter M which is maximal among all congruence filters which contain F and do not have a as an element. Then along the lines of the proof of (i) we can prove that the maximality condition mentioned above implies that M is prime.
We proceed with the proof that if V is any of W, CH, WH, and P, then V has the a.p. By Lemmas 3.3 and 3.4, it is sufficient to prove that Vlin has the a.p. We recall the following result, cf. [2,32]. Lemma 3.5. The classes LAG and LAGlin have the a.p. Corollary 3.6. The classes Wlin , CHlin, WHlin, and Plin have the a.p. Hence (Lemmas 3.3 and 3.4), the varieties W, CH, WH, and P have the a.p. Proof. The result for Wlin (and more generally for W) is due to Mundici [30], but since a similar argument will be used also for other varieties, we briefly sketch a proof. Let A, B and C be linearly ordered Wajsberg algebras as in Lemma 3.2. Let (G, u) = Γ −1 (A), (H, v) = Γ −1 (B) and (K, w) = Γ −1 (C). Note that G, H and K are linearly ordered. Moreover modulo isomorphism we can suppose that u = v = w and that G = H ∩ K. By Lemma 3.5 there are a lattice ordered abelian group F and monomorphisms h and k from H and K respectively into F such that h and k coincide on the domain of G. Thus in particular h(u) = k(u). After replacing F by its convex lattice ordered subgroup generated by h(u), we may suppose that h(u) is a strong unit of F . Then (Γ (F , h(u)), Γ (h), Γ (k)) is the desired amalgam, and the claim is proved. For CHlin we can use a similar argument, that is, we can reduce a.p. for CHlin to a.p. for LAGlin using the equivalence Υ . We now discuss WHlin. Let A, B, C, be linearly ordered Wajsberg hoops satisfying the conditions of Lemma 3.2. If A is the reduct of a Wajsberg algebra, then B and C must also be reducts of Wajsberg algebras (as the image of 0 must be an idempotent element different from 1), and the claim follows from the a.p. for Wlin . Now suppose that A is a cancellative hoop. If B and C are also cancellative hoops, then the claim is obvious because CHlin has the a.p. If one of B and C (or both) is the reduct of a Wajsberg algebra, then let A = Λ−1 (A), and let B  = B if B is the reduct of a Wajsberg algebra, and B  = Λ−1 (B) otherwise. Similarly, let C  = C if C is the reduct of a Wajsberg algebra, and C  = Λ−1 (C) otherwise. Then A, B and C are (isomorphic to) subalgebras of (the hoop reducts of) A , B  and C  respectively. Moreover, up to isomorphism we may assume that the conditions of Lemma 3.2 are satisfied by A , B  and C  . For instance, if B is a Wajsberg algebra, then A is included in the radical of B, therefore A is a subalgebra of B. Since Wlin has the a.p., there is an amalgam (D, h, k) of (A , B  , C  ) such that D ∈ Wlin . Let h  and k  be the restrictions of h and k respectively to B and to C. Since A is a subalgebra of A , (D, h  , k  ) is the desired amalgam, and the claim follows. Finally, we come to Plin . It follows from [11] that a linearly ordered product algebra is the ordinal sum of the two-element hoop 2 and a cancellative hoop. Thus the claim follows from the a.p. for CHlin.
Theorem 3.7. BL and BH have the a.p. Proof. We prove the claim for BL, because the proof for BH is similar. Once again, it is sufficient to prove that has the a.p. Let BLlin A, B and C be linearly ordered BL-algebras such that A = B ∩ C. We can write B and C as B = i∈I Wi , C = j ∈J U j , where Wi and U j are linearly ordered Wajsberg hoops, I and J are linearly ordered sets with minimum i 0 and j0 respectively, and Wi0 and U j0 are bounded. Let ≤ I and ≤ J denote the orders of I and of J respectively, and let I (A) = {i ∈ I : A ∩ Wi = {1}}, and let J (A) = { j ∈ J : A ∩ U j = {1}}. Without loss of generality, we can suppose that I ∩ J = I (A) = J (A) = M. Thus i 0 = j0 ∈ M (as 0 ∈ (Wi0 ∩ A) \ {1} and 0 ∈ (U j0 ∩ A) \ {1}). Now we can represent A, B and C as ordinal sums with reference to the same index set Y = I ∪ J . Define, for a, b ∈ Y , a ≤ b iff either a, b ∈ I and a ≤ I b, or a, b ∈ J and a ≤ J b, or a ∈ J , b ∈ I and there is m ∈ M such that a ≤ J m ≤ I b, or a ∈ I , b ∈ J and there is no m ∈ M such that b ≤ J m ≤ I a. Moreover let for a ∈ Y , Wa = Wa if a ∈ I , and let Wa be a trivial hoop otherwise. Similarly, let Ua = Ua if a ∈ J , and let Ua  be a trivial hoop  otherwise. Finally let Va = Va    if a ∈ M, and let Va be a trivial hoop otherwise. Then we have A = a∈Y Va , B = a∈Y Wa and C = a∈Y Ua . Moreover for all a ∈ Y , Va = Wa ∩ Ua . Since WHlin has the a.p., for a ∈ Y , there is an amalgam (Ha , h a , ka ) of

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 h k (Va , Wa , Ua ) such that Ha ∈ WHlin . Now let H = a∈Y Ha , and let h : B → H and k : C → H be defined as follows:   ka (x) if x ∈ Um \ {1} h a (x) if x ∈ Wa \ {1} k(x) = h(x) = 1 if x = 1 1 if x = 1. It is readily seen that (H, h, k) is an amalgam of (A, B, C).
Corollary 3.8. The varieties G, BLδ , Wδ , Pδ and Gδ have the a.p. Proof. The proof for G can be found in [17], Proposition 6.20 or in [25]. Now let K be any of BLδ , Wδ , Pδ and Gδ . By Lemma 3.3, it is sufficient to prove the claim for Klin . Note that for any linearly ordered BL-algebra A, the only operator δ which makes A a B L δ -algebra is  1 if x = 1 δ(x) = 0 otherwise. Thus the claim follows from Theorem 3.7 and Corollary 3.6.




Theorem 3.7 and Corollary 3.8 tell us that the most interesting varieties of BL-algebras have the a.p. There are however many subvarieties of BL without the a.p. For instance, in [13] it is shown that a variety of MV-algebras (hence a variety of Wajsberg algebras) has the a.p. iff it is generated by a single linearly ordered algebra. Thus for instance the variety generated by the Chang algebra Λ(Υ −1 (Z)) and by the three element Wajsberg algebra does not have the a.p. Moreover: Theorem 3.9. There are uncountably many varieties of BL-algebras which are all generated by a single linearly ordered BL-algebra and do not have the a.p. Proof. Let for a ∈ N, Wa be the (unique up to isomorphism) linearly ordered Wajsberg algebra with a + 1 elements. Let X be a set of prime numbers, and let V X be the variety generated by all Wajsberg algebras plus all ordinal sums of the form W0 ⊕W1 , where W1 is any linearly ordered Wajsberg algebra, and W0 is a linearly ordered Wajsberg algebra such that for all p ∈ X, W p does not embed into W0 . In [3] it is shown that if X and Y are different sets of primes, then V X = VY . Thus there are uncountably many varieties of this form. Moreover, each variety V X is generated by the single linearly ordered algebra W X ⊕ [0, 1]W , where W X is the subalgebra of [0, 1] X consisting of 0 and 1 plus all rationals of the form mn , where 0 ≤ n < m and for every prime p ∈ X, p does not divide m. Thus in order to prove the claim it is sufficient to show that if X = ∅, then V X does not have the a.p. Now let A = 2, B = [0, 1]W and C = 2 ⊕ [0, 1]W . Clearly A, B and C are in V X , and A is a subalgebra of both B and C. Moreover, if both B and C embed into D, then D must have a subalgebra isomorphic to [0, 1]W ⊕ [0, 1]W . Thus in order to prove that V X does / V X . Now define x⊕y = (x ⇒ (x · y)) ⇒ y, and let not have the a.p., it is sufficient to show that [0, 1]W ⊕ [0, 1]W ∈ (n)x = x⊕ · · · ⊕x (since the restriction of ⊕ to a single Wajsberg component is associative, we can omit parentheses). n times

Note that if x and y are both in a component which is a Wajsberg algebra, then ⊕ is the Łukasiewicz sum ⊕ relative to that component. Now for p ∈ X, consider the identity ((y ⇒ x) ⇒ x) · (( p − 1)x ⇔∼ x) ≤ x  y.

(ν p )

In [3] it is shown that (ν p ) holds in V X . On the other hand, (ν p ) does not hold in [0, 1]W ⊕ [0, 1]W . Indeed, let y be the copy of 12 relative to the second component of [0, 1]W ⊕ [0, 1]W , and let x be the copy of 1p relative to the first component. Then ((y ⇒ x) ⇒ x) · (( p − 1)x ⇔∼ x) = (x ⇒ x) · (( p − 1)x ⇔∼ x) = 1 / V X , and V X does not have the a.p.
and x  y = 1. Thus (ν p ) does not hold. So [0, 1]W ⊕ [0, 1]W ∈ The above varieties of BL-algebras without the a.p. are constructed ad hoc. One may wonder if there are more interesting varieties of BL-algebras without the a.p. The answer is positive. First of all, let for n > 1, Gn denote the

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variety generated by the n-element linearly ordered G¨odel algebra. As a consequence of a very general result about a.p. in superintuitionistic logics, cf. [17], Theorem 6.39, or [25], we obtain: Theorem 3.10. For n > 3, Gn does not have the a.p. We now introduce other examples of varieties of BL-algebras without the a.p. In [9], the authors prove that, even though the logic BL is a common generalization of Łukasiewicz logic, G¨odel logic and product logic, the variety of BL-algebras is not just the join of the corresponding varieties W, G and P. There the authors axiomatize the joins of W and G, of W and P, of G and P, and of W, G and P. Let us denote these varieties by WG, WP, GP, and WGP respectively. Theorem 3.11. Let V be any variety such that V ⊆ WGP and V contains either WG or WP. Then V does not have the a.p. In particular, none of WG, WP and WGP has the a.p. Proof. Let V be as in the claim of Theorem 3.11. Let A be the two-element BL-algebra and B = [0, 1]W . Moreover, if V ⊇ WG, then let C = [0, 1]G , otherwise let C = [0, 1]π . Suppose by contradiction that there is an amalgam (D, h, k) of (A, B, C) with D ∈ V. Let b and c be the images of 12 under h and k respectively. Then b =∼ b, whereas c < 1, and ∼ c = 0. Now let us decompose D as a subdirect product of an indexed family of subdirectly irreducible BL-algebras (Di : i ∈ I ). As shown in [9], every subdirectly irreducible member of WGP is either a Wajsberg algebra or a G¨odel algebra or a product algebra. Now take an index i such that ci < 1. Since bi =∼ bi , Di needs to be a Wajsberg algebra, because in a product algebra or in a G¨odel algebra ∼ has no fixed point. But then in Di we cannot have ci < 1 and ∼ ci = 0, and a contradiction has been reached.
Interestingly, there are varieties of BL-algebras which are not generated by a single linearly ordered BL-algebra and have the a.p. One example is constituted by GP. Since every linearly ordered element of GP is either a G¨odel algebra or a product algebra, GP is not generated by a single linearly ordered algebra. Moreover: Theorem 3.12. GP has the a.p. Proof. By the remark just following Lemma 3.3, it is sufficient to prove that given A, B and C in GPlin such that A = B ∩ C, there is an amalgam (D, h, k) of (A, B, C) such that D ∈ GP (possibly D ∈ / GPlin). Now any algebra in GPlin is either a G¨odel algebra or a product algebra. If B and C are either both product algebras or both G¨odel algebras, the claim follows from the fact that both G and P have the a.p. If, say B is a G¨odel algebra and C is a product algebra, then A must be the two-element BL-algebra. Indeed, in a G¨odel algebra every element is idempotent, whereas in a linearly ordered product algebra the only idempotents are 0 and 1. Define:   (x, 1) if x > 0 (1, y) if y > 0 D = B × C, h(x) = k(y) = (x, 0) otherwise (0, y) otherwise. It is readily seen that (D, h, k) is the desired amalgam, and that D ∈ GP.
4. Strong amalgamation Strong amalgamation is a strengthening of the amalgamation property, which is related to Beth’s property. In order to define it, let us introduce the following notation: given a function f and a subset X of its domain, f [X] denotes the set { f (x) : x ∈ X}. Definition 4.1. A class K of algebras has the strong amalgamation property (s.a.p. for short) iff whenever A, B and C are in K and i and j are monomorphisms from A into B and from A into C respectively, there is an amalgam (D, h, k) of (A, B, C, i, j ) such that D ∈ K, and in addition h[B] ∩ k[C] = (h ◦ i )[A] = (k ◦ j )[A]. In this case, (D, h, k) is said to be a strong amalgam of (A, B, C, i, j ). If A = B ∩ C and i and j are the identity embeddings, then we simply say that (D, h, k) is a strong amalgam of (A, B, C). The following lemma is almost trivial: Lemma 4.2. If K is closed under isomorphic images, then K has the s.a.p. iff whenever A, B and C are in K, and A = B ∩ C, then there is an algebra D ∈ K such that B and C are subalgebras of D.

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If D satisfies the conditions of Lemma 4.2, then we say that D is a strong amalgam of (A, B, C). We are going to prove that most varieties of BL-algebras do not have the s.a.p. We recall the following result, proved in [3]. Let for every class K of algebras of the same type, H(K), S(K), I(K) and Pu (K) denote the class of homomorphic images, of subalgebras, if isomorphic images and of ultraproducts of algebras from K respectively. Moreover, given two classes K1 , K2 of hoops, let K1 ⊕ K2 = {H1 ⊕ H2 : H1 ∈ K1 , H2 ∈ K2 }. Then: Theorem 4.3 (cf. [3]). Let W1 , W2 be linearly ordered hoops. Then ISPu (W1 ⊕ W2 ) = ISPu ({W1 }) ⊕ ISPu ({W2 }), and HSPu (W1 ⊕ W2 ) = HSPu ({W1 }) ∪ (ISPu ({W1 }) ⊕ HSPu ({W2 })). Of course, this result generalizes in an obvious way to finite ordinal sums. We are ready to prove: Theorem 4.4. Any variety V of BL-algebras which is not included in G does not have the s.a.p. Proof. If V ⊆ G, then there is a subdirectly irreducible  algebra H ∈ V which is not a G¨odel algebra. Now by / G, for some j ∈ I , Theorem 2.4, H can be represented as an ordinal sum i∈I Wi of Wajsberg hoops. Since H ∈ W j must have more than two elements. Distinguish the following cases: Case (a). W j is a non-perfect Wajsberg algebra. Then W j /Rad(W j ) is a simple Wajsberg algebra with more than two elements. Therefore HSPu ({Wi }) contains a finite Wajsberg chain U with m + 1 > 2 elements. Let U  be an isomorphic copy of U such that U ∩ U  = 2. We denote by u 0  the minimum of both U and U  , and by a   (a respectively) the unique atom of U (U respectively). Now let A = B = i∈I,i< j Wi ⊕ U, i∈I,i< j Wi ⊕ 2,   and C = i∈I,i< j Wi ⊕ U  . Clearly A, B and C are all in ISPu ( i∈I,i< j Wi ) ⊕ HSPu ( i∈I,i≥ j Wi ). Therefore, by Theorem 4.3, A, B and C are elements of V. Moreover, A = B ∩ C. We claim that if (D, h, k) is an amalgam / h[A]. This clearly shows the failure of strong amalgamation. of (A, B, C) such that D ∈ V, then h(a) = k(a  ) ∈ Note that in U we have a = (m − 1)a ⇒ u 0 , and that h(u 0 ) = k(u 0 ), as u 0 ∈ A. Let d0 = h(u 0 ) = k(u 0 ). Thus h(a) = h((m − 1)a) ⇒ d0 , and k(a  ) = k((m − 1)a  ) ⇒ d0 . Let us decompose D into a subdirect product of a family (D j : j ∈ J ) of linearly ordered BL-algebras. For all j ∈ J we have h(a) j = (m − 1)h(a) j ⇒ j (d0 ) j , and k(a  ) j = (m − 1)k(a  ) j ⇒ j (d0 ) j . Now if (d0 ) j = 1 j , then h(a) j = k(a) j = 1 j . Otherwise, h(a) j > (d0 ) j (if not, then, observing that h(a) j = (m − 1)h(a) j ⇒ (d0 ) j ≥ (d0 ) j , and that (d0 ) j is an idempotent, we would get (d0 ) j = h(a) j = (m − 1)h(a) j , and h(a) j = (m − 1)h(a) j ⇒ j (d0 ) j = 1 j ). Note also that if (d0 ) j = 1 j , then h(a) j and (d0 ) j belong to the same Wajsberg component (if not, then h(a) j = (m − 1)h(a) j ⇒ j (d0 ) j = (d0 ) j ). Moreover (d0 ) j , being an idempotent different from 1 j , is the minimum of that component. Similarly, if (d0 ) j = 1 j , then (d0 ) j < k(a  ) j and (d0 ) j and k(a  ) j belong to the same Wajsberg component. Now in any linearly ordered Wajsberg algebra with minimum c there can be at most one element x satisfying x = (m − 1)x ⇒ c. So h(a) j = k(a  ) j , and by the arbitrariness of j , h(a) = k(a  ). That h(a) ∈ / h[A] is clear, and the claim is proved. Case (b). W j is a perfect Wajsberg algebra. Let N = Υ (Z), where Z is the lattice ordered abelian group of integers, and let U = Λ−1 (N ). Then U can be represented as {1 − k : k ∈ N} ∪ {k : k ∈ N} for some positive infinitesimal , with monoid operation defined by x · y = max{x + y − 1, 0} and residual defined by x ⇒ y = min{1 − x + y, 1}. Let  copy of U 2U be the isomorphic copy of U consisting of {1−2k : k ∈ N}∪{2k  : k ∈ N}, and let U be an isomorphic   such that U ∩ U = 2U. Then U, 2U ∈ ISPu ({W j }). Now let A = i∈I,i< j Wi ⊕ 2U, and let B = i∈I,i< j Wi ⊕ U,    and C = i∈I,i< j Wi ⊕ U  . Once again, A, B and C are all in ISPu ( i∈I,i< j Wi ) ⊕ HSPu ( i∈I,i≥ j Wi ), therefore, by Theorem 4.3, they are in V. Moreover, A = B ∩ C. Let a = 1 − , and let a  be its isomorphic copy in U  . We prove that if (D, h, k) is an amalgam of (A, B, C) such that D ∈ V , then h(a) = k(a  ) ∈ h[B] ∩ k[C] \ h[A]. This clearly / h[A] is clear. In order implies the failure of strong amalgamation. First note that a 2 = (a  )2 = 1−2 ∈ A. That h(a) ∈ to prove that h(a) = k(a  ), let us decompose D into a subdirect product of a family (D j : j ∈ J ) of linearly ordered

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BL-algebras. Then the set {(h(a))nj : n ∈ N}, being a quotient of a cancellative hoop, is either a singleton or the domain of a non-trivial cancellative subhoop of D j . In the first case, 1 j = (h(a)) j = (h(a 2 )) j = (k((a  )2 )) j = (k(a  )) j . Otherwise, (h(a)) j and (k(a  )) j have the same square, so they are in the same Wajsberg component H j of D j . Moreover (h(a))2j is not the minimum of such a component, as {(h(a))nj : n ∈ N} is the domain of a non-trivial cancellative hoop. Now if c, d are elements of a linearly ordered Wajsberg hoop such that c2 = d 2 is not the minimum of that hoop, then c = d. Therefore, (h(a)) j = (k(a  )) j , and by the arbitrariness of j , h(a) = k(a  ). Case (c). W j is a cancellative hoop. The proof is quite similar to that of Case (b), the only difference being that in this case we take U = Υ (Z) = {1 − k : k ∈ N} and 2U = {1 − 2k : k ∈ N}. Thus the claim is proved.
In [17], Theorem 6.1 (cf. also [25]), it is shown that a variety of pseudoboolean algebras has the s.a.p. iff it has the a.p. Now it follows from [25] or from [17], Theorem 6.39 that the only subvarieties of G with the a.p. are G itself, G2 and G3 . Therefore: Theorem 4.5. A variety V of BL-algebras has the s.a.p. iff it is either the trivial variety, or G, or G2 or G3 .




In the next section we will prove that under suitable assumptions if a variety V of BL-algebras with operators is such that Vlin has the s.a.p., then the corresponding logic satisfies a weak form of Beth’s property. Thus we will look for varieties V of BL-algebras, possibly with operators, such that Vlin has the s.a.p. Unfortunately, there are only two non-trivial subvarieties of BL with such a property: Theorem 4.6. Let V be a non-trivial variety of BL-algebras. Then Vlin has the s.a.p. iff either V = G or V is the variety of Boolean algebras. Proof. We first prove that Glin has the s.a.p. Let A, B and C be linearly ordered G¨odel algebras such that A = B ∩ C. Define D as follows: the domain of D is the union of the domains of B and C; the order of D is any linear order which extends both the order of B and the order of C. Finally, the operations of D are uniquely determined by the order. It is easily seen that B and C are subalgebras of D, and the claim is proved. For the variety of Boolean algebras the claim is trivial, because the only linearly ordered Boolean algebra is 2. If V is not a subvariety of G, then observe that the algebras A, B and C occurring in the proof of Theorem 4.4 are linearly ordered, therefore the above proof implicitly shows that Vlin does not have the s.a.p. Finally, suppose that V = Gn with n > 2. Let Gn be the linearly ordered G¨odel algebra with n elements, and let A = 2, and B = C = Gn . Clearly, the only algebra in (Gn )lin in which B and C embed is Gn itself, and the embedding must be a surjection. Thus if h and k embed B and C respectively into D ∈ (Gn )lin, then D = Gn , h[B] = k[C] = D = h[A], as A has only two elements.
In the light of the above mentioned negative results for subvarieties of BL, we will try to add expressive power to BL in order to obtain varieties V of BL-algebras with operators such that Vlin has the s.a.p. To this purpose, we will introduce a variety BLdiv , whose members will be called called divisible BL-algebras, which is obtained from BL be the adding of division operators, and which is a natural generalization of the variety Wdiv of divisible Wajsbergalgebras (mostly known as divisible MV-algebras, cf. [19]). We recall the definition for the reader’s sake. Definition 4.7 (cf. [19]). A divisible Wajsberg algebra is a Wajsberg algebra A equipped with operations dn : n ∈ N, n > 1 such that for every x ∈ A and for every n ∈ N, n > 1, one has: [n − 1]dn (x) =∼ (x ⇒ dn (x)), where [n]x denotes x ⊕ · · · ⊕ x. n times

The variety of divisible Wajsberg algebras will be denoted by Wdiv , and the unique divisible Wajsberg algebra whose Wajsberg reduct is [0, 1]W will be denoted by ([0, 1]W )div . Note that in any divisible Wajsberg algebra one has [n]dn (x) = x. Moreover, the Wajsberg algebra of rational numbers in [0, 1] embeds into any non-trivial divisible Wajsberg algebra under the embedding mn → [m]dn (1). Definition 4.8. A rootable product algebra is a product algebra A equipped with unary operations rn : n ∈ N, n > 1 such that for every x ∈ A one has rn (x)n = x. The variety of divisible product algebras is denoted by Proot .

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Before introducing divisible BL-algebras, let us recall some properties of the operation ⊕ defined in the proof of Theorem 3.9. In any linearly ordered BL-algebra A, the following conditions hold: If x, y belong both to a cancellative component of A, then x⊕y = 1. If either A is a Wajsberg algebra, or x, y belong to the same Wajsberg component W of A and W is bounded, then x⊕y is the Łukasiewicz sum in W. If x and y do not belong to the same Wajsberg component and x < y, then x⊕y = x  y = y. If x and y do not belong to the same Wajsberg component and y < x, then x⊕y = 1. We now introduce divisible BL-algebras. Definition 4.9. A divisible BL-algebra is a BL-algebra equipped with unary operations χ and dn : n ∈ N, n > 1 such that, letting ι(x) = (d2 (x))2 , the following conditions hold: (i) (ii) (iii) (iv) (v) vi) (vii) (viii)

(n)dn (x) = x. dn (1) = 1. ι(x ⇒ y) ≤ ((dn (x) ⇒ dn (y)) · (χ(x) ⇒ χ(y))). Axioms of the modal logic S4 for ι, that is ι(x ⇒ y) ≤ (ι(x) ⇒ ι(y)), and ι(ι(x)) = ι(x). (Note that ι(1) = 1 and ι(x) ≤ x follow from (ii) and (iii).) ι(x)2 = ι(x). ι(x  y) = ι(x)  ι(y). (x ⇒ ι(x)) ⇒ ι(x) = x. χ(x) = χ(x) ⇒ ι(x).

Let us briefly comment on Definition 4.9. First note that the intended meaning of division operators dn in BLalgebras slightly differs from the meaning of the analogous operations in divisible Wajsberg algebras. If x = 1, then dn (x) represents once again the division of x by n in the Wajsberg component which x belongs to. However, we have chosen to define dn (1) = 1 instead of dn (1) = n1 , as there is no natural choice for the component which n1 should belong to. In order to represent the rationals in any Wajsberg component, we have introduced an operation χ(x) which roughly represents a copy of 12 , referred to the component which x belongs to. In this way, for every m < n, we can express mn , referred to the component which x belongs to, as (2m)dn (χ(x)). Note that the operator ι(x) = (d2 (x))2 represents, for every x, the greatest idempotent below x. Axioms (i) and (ii) describe the behavior of the operators dn , axioms (iv) . . . (viii) describe the behavior of the operators χ and ι. Finally, axiom (iii) will be used in Lemma 4.11 in order to prove that congruence filters of divisible BL-algebras are precisely the implicative filters closed under ι. The variety of divisible BL-algebras will be denoted by BLdiv . We prove some basic properties of such variety. Lemma 4.10. The equational logic of BLdiv extends the equational logic of BL conservatively. In other words, if a quasi-equation in the language of BL is valid in BLdiv , then it is valid in BL. Hence the logic B L div associated with BLdiv is a conservative extension of BL with respect to the finite consequence relation. Proof. The BL-algebra (ω)[0, 1]W generates BL as a quasivariety (Theorem 2.5). Now for every real number α ∈ [0, 1) and for every natural number n, let (α)n denote the copy of α in the n + 1th component of (ω)[0, 1]W (thus (α)0 denotes the copy of α in the first component). Then let for every n, m ∈ N and for every α ∈ [0, 1), dm ((α)n ) = ( mα )n , and χ((α)n ) = ( 12 )n . Moreover, let dm (1) = χ(1) = 1. Let (ω)[0, 1]div denote the resulting algebra. It is readily seen that (ω)[0, 1]div satisfies axioms (i) . . . (viii) in Definition 4.9, therefore it is a divisible BL-algebra. Thus if a quasi-equation in the language of BL can be invalidated in some BL-algebra, then it can be invalidated in (ω)[0, 1]W , therefore it can be invalidated in (ω)[0, 1]div, and it is not valid in all divisible BLalgebras.
Lemma 4.11. (a) The congruence filters of a divisible BL-algebra are precisely the filters which are closed under the operator ι. (b) Let a be any element of a divisible BL-algebra. Then the congruence filter generated by a is the lattice filter generated by ι(a).

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Proof. (a) That a congruence filter must be a filter is obvious. Moreover, if θ is any congruence and xθ 1, then ι(x)θ ι(1) = d2 (1)2 = 1. Thus a congruence filter must be closed under ι. Conversely, given a filter F closed under ι, the relation θ defined by xθ y iff (x ⇔ y) ∈ F is a congruence of BL-algebras. Moreover xθ y implies ι(x ⇔ y) ∈ F, and by axiom (iii) in Definition 4.9, this implies dn (x) ⇔ dn (y) ∈ F and χ(x) ⇔ χ(y) ∈ F, that is dn (x)θ dn (y) and χ(x)θ χ(y). Thus F is a congruence filter. (b) By axioms (iv) and (v), ι(a) is an idempotent below a. Now let Fι(a) be the lattice filter generated by ι(a), and let Fg(a) be the congruence filter generated by a. Then a ∈ Fg(a), therefore ι(a) ∈ Fg(a) (by part (a)), and Fι(a) ⊆ Fg(a), as a congruence filter is upwards closed. For the other direction, since obviously a ∈ Fι(a) , it is sufficient to prove that Fι(a) is a congruence filter, i.e., that Fι(a) is a BL-filter closed under ι. Now suppose b, b ⇒ c ∈ Fι(a). Then ι(a) ≤ b and ι(a) ≤ b ⇒ c, therefore ι(a) = ι(a)2 ≤ b · (b ⇒ c) ≤ c, and c ∈ Fι(a) . Thus Fι(a) is a filter. Moreover if b ∈ Fι(a) , then ι(a) ≤ b, and by axiom (iii) in Definition 4.9, ι(a) = ι(ι(a)) ≤ ι(b), therefore ι(b) ∈ Fι(a), and Fι(a) is closed under ι.
Lemma 4.12. Every divisible BL-algebra is isomorphic to a subdirect product of linearly ordered divisible BLalgebras. Proof. It is sufficient to prove that every subdirectly irreducible divisible BL-algebra A is linearly ordered. Now let F be the minimum non-trivial filter of A. By its minimality, F is generated by a single element, a say. Thus F = {x ∈ A : ι(a) ≤ x}. Suppose by contradiction that there are b, c ∈ A such that b ≤ c and c ≤ b. Then both b ⇒ c and c ⇒ b generate a non-trivial filter, which contains F. It follows that ι(c ⇒ b) ≤ a and ι(b ⇒ c) ≤ a. Thus 1 = ι(1) = ι((b ⇒ c)  (b ⇒ c)) = ι(b ⇒ c)  ι(c ⇒ b) ≤ a  a = a, and a contradiction has been reached.
Theorem 4.13. Every linearly ordered divisible BL-algebra A is the ordinal sum of a family of linearly ordered divisible Wajsberg algebras (Wi : i ∈ I ), that is:  The BL-reduct A− of A is the BL-reduct of the ordinal sum i∈I Wi− of the hoop reducts Wi− of Wi . Let the superscript i denote the interpretation of symbols in Wi , and let h i = d2i (1i ). Then dn (1) = χ(1) = 1, and if x ∈ Wi \ {1}, then dn (x) = dni (x) and χ(x) = h i . Proof. By Theorem 2.4, the BL-reduct A− of A is the ordinal sum of a family (Wi− : i ∈ I ) of linearly ordered Wajsberg hoops, therefore it is sufficient to prove that each Wi− is bounded, that if 0i denotes the bottom element of Wi , the restriction dni of dn to each Wi− \ {1} satisfies the axiom (n − 1)dni (x) = (x ⇒ dni (x)) ⇒ 0i (hence Wi− is the reduct of a divisible Wajsberg algebra), and that if x ∈ Wi− \ {1} then χ(x) = χ(x) ⇒ 0i . Now by equation (v) of divisible BL-algebras, ι(x) is an idempotent. Moreover equations (ii) and (iii) imply that ι(x) ≤ x. Note that in any BL-algebra, if x < y and x and y do not belong to the same Wajsberg component, then (y ⇒ x) ⇒ x = 1, therefore equation (vii) implies that x and ι(x) belong to the same Wajsberg component. Thus if x ∈ Wi− \ {1}, then ι(x) is the bottom 0i of Wi− , and Wi− is boundend. Now axiom (viii) tells us that Wi− has more than two elements, and that if x ∈ Wi− \ {1}, then χ(x) = χ(x) ⇒ 0i , that is, χ(x) is the unique fixed-point of the negation function in Wi− . Finally, (i) tells us that Wi− is the reduct of a divisible Wajsberg algebra such that for x ∈ Wi− \ {1} one has [n]dni (x) = x, therefore [n − 1]dni (x) = (x ⇒ dni (x)) ⇒ 0i .
By Theorem 4.13, the structure of a linearly ordered divisible BL-algebra is uniquely determined by its BL-reduct. So if A is the BL-reduct of a divisible BL-algebra, we denote by Adiv the unique divisible BL-algebra whose BL-reduct is A. Theorem 4.14. BLdiv is generated as a quasivariety by (ω)[0, 1]div. Proof. Since any algebra embeds into an ultraproduct of its finitely generated subalgebras, BLdiv is generated as a quasivariety by its linearly ordered and finitely generated members. Now it follows from the definition of ordinal sum that if a1 , . . . , an belong to a linearly ordered BL-algebra A, then the subalgebra of A generated by a1 , . . . , an is a subalgebra of the ordinal sum of the (finitely many) Wajsberg components which 0, a1 , . . . , an belong to. Now for

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every x = 1, dn (x) and χ(x) belong to the Wajsberg component which x belongs to. It follows that the BL-reduct of a finitely generated linearly ordered divisible BL-algebra A is the ordinal sum of finitely many divisible Wajsberg algebras W1 , . . . , Wn . The Wajsberg reduct of each Wi embeds into an ultraproduct of [0, 1]W , and since division operators are uniquely determined by the Wajsberg structure, the embedding also preserves division operators. Thus B embeds into the ordinal sum of n copies of ultraproducts of ([0, 1]W )div , which in turn embeds into an ultraproduct of (ω)[0, 1]div.
We are going to prove that (BLdiv )lin has the s.a.p. We start from the following: Lemma 4.15. The class of divisible linearly ordered abelian groups has the s.a.p. Proof. Let G, H and K be divisible linearly ordered abelian groups, and suppose that G = H ∩ K. Since linearly ordered abelian groups have the a.p., there is an amalgam (F , h, k) of (G, H, K) such that F is a linearly ordered abelian group. Now any linearly ordered abelian group can be embedded into a divisible linearly ordered abelian group, so we may assume without loss of generality that F is divisible. Then F can be regarded as a linearly ordered vector space over Q. Let B0 be a base of h[G], and let us extend B0 to a base B1 of h[H] and to a base B2 of k[K]. Now consider F ×lex F , that is, the linearly ordered group whose group reduct is the direct product F × F , and whose order  is defined by (x, y)  (u, v) iff either x < u or x = u and y ≤ v. Next consider the subgroup F  of F ×lex F consisting of all linear combinations with coefficients in Q of elements of the form (b, 0) with b ∈ B1 or of the form (c, c) with c ∈ B2 \ B1 = B2 \ B0 . Let g be the unique linear map from k[K] into F  such that for b ∈ B0 , g(b) = (b, 0), and for b ∈ B2 \ B0 , g(b) = (b, b). Finally let, for x ∈ H, h  (x) = (h(x), 0), and for y ∈ K, k  (y) = g(k(y)). It is easy to verify that (F  , h  , k  ) is a strong amalgam of (G, H, K).
Corollary 4.16. (Wdiv )lin and (Proot )lin have the s.a.p. Proof. We first prove the claim for (Wdiv )lin . Let A, B and C be linearly ordered divisible Wajsberg algebras, and assume that A = B ∩ C. Then A = Γ (G, u), B = Γ (H, v) and C = Γ (K, w) for some linearly ordered abelian groups G, H and K with strong units u, v and w respectively. Clearly, u = v = w, because A, B and C have the same top element. Moreover, up to isomorphism we can assume that G = H ∩ K. Furthermore, G, H and K are divisible. For instance, every element g ∈ G is the sum of finitely many elements of the form ±a1 , . . . , ±ak for some a1 , . . . , ak ∈ A. Therefore ng is the sum of ±dn (a1 ), . . . , ±dn (ak ), which is clearly an element of G. Then we may obtain a linearly ordered divisible abelian group F which is a strong amalgam of (G, H, K). After replacing F by its convex subgroup generated by u, we can assume without loss of generality that u is a strong unit of F . Then Γ (F , u) is a strong amalgam of (A, B, C), and the claim is proved. Now we prove the claim for (Proot )lin. Every linearly ordered product algebra A is the ordinal sum of 2 and a cancellative hoop C [11]. It is easily seen that if A is rootable, then C is divisible, that is, C = Υ (G) for some divisible linearly ordered abelian group. Thus the s.a.p. for (Proot )lin follows from the s.a.p. for divisible linearly ordered abelian groups.
Theorem 4.17. (BLdiv )lin has the s.a.p. Proof. Let A, B and C be linearly ordered divisible BL-algebras such that A = B ∩C. Let A− , B − and C − denote their BL-reducts. Along the lines of the proof of Theorem 3.7, possibly introducing trivial components, we can decompose of) possibly trivial A− , B − and C − as ordinal sums of (Wajsberg reducts  divisible bounded Wajsberg hoops with reference to the same index set I . Thus we have A− = i∈I Wi , B − = i∈I Ui , and C − = i∈I Vi , where we can assume without loss of generality that for every i ∈ I , Wi = Ui ∩ Vi . Since (Wdiv )lin has the s.a.p., we can find a sequence (Hi : i ∈ I ) such that for every i ∈ I , Hi ∈ (Wdiv )lin and Hi is a strong amalgam of (Vi , Wi , Ui ). Then,  H (thought of as a divisible BL-algebra) is a strong amalgam of (A, B, C). This concludes the proof.
i i∈I If B is a linearly ordered B L δ -algebra, possibly with operators, then every element of the form δ(x) is either 0 or 1, hence it belongs to any subalgebra of B. Thus δ does not cause problems for strong amalgamation: if a variety V of BL-algebras, possibly with operators, is such that Vlin has the s.a.p., then (Vδ )lin has in turn the s.a.p. Therefore: Corollary 4.18. If V is any of Gδ , (Wdiv )δ , (Proot )δ or (BLdiv )δ , then Vlin has the s.a.p.


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Unfortunately, we have not been able to extend the above results to the full varieties WAdiv , Proot and BLdiv . Note that given A, B and C such that A = B ∩ C, and given a subdirect decomposition of A, B and C into families (Ai : i ∈ I ), (Bi : i ∈ I ) and (Ci : i ∈ I ) of linearly ordered algebras, the existence for every i ∈ I of a strong amalgam of (Ai , Bi , Ci ) does not immediately imply the existence of a strong amalgam of (A, B, C). However, any of WAdiv , Proot and BLdiv satisfies conditions (i) and (ii) of Lemma 3.3. Therefore from the fact that Vlin has the s.a.p. we can deduce that any of the above varieties has the a.p. The same result can be proved for the extensions by δ, that is: Theorem 4.19. If V is any of WAdiv , Proot and BLdiv , then V and Vδ have the a.p.
5. Deductive interpolation and deductive Beth’s property In [31], various connections between amalgamation and interpolation have established. There, the author introduces a variant, called ROB of Robinson’s joint consistency property, namely: Definition 5.1. A logic L has ROB iff for any two sets Γ1 , Γ2 of formulas of L, if for every formula A in the common language one has: Γ1  L A iff Γ2  L A, then for i = 1, 2 and for every formula B in the language of Γi one has Γ1 ∪ Γ2  L B iff Γi  L B. In [31], Theorem 7, the author shows that an equational logic has ROB iff its associate variety has the a.p. Now the logics considered in this paper are algebraizable, therefore they are equivalent to the equational logics of their corresponding varieties. Moreover, in [31], Proposition 6 and Corollary 9, it is shown that if the variety corresponding to an algebraizable logic has the congruence extension property, then ROB implies a property, called G I N T , which in turn implies deductive interpolation. Finally, Ono and Galatos ([18], Theorem 5.8) prove the following: Theorem 5.2. If V is any variety of commutative residuated lattices, then V has the a.p. iff its corresponding logic L V has deductive interpolation. Thus if G, Ł, Π , ŁG, ŁΠ , GΠ , ŁGΠ , BL, B L div denote the logics associated with G, W, P, WG, WP, GP, WGP, BL and BLdiv , then we have: Theorem 5.3. G, Ł, Π , GΠ , BL, B L div, as well as their extensions with ∆ have deductive interpolation.
Let for any non-empty set X of prime numbers, V(X) be defined as in Theorem 3.9, and let L X be its corresponding logic. According to Theorem 5.2, none of L X Gn , with n > 3, ŁΠ , ŁG and ŁGΠ has deductive interpolation. For Gn , with n > 3, the claim also follows from [25] or from [17], Theorem 6.42. For the remaining logics we give a constructive proof of the failure of deductive interpolation by providing explicit counterexamples. Theorem 5.4. None of L X (X a non-empty set of primes), ŁΠ , ŁG and ŁGΠ has deductive interpolation. Thus there are uncountably many axiomatic extensions of BL without deductive interpolation. Proof. We first prove the claim for L X . Let m ∈ X. Let A( p, q) = ( p → p)&(q ↔ ¬(q m−1 )), and let B( p, r ) = ( p → p)&(r ↔ ¬¬r ). ( p → p has been added just because some authors in the definition of interpolation require A and B to have at least one common variable, cf. [31]). We first prove that A( p, q)  L X B( p, r ). Let A ∈ V(X), and let e be an evaluation in A such that e(A( p, q)) = 1. Then e(q) =∼ (e(q)m−1 ), and this is only possible if Wm embeds in A and e(q) is equal to (the isomorphic copy of) m−1 m . Since A ∈ V(X), it follows that A is a Wajsberg algebra. Therefore, e(r ) = e(¬¬r ) and e(B( p, r )) = 1. Now we prove that for any formula D( p) in the variable p only, we cannot have A( p, q)  L X D( p) and D( p)  L X B( p, r ). Note that either L X  D(⊥) or L X  ¬D(⊥). In the first case, let A = 2 ⊕ [0, 1]W . Note that A ∈ V(X). Take an evaluation e such that e( p) = 0 and e(r ) belongs to the second component and is different from 1. Then e(D( p)) = 1, e(¬¬r ) = 1 and e(r ) = 1. Therefore, e(B( p, r ) = 1 and D( p)  L X B( p, r ). If L X  ¬D(⊥), then let A = [0, 1]W , and let e be an evaluation in A such that e( p) = 0 and e(q) = m−1 m . Then e(A( p, q)) = 1 and e(D( p)) = 0, therefore A( p, q)  L X D( p). Now assume that L is any of ŁΠ , ŁG or ŁGΠ . Let A = ¬ p&(q ↔ (q → p)), and B = ¬ p&(r ↔ ((r → p) → p)). We claim that A( p, q)  L B( p, r ). Let A be a linearly ordered algebra in the variety associated with L and e be an evaluation in A such that e(A( p, q)) = 1. Now A is either a Wajsberg algebra, or a G¨odel algebra or a product algebra. If e(A( p, q)) = 1, then e( p) = 0, and e(q) = e(q) ⇒ e( p) = e(¬q). It follows that A must be a Wajsberg

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algebra and e( p) must be the fixed point of negation in that algebra. But then e((r → p) → p) = e(¬¬r ) = e(r ), therefore, e(B( p, r )) = 1. We claim that there is no formula D( p) in the variable p only such that A( p, q)  L D( p) and D( p)  L B( p, r ). Suppose by contradiction A( p, q)  L D( p) and D( p)  L B( p, r ). Then a fortiori A( p, q) Ł D( p), and either D( p) G B( p, r ) or D( p) Π B( p, r ). We claim that ¬ p Ł D( p). Let e be an evaluation in [0, 1]W such that e(¬ p) = 1 (hence e( p) = 0), and let e be an evaluation such that e (q) = 12 and e ( p) = e( p) = 0. Then e (A( p, q)) = 1, and since A( p, q)  L D( p), we obtain e (D( p)) = e(D( p)) = 1. Thus ¬ p Ł D( p). We now claim that D( p) is inconsistent either with G or with Π , that is, either D( p) Π ⊥ or D( p) G ⊥. Suppose not. Then there are evaluations e, e in [0, 1]G and in [0, 1]Π respectively such that e(D( p)) = 1 and e (D( p)) = 1. Now let i be an evaluation in [0, 1]G (in [0, 1]Π respectively) such that i ( p) = 0 and 0 < i (r ) < 1. Then i (¬¬r ) = 1, and i (r ) < 1, therefore i (B( p, r )) < 1. Moreover, since i ( p) = e( p) = 0 (i ( p) = e ( p) = 0 respectively), we also have i (D( p)) = e(D( p)) = 1 (i (D( p)) = e (D( p)) = 1 respectively). However, we have seen that either D( p) G B( p, r ) or D( p) Π B( p, r ), which is a contradiction. Summing up, we have shown that if A( p, q)  L D( p) and D( p)  L B( p, r ), then ¬ p  L D( p) and D( p) is inconsistent either with G of with Π . Now ¬ p  L D( p) implies that for any evaluation e in any Wajsberg algebra e(D( p)) = 1 if e( p) = 0. Since for every subformula E of D( p), e(E) is a Boolean value, D( p) is consistent with classical logic, hence a fortiori it is consistent with both G and Π , and a contradiction has been reached.
We now investigate the deductive Beth property. We first consider the extensions of BL in the language of BL, and we prove that almost all such extensions do not have the deductive Beth property. In the proof of this fact, we will borrow some ideas from the proof of Theorem 4.4. Indeed, as we will see later, there are connections between strong amalgamation and the deductive Beth property. First of all, every superintuitionistic logic has Beth’s property, cf. [24] or [17], Theorem 4.38. In this theorem, Beth’s property is formulated in terms of implication, but in G¨odel logics this formulation is equivalent to the deductive Beth property, because the deduction theorem holds in any extension of G in the language of G. Therefore: Theorem 5.5. Any logic extending G¨odel logic G (in the same language as G) has both the deductive and the implicative Beth property.
However: Theorem 5.6. If L is an axiomatic extension of BL in the language of BL which is not an extension of G, then L does not have the deductive Beth property. Thus an axiomatic extension of BL has the deductive Beth property iff it extends G. Proof. Let V be the variety associated with L. Then V is a variety of BL-algebras which is not contained in G, therefore there is a linearly ordered algebra A ∈ V which is the ordinal sum of an indexed family of Wajsberg components, one of which, W say, has more than two elements (otherwise every element of V would be a G¨odel algebra). Taking the subalgebra of A generated by the first Wajsberg component of A and by W, we can assume that A = H ⊕ W for some (possibly trivial) bounded Wajsberg hoop H. Now HSPu (A) contains all algebras of the form H ⊕ V with V ∈ HSPu (W). If W is a cancellative hoop, then HSPu (W) contains all subdirectly irreducible cancellative hoops (cf. [16] or [3]), hence it contains the (0, 1]Π , the 0-free subreduct of [0, 1]Π with domain (0, 1]. Using the equivalence Λ−1 , we see that if W is a perfect Wajsberg algebra then HSPu (W) contains Λ−1 ((0, 1]Π ). Finally, if W is a non-perfect Wajsberg algebra, then HSPu (W) contains a finite Wajsberg algebra with more than two elements. Thus V contains an algebra of the form H ⊕ U, where U is either (0, 1]Π , or Λ−1 ((0, 1]Π ), or a finite linearly ordered Wajsberg algebra with n > 2 elements. Now consider the formula A(a, p) = p ↔ ( pn−2 → a), where n is the cardinality of U if U is finite, and n = 3 (hence n − 2 = 1) otherwise. We first prove that A(a, p), A(a, q)  L p ↔ q. Since any variety of BL-algebras is generated as a quasivariety by its linearly ordered members [21,1], it is sufficient to check that for every linearly ordered algebra B ∈ V and for every evaluation i in B such that i (A(a, p)) = i (A(a, q)) = 1, one has i ( p) = i (q). Suppose i (A(a, p)) = 1. If i (a) = 1, then i ( p) = i (q) = 1. Now assume that i (a) = 1; then i ( p) > i (a) (otherwise i ( p) = i ( p n−2 → a) = 1, and i (a) = 1). Moreover, i (a) and i ( p) are in the same Wajsberg component, otherwise we would obtain i ( p n−2 → a) = i (a), and finally i ( p) = i (a) = 1. Consider the function F(i ( p)) = i ( pn−2 → a) as a function of i ( p) (for i (a) < 1 fixed), when i ( p) ranges over the Wajsberg component Z which i (a) belongs to.

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If i ( p) = i (a), then F(i ( p)) > i ( p), and F(i ( p)) is weakly decreasing in i ( p). Thus there can be at most one value i ( p) in Z (and at most one value of i ( p) in A) such that F(i ( p)) = i ( p). Hence, i ( p) is uniquely determined from i (a). Therefore, if i (A(a, p)) = i (A(a, q)) = 1 , then i ( p) = i (q). We now prove that there is no formula D(a) such that A(a, p)  L p ↔ D(a). Consider the algebra H ⊕ U introduced above. Define an evaluation e in H ⊕ U as follows: if U is finite and has cardinality n > 2, then let e(a) = min(U); note that e(A(a, p)) = 1 iff e( p) is the unique coatom of U. Then, for any formula D(a) in the variable a only, e(D(a)) is in the subalgebra of H ⊕ U generated by e(a), which consists of e(a), 0 and 1. Thus  e( p) = e(D(a)). If U is (0, 1]Π , let e(a) = 12 . Note that in this case n = 3 and e(A(a, p)) = 1 iff e( p) = 12 . Moreover for any formula D(a) in the variable a only, e(D(a)) is in the the  subalgebra of H ⊕ U generated by e(a), which consists of 0, 1 and of some rationals in (0, 1]Π , whereas e( p) = Λ−1 ((0, 1]

1 2

is an irrational in (0, 1]Π . Once more,

e(D(a)) = e( p). Finally, suppose that U = ∈ Rad(U)) (note that Rad(U) is the domain Π ). Let e(a) = of a subhoop of H ⊕ U which is isomorphic to (0, 1]Π ). Once more, for any formula D(a) in the variable a only, e(D(a)) must belong to the subalgebra of H ⊕ U generated by e(a). The elements common  to such a subalgebra 1 2

and to Rad(U) are rationals in (0, 1]Π , whereas if e(A(a, p)) = 1, then it must be e( p) = proof.


1 2.

This concludes the

Now we investigate the deductive Beth property in the extensions of BL with additional operators. Our aim is to find a conservative extension of BL which has such a property. We start from the following result, which relates Beth’s property to strong amalgamation. Theorem 5.7. Let V be a variety of BL-algebras, possibly with additional operators, let L = L V and L ∆ be its extension with ∆. Then: (i) If V has the s.a.p., then L has the deductive Beth property. (ii) If V is generated as a quasivariety by Vlin and Vlin has the s.a.p., then L has the weak deductive Beth property. (iii) If Vlin has the s.a.p. and Vδ is generated as a quasivariety by (Vδ )lin , then L ∆ has the deductive Beth property. Proof. (i) We argue contrapositively. Suppose that for every formula D( a ) in the propositional variables a , A( a , p)  L p ↔ D( a ). Let F be the set of formulas of L whose variables are among a . For every D ∈  F, choose an algebra a , p)) = 1 and e D ( p ↔ D) = 1. Let B = D∈F A D . Define an A D ∈ V and an evaluation e D in A D such that e D (A( evaluation e in B letting for every formula C and for every D ∈ F, (e(C)) D = e D (C). Next, let A be the subalgebra of B whose domain is {e(D) : D ∈ F}. Note that for every D ∈ F, e D ( p) = e D (D), therefore e( p) ∈ / A. Finally, let C be an isomorphic copy of B such that A = B ∩ C, and there is an isomorphism j from B onto C whose restriction to A is the identity. Let D ∈ V be a strong amalgam of (A, B, C). Let q be a propositional variable not occurring in A( a , p), a , p)) = e (A( a , q)) = 1, and define an evaluation e on D by e (r ) = e(r ) if r = q, and e (q) = j (e( p)). Then e (A( but e ( p) = e (q), therefore e ( p ↔ q) = 1. It follows that A( a , p), A( a , q)  L p ↔ q.
n (ii) Let D0 , . . . , Dn , . . . be an enumeration of all formulas in F, and let E n = i=0 ( p ↔ Di ). We argue contrapositively. Suppose that for every n ∈ N, A( a , p)  E n . Then for every n there are a linearly ordered model En of V and an evaluation en in En such that en (A( a , p)) = 1 and e(E n ) = 1. Now let U be a non-principal ultrafilter  on N, let E = n∈N En , and EU = E/U . Let for a ∈ E, a/U denote the equivalence class of a modulo the ultrafilter U . Define an evaluation e in E and an evaluation eU in EU by e(B)n = en (B), and eU (B) = e(B)/U . Then by the a , p)) = 1 and that eU (E n ) = 1 for every n ∈ N. ultraproduct theorem we have that EU is linearly ordered, that eU (A( At this point, the proof is completely parallel to the proof of (i). Let B = EU , let A be the subalgebra of B whose domain is {eU (D) : D ∈ F}, and let C be an isomorphic copy of B such that A = B ∩ C, and there is an isomorphism j from B onto C which is identical on A. Let D ∈ Vlin be a strong amalgam of (A, B, C), let q be a variable not in A( a , p), and let e be the evaluation on D defined for every propositional variable r by e (r ) = eU (r ) if r = q, and  a , p)) = e (A( a , q)) = 1, but e ( p ↔
q) = 1. e (q) = j (eU (q)). Then e (A( a , p)  L ∆ ni=1 ( p ↔ Di ) for some (iii) By (ii), if A( a , p), A( a , q)  L ∆ p ↔ q, then A(
n formulas D1 , . . . , Dn containing only variables among a . We prove that this implies A( a , p)  L ∆ p ↔ a , Di ))&Di ). i=1 (∆(A( Clearly, it is sufficient to prove that for every B ∈ (V ) and for every evaluation e in B, if e(A( a , p)) = 1, δ lin
n

a , Di ))&Di ). Since B is linearly ordered and A( a , p)  L ∆ ( p ↔ Di ), we then e( p) = e( ni=1 ∆(A( i=1 must have e( p) = e(Di ) for some i ≤ n. Clearly if for some j = i we have e(Di ) = e(D j ) = e( p), then

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e(A( a , Di )) = e(A( a , D j )) = e(A( a , p)) = 1. Moreover, since A( a , p), A( a , q)  L ∆
p ↔ q, we have that if e(D j ) = e( p), then e(A( a , D j )) = 1, and e(∆(A( a , D j ))) = 0. It follows that e( p) = e( ni=1 ∆(A( a , Di ))&Di ), and the claim is proved.
As a corollary we obtain that BL, Π and Ł have a conservative extension with the deductive Beth property. More precisely, let B L div , Łdiv and Πroot denote the logics associated with BLdiv , Wdiv and Proot . Then Theorems 4.17 and 5.7, together with Corollary 4.16, give: Theorem 5.8. B L div , Łdiv and Πroot have the weak deductive Beth property. Moreover, (B L div)∆ , (Łdiv )∆ and (Πroot )∆ have the deductive Beth property.
We close this section with a rather surprising example showing that there are extensions L of BL such that L has the deductive Beth property, but L ∆ has not. We know (Theorem 5.6) that for every n, Gn has the deductive Beth property. However: Theorem 5.9. For n > 2, (G n )∆ does not have the deductive Beth property. Proof. Let p
< e(q), and e( p e(B(q, r )); if e(D(q)) < 1, then e(D(q)) < e(A( p, q)). So D(q) cannot be an interpolant of A( p, q) and B(q, r ). (b) W is a cancellative hoop. Let a ∈ W \ {1}, and let e be any evaluation such that e( p) = e(r ) = a, e(q) = a 2 . Then e(A( p, q)) = e(B(q, r )) = a. Now let D(q) be any formula in the variable q only. Then e(D(q)) belongs to the subalgebra of A generated by 0, 1 and e(q) = a 2 . Such a subalgebra consists of 0, 1 and all elements of the form a 2n : n ∈ N, and a is not an element of that subalgebra. Thus D(q) is not an interpolant of A( p, q) and B(q, r ).


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The negative result contained in Theorems 5.6 and 6.1 extends to the implicative Beth property: Theorem 6.2. Let L be any schematic extension of BL in the language of BL. Then L has the implicative Beth property iff L extends G. Proof. Since every superintuitionistic logic has the implicative Beth property [24], it is sufficient to prove that if V is a subvariety of BL which is not contained in G, then L V does not have the implicative Beth property. If V ⊆ G, then there is a linearly ordered algebra A ∈ V such that one of the Wajsberg components W of A has more than two elements. As in the proof of Theorem 5.6 we may assume that W is either a perfect Wajsberg algebra or a cancellative hoop, or a finite Wajsberg algebra. In the first two cases, let A( p, q) = q ↔ (q → p); in the third case let A( p, q) = q ↔ (q n−1 → p), where n + 1 > 2 is the cardinality of W. As in the proof of Theorem 5.6, we may find an algebra A ∈ V and an evaluation e in A such that e(A( p, q)) = 1 and e(q ↔ D( p)) = 1 for every formula D( p) in the variable p only. Thus there is no formula D( p) in the variable p only, such that A( p, q)  L V q ↔ D( p). In order to conclude that L V does not have the implicative Beth property, it is sufficient to prove that B L  (A( p, q)& A( p, r )) → (q ↔ r ). Let us write A( p, q) as q ↔ (q i → p), where i can be either 1 or n − 1. By symmetry reasons, it is sufficient to prove that B L  (A( p, q)& A( p, r )) → (q → r ), which by the residuation property amounts to proving that B L  (A( p, q)& A( p, r )&q) → r. Let C( p, q, r ) = A( p, q)& A( p, r )&q. We prove that for any linearly ordered BL-algebra A and for any evaluation e in A, one has e(C( p, q, r )) ≤ e(r ). This clearly implies the claim. Since e(C( p, q, r )) ≤ e(q), if e(q) ≤ e(r ), the claim is trivial. If e(r ) < e(q), then e(q i → p) ≤ e(r i → p), and since e(A( p, q)&q) ≤ e((q i → p), we get: e(C( p, q, r )) ≤ e((q i → p)& A( p, r )) ≤ e((r i → p)&(r ↔ (r i → p))) ≤ e(r ), and the claim is proved.
By Theorems 6.1 and 5.3, deductive interpolation in a many-valued logic does not imply Craig interpolation. For logics with ∆, however, deductive interpolation implies the following modal form of Craig interpolation: Theorem 6.3. Let L be a schematic extension of BL, and suppose that L ∆ has deductive interpolation. Then, if L ∆  ∆(A) → B, there is a formula C of L ∆ in the language common to A and B such that L ∆  ∆A → C and L ∆  C → B. Proof. Recall that for any two formulas A and B of L ∆ one has: A  L ∆ B iff L ∆  ∆(A) → B. Now suppose that L ∆  ∆(A) → B. Then A  L ∆ B, therefore by the deductive interpolation there is a formula D of L ∆ in the language common to A and B such that A  L ∆ D and D  L ∆ B. Using the rule ∆D (D) , we obtain A  L ∆ ∆(D), therefore, L ∆  ∆(A) → ∆(D), and L ∆  ∆(D) → B. Thus ∆(D) is the desired interpolant.
The negative results contained in Theorems 6.1 and 6.2 suggest to look for conservative extensions of the most important many-valued logics having Craig’s interpolation property and therefore having the implicative Beth property. Since the solution for G, Ł and Π has been found in [4] (namely, G has uniform interpolation, and Łdiv and Πroot are conservative extensions of Ł and Π respectively having also uniform interpolation), we will investigate the problem of finding an analogous extension for BL. As we said in the introduction, we will be only partly successful. Indeed, for every n, we will find an extension BL(n) of BL which has uniform interpolation and is conservative over BL with respect to formulas in n variables only. Even though we do not know if the expressive power of the extensions we are going to introduce is minimal in order to get interpolation, in the next examples we will motivate our choice of the operators to be added. Here below, the operators H , I and Dn represent the logical translations of the symbols χ, ι and dn of divisible BL-algebras.

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Moreover, A ⊕ B, A⊕B, [n] A and (n)A are abbreviations for ¬(¬A&¬B), (A → (A&B)) → B, A ⊕ · · · ⊕ A and · · ⊕ A (as for the algebraic counterpart of ⊕, we can omit parentheses). A⊕ ·

n times

n times

(a) Let An = p ∧ ( pn → q), and Bn = r ∨ (r n → q). Then in Łdiv , the unique interpolant of An and Bn is [n]Dn+1 (1) ⊕ Dn+1 (q). This example suggests that division operators should be introduced also in the case of BL. In fact, in B L div , an interpolant of An and Bn is (2n)Dn+1 (H (q))⊕Dn+1 (q). This example also suggests the use of the logical operator H . (b) Suppose that we want to find a right interpolant A(q, r ) of the formula p ∧ ( p → q) ∧ (r → p), that is, suppose we want to compute ∃ p( p∧( p → q)∧(r → p)). Consider any evaluation e in a linearly ordered divisible BL-algebra. If e(q) < e(r ) and e(q) and e(r ) are in the same Wajsberg component, then e(∃ p( p ∧ ( p → q) ∧ (r → p))) = D2 (q⊕r ), otherwise (i.e., if either e(r ) ≤ e(q) or e(q) < e(r ) and e(q), e(r ) are in different Wajsberg components, then e(∃ p( p ∧ ( p → q) ∧ (r → p))) = D2 (r )⊕H (r ). Thus the uniform right interpolant is defined by cases. Even though we do not exclude that all definitions by cases needed in order to compute right or left interpolants can be simulated by a formula of B L div , in such logic we do not have crisp formulas expressing e.g. that e( p) < e(q) and e( p) and e(q) are not in the same component, therefore finding a general method for computing interpolants defined by cases seems not an easy task. We will see in a moment that such crisp formulas exist in (B L div )∆ . (c) Consider the formula A( p, q, r ) = ∆(( p → q) → q) ∧ ∆((r → p) → p) ∧ ¬∆( p → q) ∧ ¬∆(r → p) ∧ ¬∆(r ). Using the definition of implication in an ordinal sum, the reader may check that for every evaluation e in a linearly ordered BL-algebra A, one has e(A( p, q, r )) = 1 iff e(q) < e( p) < e(r ) < 1, and e( p), e(q) and e(r ) belong to different components, otherwise, e(A( p, q, r ) = 0. Now let us look for the right interpolant B(q, r ) of p ∧ ( p → I ( p)) ∧ A( p, q, r ). Given any evaluation e, one has e(∃ p( p ∧ ( p → I ( p)) ∧ A( p, q, r )) = 0 unless there is a ∈ A such that e(q) < a < e(r ) and e(q), a and r are not in the same component. In this case, let a < 1 be an element of the Wajsberg component immediately below the component which e(r ) belongs to. Then e(∃ p( p ∧ ( p → ( p& p)) ∧ A( p, q, r )) = χ(a). This element is not definable by a formula of (B L div )∆ . The above examples suggest that in order to get uniform interpolation one should consider a logic which has division operators and which is able to treat definitions by cases and to express the component under consideration. Instead of a single logic, we propose for every n a logic BL(n) whose algebraic semantics consists of divisible B L δ algebras with exactly n + 1 components and with a constant 0i for i = 0, . . . , n, representing the bottom of the i + 1th component (we will identify 00 with the bottom element 0 of the algebra). In order to give a formal definition, we define x ⇑ y = (x ⇒ y) ⇒ y, and we observe that in any linearly ordered BL-algebra one has: x ⇑ y = 1 if either x = 1 or y = 1 or y < x and x, y do not belong to the same component. Definition 6.4. A BL(n)-algebra is a (B L div )δ algebra equipped with constants 01 , . . . , 0n such that, letting 00 = 0, the following additional axioms holds: (i) (ii) (iii) (iv)

0i+1 ⇑ 0i = 1 (i = 0, . . . , n). i δ(0

n ) = 0 (i =i 0, . . . , in). i=0 (δ(x ⇑ 0 )  δ(0 ⇑ x)) ≤ δ(x). 0i = 0i · 0i (i = 1, . . . , n).

The variety of BL(n)-algebras is denoted by BL(n), and the associated logic is denoted by BL(n). We briefly discuss the properties of BL(n)-algebras. Since the introduction of the new constants does not change the congruences, we still have that every subdirectly irreducible BL(n) algebra is linearly ordered. Therefore every BL(n)-algebra decomposes as a subdirect product of linearly ordered BL(n) algebras. Now axioms (ii) and (iv) imply that for i = 1, . . . , n, 0i is an idempotent less than 1. Axiom (i) says that for i = 0, . . . , n − 1 one has 0i < 0i+1 , and 0i and 0i+1 belong to different components. Thus any non-trivial BL(n)-algebra has n +1 components at least. Finally, axiom (iii) says that if x = 1 (that is, if δ(x) = 0), then there is an i ≤ n such that x is in the same component as 0i (that is, δ(x ⇑ 0i )  δ(0i ⇑ x) = 0). In other words, every BL(n)-algebra has precisely n + 1 components, which by Theorem 4.13 are all divisible Wajsberg algebras, and whose minimal elements are 00 = 0, 01 , . . . , 0n respectively. Clearly, any BL(n)-algebra is uniquely determined by its B L div reduct, since the interpretation of the constants 0i and of δ are uniquely determined. Now every divisible linearly ordered Wajsberg algebra is in ISPu (([0, 1]W )div ), therefore by Theorem 4.3, the B L div reduct of any linearly ordered BL(n)-algebra is in ISPu ((n + 1)([0, 1]W )div ),

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where (n+1)([0, 1]W )div denotes the ordinal sum of n+1 copies of the divisible Wajsberg algebra on [0, 1]. Moreover, (n + 1)([0, 1]W )div can be uniquely extended to a BL(n)-algebra, which we denote by (n + 1)[0, 1]+ W : it suffices to interpret 0i as the bottom element of the i + 1th component. It follows: Theorem 6.5. BL(n) is generated as a quasivariety by (n + 1)[0, 1]+ W , and BL(n) is complete with respect to the algebra (n + 1)[0, 1]+ , also relatively to the consequence relation.
W Unfortunately, BL(n) is not a conservative extension of BL: the equation n i=0

(x i ⇑ x i+1 ) ≤

n+1 

xi

(λn+1 )

i=0

is valid in all BL-algebras with n + 1 components but not in all BL-algebras (cf. [3]), therefore the BL-formula corresponding to (λn+1 ) is valid in all BL(n) algebras (thus it is a theorem of BL(n)), but not in all BL-algebras (thus it is not provable in BL). However, if a linearly ordered BL-algebra A is generated by elements a1 , . . . , an , then by an easy induction we see that the components of A are precisely those which a1 , . . . , an and 0 belong to, therefore A has n + 1 components at most. Now it follows from Theorem 4.3 that any linearly ordered BL-algebra with n + 1 components at most is in ISPu ((n + 1)[0, 1]W ), therefore it can be embedded in the BL-redcuct of a BL(n)-algebra. Therefore: Theorem 6.6. (i) Every n-generated BL-algebra is a subreduct of a BL(n)-algebra. (ii) If an equation of BL-algebras in n variables is not valid in BL, then it is not valid in BL(n). (iii) If A( p1, . . . , pn ) is a formula of BL in the propositional variables shown which is not provable in BL, then it is not provable in BL(n).
Thus roughly speaking, even if BL(n) is not a conservative extension of BL, we may choose n large enough in such a way that all formulas we want to deal with only contain variables among p1 , . . . , pn . When dealing with such formulas, it is safe to work in BL(n) rather than in BL. Our goal is to prove that BL(n) has uniform interpolation. To this purpose, we start from the algebraic counterpart of a result contained in [4]. Let ([0, 1]W )div,δ denote the divisible Wajsberg algebra on [0, 1] with δ, let (Wdiv )δ denote the variety of divisible Wajsberg algebras with δ, and let for every term s, sW,δ denote its realization in ([0, 1]W )div,δ . Then: Theorem 6.7 (cf. [4]). Let t ( x , y) be a term in the language of (Wdiv )δ , where x denotes a k-tuple of variables. Then, x ) and t − ( x ) such that for all a ∈ [0, 1]k one has: there are terms t + ( + tW,δ ( a ) = sup{tW,δ ( a , y) : y ∈ [0, 1]},

− tW,δ ( a ) = inf{tW,δ ( a , y) : y ∈ [0, 1]}.

Our plan is to prove a similar result for BL(n)-algebras. That is, let for every term s, s(n) denote its realization in (n + 1)[0, 1]+ W . Then we want to show: x ) and t − ( x ) such that Theorem 6.8. For every term t ( x , y) in the language of BL(n)-algebras, there are terms t + ( + k for all a ∈ ((n + 1)[0, 1]W ) , one has: + t(n) ( a ) = sup{t(n) ( a , y) : y ∈ (n + 1)[0, 1]+ W }, − t(n) ( a ) = inf{t(n) ( a , y) : y ∈ (n + 1)[0, 1]+ W }.

Before proving Theorem 6.8, note that it immediately gives uniform interpolation. Indeed, if for every term t there are terms t + and t − as described above, then the left and right interpolants of a formula A( p, q ) ( p a k-tuple of variables and q an h-tuple of variables distinct from those in p) can be constructed as follows: consider the term T A ( x , y) associated with A. Iterating the construction of t + and t − , we may obtain terms (T A )+ ( x ) and (T A )− ( x) k , one has ) such that for all a ∈ ((n + 1)[0, 1]+ W A h a ) = sup{T(n) ( a , y) : y ∈ ((n + 1)[0, 1]+ (T A )+ W ) }, (n) ( A h a ) = inf{T(n) ( a , y) : y ∈ ((n + 1)[0, 1]+ (T A )− W ) }. (n) (

Then the formulas corresponding to (T A )+ and to (T A )− are the right and the left interpolant of A respectively.

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Proof of Theorem 6.8. Our proof consists of three steps. At the first step, we show that every term in the language of BL(n)-algebras has both some kind of disjunctive normal form and some kind of conjunctive normal form. At the second step, we show that terms in normal form can be somehow reduced to terms of (Wdiv )δ . Finally, at the third step we reduce the construction of t + and t − to the analogous construction for (Wdiv )δ given in [4]. In order to do the first step, we follow the definition by cases of the interpretation of operations in an ordinal sum of divisible Wajsberg algebras. The cases occurring in our treatment are described by the following relations on (n + 1)[0, 1]+ W: x ! y iff x < y and either x and y are not in the same component, or y = 1. x ≺ y iff x < y, x, y are in the same component, and y = 1. x ≈ y iff x and y are equal. x∼ = y iff either x ≈ 1, or y ≈ 1, or x ≈ y, or x ≺ y, or y ≺ x, that is, if x and y are in the same component. x ≡ y iff x ∼ = y and x ≈ 1 iff y ≈ 1. Ci (x) (where 0 ≤ i ≤ n) iff x ≡ 0i . ∼ Note that the strict order < and the large order ≤ on (n + 1)[0, 1]+ W ), as well as =, ≡ and Ci (x), can be defined as i Boolean combinations of !, ≺ and ≈ (using also the constants 1 and 0 ). E.g., x < y is equivalent to the disjunction of x ≺ y and x ! y; x ≡ y is equivalent to the disjunction of x ≺ y, x ≈ y and y ≺ x, and so on. In the following, we use the following notation: K ≈ (x, y) denotes δ(x ⇔ y). K ! (x, y) denotes ∼ δ(x)  δ(y ⇑ x). K ≺ (x, y) denotes ∼ δ(y ⇒ x) ∼ K ! (x, y). ∼ K = (x, y) denotes δ(x)  δ(y)  K ≈ (x, y)  K ≺ (x, y)  K ≺ (y, x). ∼ K ≡ (x, y) denotes K = (x, y)  (δ(x) ⇔ δ(y)). For i = 0, . . . , n, K Ci (x) denotes K ≡ (x, 0i ). The following lemma is almost immediate: % (a, b) = 1 if a % b, and Lemma 6.9. For % ∈ {≈, !, ≺, ∼ =, ≡} and for a, b ∈ ((n + 1)[0, 1]W )+ , one has: K (n) % (a, b) = 0 otherwise. Moreover for a ∈ (n + 1)[0, 1]+ and for i = 0, . . . , n, one has: K C i (a) = 1 if C (a) and K (n) i W (n)

Ci K (n) (a) = 0 otherwise.


We continue the proof of Theorem 6.8. Let for every term t, Sub(t) denote the set of all subterms of t, plus the set of all constants of BL(n). We define a condition to be an expression of the form s % u, where s and u are terms and % ∈ {!, ≺, ≈}. A condition of the form s % u where s, u ∈ Sub(t) will be called a t-condition. A condition of the form x ! y will be called external, and a condition of the form x ≺ y or x ≈ y will be called internal. A set S of t-conditions is said to be t-saturated if there is an evaluation e such that S is precisely the set of all t-conditions which are true in (n + 1)[0, 1]+ W under the evaluation e. We denote by Sat (t) the family of all t-saturated sets of conditions. Clearly, Sat (t) is finite and its elements are in turn finite. If S is a set of conditions and C is either a condition or a Boolean combination C of conditions, we write S |= C to mean that every evaluation which satisfies S satisfies C. Thus e.g. if S ∈ Sat (t) and s, u ∈ Sub(t), then S |= s ≡ u iff one of s ≈ u, s ≺ u or u ≺ s is in S. Let S ∈ Sat (t). We define for all w ∈ Sub(t), a term w S ∈ Sub(t), which intuitively represents a simplification of w when all conditions in S are satisfied. The definition is by induction on w as follows: If w is a constant, then w S = w. If w is a variable, then distinguish the following cases: if w ≈ c is in S for some constant term c, then w S = c; otherwise, w S = w. If w = s · u, then, distinguish the following cases: if s ≈ 1 and u ≈ 1 are in S, then w S = 1; if s ! u is in S, then w S = s S ; if u ! s is in S, then w S = u S ; otherwise w S = s S · u S . If w = s ⇒ u, then distinguish the following cases: if S |= s ≤ u, then w S = 1; if u ! s is in S, then w S = u S ; otherwise (that is, if u ≺ s is in S), w S = s S ⇒ u S . If w = δ(s), then w S = 1 if s ≈ 1 is in S, and w S = 0 otherwise. If w = dn (s) and s ≈ 1 is in S, then w S = 1, otherwise, w S = dn (s S ).

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If w = χ(s), distinguish the following cases: if s ≈ 1 is in S, then w S = 1; otherwise, there is a unique i such that S |= Ci (s). Then let w S = χ(0i ). Lemma 6.10. Let S ∈ Sat (t), and let S− be the set of all conditions in S which are of the form s % u, where % ∈ {≈, ≺, !}, and s, u are either both atomic terms or both proper subterms of t. (i) For every s ∈ Sub(t), S− |= s ≈ s S . (ii) If x is a variable or a constant occurring in s S , then S− |= x ≡ s S . Proof. Both claims are easily proved by induction on s. If s is atomic, both claims are obvious. We check the induction steps corresponding to ⇒, to · and to χ, and we leave the remaining cases to the reader. Suppose that s = u ⇒ w. Then note that u and w are both proper subterms of t, therefore exactly one of w ≈ u, w ≺ u, u ≺ w, w ! u or u ! w is in S− . We distinguish the following cases: if S− |= u ≤ w, then S− |= s ≈ 1. Moreover, s S = 1. This proves (i) and also (ii), as there are no variables in s S . If w ! u is in S− , then S− |= s ≈ w. Moreover, s S = w S , and by the induction hypothesis S− |= w ≈ w S . It follows that S− |= s S ≈ s. This shows (i). Moreover, since s S = w S , by the induction hypothesis, if x is a variable or a constant in s S , then S− |= x ≡ s S . This gives (ii). Finally, if none of the cases above occurs, then w ≺ u is in S− . In this case, using the induction hypothesis, we derive that S− |= s ≈ u S ⇒ w S , and finally S− |= s ≈ s S . This shows (i). Moreover, since w ≺ u is in S− , S− |= s ≡ w and S− |= s ≡ u. Now let x be a variable or a constant in s S . Since S− |= u ≡ w, we can assume without loss of generality that x occurs in u S . Then by the induction hypothesis, S− |= x ≡ u S . Since S− |= u S ≡ s S , we conclude that S− |= x ≡ s S . This proves (ii). Next suppose that s = u · w. Once more, u, w are proper subterms of t, and exactly one of w ≈ u, w ≺ u, u ≺ w, w ! u or u ! w is in S− .We distinguish the following cases: if u ! w is in S− , then, then S− |= s ≈ u. Moreover, s S = u S , and by the induction hypothesis S− |= u ≈ u S . Thus (i) holds. As regards to (ii), if x is a variable or a constant occurring in s S (hence in u S ), then by the induction hypothesis, S− |= x ≡ u S . This implies (ii). The case where w ! u is in S− is symmetric. If none of the above conditions holds, then S− |= u ≡ w. Thus by the induction hypothesis we derive that S− |= s ≈ u S · w S , and finally S− |= s ≈ s S . This shows (i). Now let x be a variable or a constant in s S . Assume e.g. that x is in u S . Then by the induction hypothesis, S− |= x ≡ u S . Since S− |= u S ≡ s S , (ii) holds. Now suppose s = χ(u). Then, u is a proper subterm of t, and either S− |= u ≈ 1, or S− |= Ci (u) for some i ≤ n. In the first case s S = 1 and S− |= s ≈ 1, therefore the claim is obvious. If S− |= Ci (u), then s S = χ(0i ). Moreover S− |= χ(u) ≈ χ(0i ) and (i) follows. Since in this case s S has no variables, (ii) is obvious.
We now prove that any S ∈ Sat (t) is semantically equivalent to a simpler set of conditions. Lemma 6.11. Let S ∈ Sat (t), and let S be the set of all conditions C such that either C ∈ S and C has the form x ! y with x, y variables or constants, or C has the form w S % s S for some w, s ∈ Sub(t) such that w % s is in S and % ∈ {≺, ≈}, or C is of the form x ≈ c S where x ≈ c is in S, x is a variable and c a closed term. Then S is equivalent to S (i.e., S and S are satisfied by the same evaluations). Proof. By Lemma 6.10, we can easily derive that S |= C for every condition C in S . Thus we only need to prove that S |= C for every condition C in S. Let for every term s, K 1 (s) denote the number of symbols in s. Moreover, let K 2 (u) = K 1 (u) if u = u S , and K 2 (u) = 1 if u = u S . If C is a condition of the form u ! v, then we define K (C) = 2 max{K 1 (u), K 1 (v)} if K 1 (u) = K 1 (v), and K (C) = 2K 1 (u) + 1 if K 1 (u) = K 1 (v); if C is a condition of the form u ≺ v or u ≈ v, then we define K (C) = 2 max{K 2 (u), K 2 (v)} if K 2 (u) = K 2 (v), and K (C) = 2K 2 (u)+1 if K 2 (u) = K 2 (v). Note that K (C) ≥ 3 for every condition C. Let for any set T of conditions, a(T ) = max{K (C) : C ∈ T }, and let b(T ) be the number of conditions C ∈ T with K (C) = a(T ). We prove that if T is a set of conditions for which there is a term w such that T is equivalent to a set U ∈ Sat (w), then for every condition C in T , U |= C. This will clearly suffice. We argue by induction on (a(T ), b(T )) with respect to the lexicographic order. If a(T ) = 3, then T ⊆ U , and the claim is trivial. Now suppose a(T ) > 3. Let C be a condition with K (C) = a(T ). Distinguish the following cases: (a) C is u ⇒ w ! z, with K 1 (u ⇒ w) ≥ K 1 (z). Then either (a1): U |= C1 and U |= C2 , where C1 is w ! u and C2 is w ! z, or (a2): U |= C2 and U |= C3 , where C3 is w ≺ u. In case (a1), replace C by C1 and C2 , and

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in case (a2) replace C by C2 and C3 . Note that C1 and C2 , as well as C2 and C3 imply C, and that for i = 1, 2, 3, K (Ci ) < K (C). Thus the claim follows from the induction hypothesis. (b) C is z ! u ⇒ w, with K 1 (u ⇒ w) ≥ K 1 (z). Then either (b1): U |= C1 and U |= C2 , where C1 is one of u ≈ w, u ≺ w or u ! w, and C2 is z ! 1, or (b2): U |= C3 and U |= C4 , where C3 is one of w ≺ u or w ! u, and C4 is z ! w. In case (a1), replace C by C1 and C2 , and in case (a2) replace C by C3 and C4 . Note that C1 and C2 , as well as C3 and C4 imply C, and that for i = 1, 2, 3, 4, K (Ci ) < K (C). Once more, the claim follows from the induction hypothesis. (c) C is u · w ! z, with K 1 (u · w) ≥ K 1 (z). Then either U |= C1 or U |= C2 , where C1 is u ! z and C2 is w ! z. Accordingly, replace C by C1 or by C2 respectively, and apply the induction hypothesis. (d) C is z ! u · w, with K 1 (u · w) ≥ K 1 (z). Then replace C by z ! u and z ! w and apply the induction hypothesis. (e) C is u 0 ! u 1 , where for some i ∈ {0, 1} we have K (u i ) ≥ K (u 1−i ), and u i is either χ(z) or dn (z) for some n. Then replace C by the condition C  obtained by substituting u i by z. (f) C is δ(u) ! v, where K 1 (δ(u)) ≥ K 1 (v). Then U |= u ! 1 and U |= 0 ! v, therefore we may replace C by these conditions and apply the induction hypothesis. (g) C is u ! δ(v). Then U |= v ≈ 1 and U |= u ! 1, therefore we may replace C by these conditions and apply the induction hypothesis. (h) C is of the form u ≺ v or u ≈ v. Suppose without loss of generality that K 2 (u) ≤ K 2 (v). Since we are assuming that a(T ) > 3, v is not an atomic term. Let U− (v) be the set of all conditions s % w in U , where % ∈ {≈, ≺, !}, s, w ∈ Sub(v) and either s, w are both atomic, or s, w are both proper subterms of v. Then by Lemma 6.10, U− (v) |= v ≈ v S . Now let C1 be obtained from C by replacing v by v S . Then K (C1 ) < K (C), and for every C  ∈ U− (v), K (C  ) < K (C). Thus we may substitute C by C1 and by all conditions in U − (v), and then we may apply the induction hypothesis.
We continue the proof of Theorem 6.8. Let S ∈ Sat (t), and let S be as in Lemma 6.11. For every condition

C ∈ SC C % S of the form s % w, where % ∈ {≈, ≺, !}, let K denote the term K (s, w) (cf. Lemma 6.9). Let K = C∈S K . Then Lemmas 6.9–6.11, we can write any term t in the language of BL(n)-algebras either as  (K S  t S ) (disjunctive normal form of t) tD = S∈Sat (t )

or as tC =



(∼ K S  t S )

(conjunctive normal form of t).

S∈Sat (t )

Note that by Lemma 6.10, if u is a variable or a constant in t S , then S |= u ≡ t S . Moreover, δ does not occur in t S , and there are no nested occurrences of δ, either in t D or in t C . Thus we have obtained the desired normal forms, and the first step of the proof of Theorem 6.8 is concluded. k Now let S ∈ Sat (t), and let for all a ∈ ((n + 1)[0, 1]+ W) , S( a ) = {b ∈ (n + 1)[0, 1]+ a , b) satisfies S}.1 W ) : ( k Then our next goal is to define two terms t S+ ( x ) and t S− ( x ) such that for all a ∈ ((n + 1)[0, 1]+ W) ,

(t S+ )(n) ( a ) = sup{t(n) ( a , b) : b ∈ S( a )},

(t S− )(n) ( a)

(6.7.1)

= inf{t(n) ( a , b) : b ∈ S( a )}

(if S( a ) = ∅, then (t S+ )(n) ( a ) = 0 and (t S− )(n) ( a ) = 1). Clearly, if we reach this goal, then the terms t + ( x ) and t − ( x) satisfying Theorem 6.8 can be defined as  x) = {t S+ ( x ) : S ∈ Sat (t)} and t − ( x) = {t S− ( x ) : S ∈ Sat (t)}. t + ( 1 That is, b ∈ S( a ) iff every condition C in S is satisfied by any evaluation e such that e( x ) = a and e(y) = b.

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The rest of the proof is devoted to the definition of the terms t S+ and t S− and to the proof that for all a ∈ S( a ) they satisfy conditions (6.7.1). Our plan is to reduce the construction of t S+ and t S− to the construction of analogous terms for (Wdiv )δ whose existence is granted by Theorem 6.7. To this purpose, in the second step of our proof we introduce some translations of BL(n) terms into (Wdiv )δ terms and vice versa. For every BL(n) term s, we define a (Wdiv )δ term s W by induction as follows: if s is either a variable or 0 or 1, then W s = s; for i = 1, . . . , n, (0i )W = 0; for every binary operation ◦, (u◦w)W = u W ◦w W ; (dn (u))W = δ(u W )dn (u W ); (δ(u))W = δ(u W ); (χ(u))W = d2 (1)  δ(u W ). Conversely, define for i = 0, . . . , n and for every (Wdiv )δ term s, a term s (i) as follows: if s is a variable or 1, then s (i) = s; 0(i) = 0i ; for every binary operation ◦, (u ◦ w)(i) = u (i) ◦ w(i) ; (dn (u))(i) = (δ(u)  (2)dn (χ(0i )))  (∼ δ(u)  dn (u (i) )); (δ(u))(i) = δ(u (i) )  0i . Now let for every real number α ∈ [0, 1] and for i = 0, . . . , n, (α)i denote the copy of α in the i + 1th component of ((n + 1)[0, 1]+ W . For every W in (([0, 1] ) ) letting for every variable x and for every , define an evaluation e evaluation e in (n + 1)[0, 1]+ W div δ W α ∈ [0, 1), e W (x) = 1 iff e(x) = 1 and e W (x) = α iff there is an i ≤ n such that e(x) = (α)i . Conversely, for every evaluation e in (([0, 1]W )div )δ , define an evaluation e(i) in (n + 1)[0, 1]+ W letting for every variable x and for every (i) (i) α ∈ [0, 1), e (x) = 1 iff e(x) = 1 and e (x) = (α)i if e(x) = α < 1. Say the term w is of type i if δ does not occur in w, and w does not contain occurrences of constants different from 0i and 1. Then we can prove: Lemma 6.12. Let 0 ≤ i ≤ n, let s(x 1 , . . . , xr ) be a (Wdiv )δ term, and let u(y1 , . . . , ys ) be a BL(n) term of type i . Let e be an evaluation in (([0, 1]W )div )δ , and let h be an evaluation in (n + 1)[0, 1]+ W such that for j = 1, . . . , s, either i e(y j ) = 1 or e(y j ) ≡ 0 . Then: (i) e(i) (s (i) ) = 1 iff e(s) = 1, and for all α ∈ [0, 1), e(i) (s (i) ) = (α)i iff e(s) = α. (ii) h W (u W ) = 1 iff h(u) = 1, and for α ∈ [0, 1), h W (u W ) = α iff h(u) = (α)i . Proof. (i) Induction on s. If s is an atomic term, the claim is trivial. If s is a term of the form s1 ◦ s2 , where ◦ is a binary operation, the claim follows from the definition of · and ⇒ in an ordinal sum. The induction step corresponding to δ is trivial. Assume that s = dn (w). If e(w) = 1, e(s) = n1 . Moreover, e(wi ) = 1 by the induction hypothesis, therefore ei (∼ δ(wi )) = 0, and ei (s i ) = ei (δ(wi ))  ei ((2)dn (χ(0i ))), that is, ei (s i ) = ( n1 )i . If e(w) = α = 1, then e(dn (w)) = αn . Moreover, e(wi ) = (α)i by the induction hypothesis, therefore ei (δ(wi )) = 0, and ei (s i ) = ei (∼ δ(wi )  dn (w)), that is ei (s i ) = ( αn )i . (ii) Since δ does not occur in u, by an easy induction we see that for every subterm w of u, either h(w) ≡ 0i , or h(w) = 1. Now the proof proceeds by induction on u. The only non-trivial cases are the induction steps corresponding to χ and to dn . Suppose u = χ(w). If h(w) = 1, then h(u) = 1. Moreover by the induction hypothesis, h W (w W ) = 1, therefore, h W (δ(w W )) = 1, and finally h W (u W ) = h W (d2 (1)  δ(w W )) = 1. If h(w) = 1, then h(w) = (α)i for some α ∈ [0, 1), and h(u) = ( 12 )i . By the induction hypothesis, h W (w W ) = α = 1, therefore h W (δ(w W )) = 0, and finally h W (u W ) = h W (d2 (1)) = 12 . Finally, suppose u = dn (w). Once again, if h(w) = 1, then h(u) = 1. Moreover by the induction hypothesis, h W (w W ) = 1, therefore, h W (δ(w W )) = 1, and finally h W (u W ) = h W (d2 (1)  δ(w W )) = 1. If h(w) = 1, then h(w) = (α)i for some α ∈ [0, 1), and h(u) = ( αn )i . By the induction hypothesis, h W (w W ) = α = 1, therefore h W (δ(w W )) = 0, and finally h W (u W ) = h W (dn (w W )) = αn .
Recalling that (δ(s))W = δ(s W ), as a corollary of Theorem 6.12, we obtain: Corollary 6.13. Let s be a term of type i , and let w be a Boolean combination of terms of the form δ(s1 ), . . . , δ(sn ), where each si is of type i . Let e be any evaluation in (n + 1)[0, 1]+ W such that for every variable x in s or in w, either e(x) = 1 or e(x) ≡ 0i . The following properties hold: (a) if e W (w) = 1 then e(w) = 1, otherwise, e(w) = 0; (b) if either u = w  s or u =∼ w  s, then e(u) = 1 iff e W (u W ) = 1, and e W (u W ) = α ∈ (0, 1) iff e(u) = (α)i ; (c) if e W (w W ) = 0, then e(w  s) = 0 and e(∼ w  s) = 1, and if e W (w W ) = 0 and e W (s W ) = 0, then e(w  s) = e(∼ w  s) = 0i .
Thanks to Lemma 6.12 and Corollary 6.13 we can somehow translate terms of BL(n) into terms of (Wdiv )δ and vice versa. Thus the second step is concluded. In the next step we use the translations above in order to obtain the desired terms t S+ and t S− from the analogous terms for (Wdiv )δ whose existence is ensured by Theorem 6.7. This third step is covered by the following lemma, which will conclude the proof of Theorem 6.8:

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Lemma 6.14. For any BL(n) term t ( x , y) and for any S ∈ Sat (t), there are terms t S+ ( x ) and t S− ( x ) such that for all + k a ∈ ((n + 1)[0, 1]W ) the formulas (6.7.1) hold. Proof. We distinguish the following cases: Case (a): for some constant term c, y ≈ c is in S. Then it is sufficient to define t S+ ( x ) = K S ( x , c)  t S ( x , c), and − + k S S S t S ( x ) =∼ K ( x , c)  t ( x , c), because for a ∈ ((n + 1)[0, 1]W ) , one has: S( a ) = ∅ iff (K )(n) ( a , c) = 1, and in S a , y) is constantly equal to t(n) ( a , c). this case, when y ranges over S( a ), t(n) ( S Case (b): case (a) does not apply, and y does not occur in t . Then y ! 1 is in S, therefore there is exactly one i ≤ n such that S |= y ≡ 0i . Let S x denote the set of all conditions in S in which y does not occur,

and let S y denote the x C y x ) = C∈S x K , and K ( x , y) = C∈S y K C . Note that set of internal conditions in S in which y occurs. Let K ( x the external conditions involving y are uniquely determined by S and by the component which y belongs to. Indeed, assume that S |= y ≡ 0i . Then one of y ≈ 0i or 0i ≺ y is in S y , and for any variable or constant term z ∈ Sub(t), one has: S |= z ! y iff z ! 0i is in S x , and S |= y ! z iff 0i ! z is in S x . In other words, S x and S y contain enough information to determine all conditions involving y. Moreover by Lemma 6.11, if z is a variable or a constant occurring in a condition C in S y , then S |= z ≡ 0i . Thus in general K y does not contain all variables x . Note that for k a ∈ ((n + 1)[0, 1]+ W ) we have y

x S sup{t(n) ( a , y) : y ∈ S( a )} = K (n) ( a )  sup{K (n) ( a , y) : y ≡ 0i }  t(n) ( a ), y

x S a , y) : y ∈ S( a )} =∼ K (n) ( a ) ∼ sup{K (n) ( a , y) : y ≡ 0i }  t(n) ( a ). inf{t(n) ( y

x ( If K (n) a ) = 1 and there is y ≡ 0i such that K (n) ( a , y) = 1, then S( a ) = ∅, and S sup{t(n) ( a , y) : y ∈ S( a )} = inf{t(n) ( a , y) : y ∈ S( a )} = t(n) ( a ).

Otherwise, S( a ) = ∅, sup{t(n) ( a , y) : y ∈ S( a )} = 0 and inf{t(n) ( a , y) : y ∈ S( a )} = 1. In order to express the desired supremum and infimum by means of terms, note that K y ( x , y) is a Boolean combination of terms of the form i δ(w) with w of type i . Hence, by Corollary 6.13(a), for every evaluation e in (n + 1)[0, 1]+ W such that e(z) ≡ 0 for y y W y W y every variable z occurring in K ( x , y), one has e(K ( x , y)) = 1 if e ((K ) ( x , y)) = 1, and e(K ( x , y)) = 0 otherwise. By Theorem 6.7, there is a term (K y )+ ( x ) expressing sup{(K y )W ( x , y) : y ∈ [0, 1]} in (([0, 1]W )div )δ . k x ) = δ(((K y )+ )(i) ( x )), let a ∈ ([0, 1]+  W ∈ [0, 1]k be defined as a W = e W ( x ), where Now let K + ( W ) , and let a + W ) = 1 iff there is e is any evaluation such that e( x ) = a . By Corollary 6.13, we have K (n) ( a ) = 1 iff (K y )+ ( a W,δ y y ∈ [0, 1] such that ((K y )W )W,δ ( a , y) = 1. Since S |= y ≡ 0i , 1 ∈ / S( a ), therefore K (n) ( a , 1) = 0. + So, K (n) ( a ) = 1 iff there is y ∈ [0, 1) such that ((K y )W )W,δ ( a , y) = 1. By Corollary 6.13, this is the case iff y + + x ( there is y ≡ 0i such that K (n) ( a , y) = 1. Otherwise, K (n) ( a ) = 0. Hence K (n) a )  K (n) ( a ) = 1 iff S( a ) = ∅, and + x K (n) ( a )  K (n) ( a ) = 0 otherwise. Now let t S+ ( x ) = K x ( x )  K + ( x )  t S ( x) and t S− ( x ) =∼ K x ( x ) ∼ K + ( x )  t S ( x ). k a ) = ∅, then We prove that t S+ and t S− satisfy the requirements of Lemma 6.14. Let a ∈ ((n+1)[0, 1]+ W ) be given. If S( + x sup{t(n) ( a , y) : y ∈ S( a )} = 0 and inf{t(n) ( a , y) : y ∈ S( a )} = 1. On the other hand, since K (n) ( a )  K (n) ( a ) = 0, we have (t S+ )(n) ( a ) = 0 and (t S− )(n) ( a ) = 1. Thus the claim holds in this case. If S( a ) = ∅, then S a , y) : y ∈ S( a )} = inf{t(n) ( a , y) : y ∈ S( a )} = t(n) ( a ). sup{t(n) ( + x ( a ) = K (n) ( a ) = 1, we have Moreover, since K (n) S (t S+ )(n) ( a ) = (t S− )(n) ( a ) = t(n) ( a ),

and once again the claim holds. So, case (b) is complete.

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Case (c): case (a) does not apply and y occurs in t S . Then by Lemma 6.11, for every variable or constant term z occurring in t S we have that S |= z ≡ t S . Since we have excluded case (a), y ! 1 is in S, therefore S |= t S ! 1. Let i k x ), K y ( x , y) and K + ( x ) be as in case (b). Note that for every a ∈ ((n + 1)[0, 1]+ be such that S |= y ≡ 0i . Let K x ( W) + x y S one has that S( a ) = ∅ iff K (n) ( a )  K (n) ( a ) = 1. Now consider the terms s( x , y) = K ( x , y)  t ( x , y) and y S w( x , y) =∼ K ( x , y)  t ( x , y). Once more, if z is a variable or constant occurring in s( x , y) or in w( x , y), then S |= z ≡ y. Let u( x , y) be any of s( x , y) or w( x , y). By Corollary 6.13, for every evaluation e such that e(z) ≡ 0i for every variable z in u( x , y), we have: (a) e(u( x , y)) = 1 iff e W (u W ( x , y)) = 1; (b) e(u( x , y)) = (α)i with W W W y W α = 0 iff e (u ( x , y)) = α; (c) if e ((K ) ( x , y)) = 0, then e(s( x , y)) = 0 and e(w( x , y)) = 1; (d) if x , y)) = 0 and e W ((t S )W ( x , y)) = 0, then e(u( x , y)) = 0i . e W ((K y )W ( By Theorem 6.7, there are terms s + ( x ) and w− ( x ) expressing sup{s W ( x , y) : y ∈ [0, 1]} and inf{w W ( x , y) : y ∈ [0, 1]} respectively. If i < n, then define x ) = K x ( x )  K + ( x )  (s + )(i) ( x )  0i+1 , t S+ ( and if i = n, then define x ) = K x ( x )  K + ( x )  (s + )(i) ( x ). t S+ ( Moreover, define x ) =∼ K x ( x ) ∼ K + ( x )  (w− )(i) ( x ). t S− ( k x ) and t S− ( x ) meet our requirements. Let a ∈ ((n + 1)[0, 1]+ We claim that t S+ ( W ) . Distinguish the following cases: + x Subcase (c1): either K (n) ( a ) = 0 or K (n) ( a ) = 0. Then S( a ) = ∅, sup{t(n) ( a , y) : y ∈ S( a )} = 0 and + + − x inf{t(n) ( a , y) : y ∈ S( a )} = 1. Moreover, K (n) ( a )  K (n) ( a ) = 0, therefore (t S )(n) ( a ) = 0 and (t S )(n) ( a ) = 1. + x Subcase (c2): K (n) ( a ) = K (n) ( a ) = 1 and i < n. Then S( a ) = ∅. Furthermore, S a , y) : y ∈ S( a )} = sup{t(n) ( a , y) : y ∈ S( a )} = sup{s(n) ( a , y) : y ≡ 0i }, sup{t(n) ( S a , y) : y ∈ S( a )} = inf{t(n) ( a , y) : y ∈ S( a )} = inf{w(n) ( a , y) : y ≡ 0i }. inf{t(n) ( S Note that for all y ∈ S( a ), t(n) ( a , y) ≡ 0i , therefore S ( a , y) : y ∈ S( a )} ≤ sup{t S ( a , y) : y ∈ S( a )} ≤ 0i+1 . 0i ≤ sup{t(n)

Now let a W be defined as in Case (b), and let α = sup{(s W )W,δ ( a W , y) : y ∈ [0, 1]} = (s + )W,δ ( a W ), β = inf{(w W )W,δ ( a W , y) : y ∈ [0, 1]} = (w− )W,δ ( a ). Note that β < 1, because there is y ≡ 0i , such that w(n) ( a , y) ≡ 0i . Thus by Corollary 6.13, inf{t(n) ( a , y) : y ∈ S( a )} = inf{w(n) ( a , y) : y ≡ 0i } (i)

a ) = (t S− )(n) ( a ). = (β)i = (w− )(n) ( Moreover, if α = 1, then by Corollary 6.13, we have sup{t(n) ( a , y) : y ∈ S( a )} = sup{s(n) ( a , y) : y ≡ 0i } (i)

a ) = (t S− )(n) ( a ). = (α)i = (s + )(n) ( a , y) : y ∈ S( a )} = sup{s(n) ( a , y) : y ≡ 0i } = 0i+1 , and If α = 1, then by Corollary 6.13 again, we have sup{t(n) ( (i) + + i+1 i+1 a ) = (s )(n) ( a)  0 =0 . (t S )(n) ( + x Subcase (c3): K (n) ( a ) = K (n) ( a ) = 1 and i = n. This subcase is treated as subcase (c2), the only difference being that if i = n, then the supremum of all y such that y ≡ 0i is 1 and not 0i+1 (which does not make sense if i = n). Therefore we have to omit 0i+1 in the expression for t S+ .


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It is clear that Lemma 6.14 implies Theorem 6.8, so its proof is concluded.
As a consequence we obtain: Theorem 6.15. For every n > 0, BL(n) has uniform interpolation and therefore it has the implicative Beth property.
7. Open problems We list some problems that we have not been able to solve in the present paper. Does Wdiv have the strong amalgamation property? Same question for BLdiv . Characterize the extensions of BL in the language of BL which have deductive interpolation. We have seen that there are uncountably many extensions without the deductive interpolation. Are there uncountably many extensions with deductive interpolation? Does B L div have Craig’s interpolation? Does B L div have the deductive Beth property? We have seen that a schematic extension L of BL in the language of BL has the deductive Beth property iff it has the implicative Beth property, because both properties hold in L iff L extends G. Is there any extension of BL in a richer language which has the deductive Beth property but not the implicative Beth property or vice versa? Find reasonable algorithms for constructing deductive interpolants in BL and uniform interpolants in BL(n). What is the minimal complexity of such algorithms? Does M T L (the monoidal t-norm logic, cf. [15]) have deductive interpolation? Acknowledgments Almost all we know about interpolation and amalgamation is due to very stimulating conversations with (in alphabetical order) Mathias Baaz, Dov Gabbay, Andrew Glass, Daniele Mundici and Hiroakira Ono. Thus we wish to thank all these people very warmly. We are also indebted to Silvio Ghilardi, to Peter Jipsen and to Larisa Maksimova (and we are indebted twice to Andrew Glass, to Daniele Mundici and to Hiroakira Ono) for showing us some very useful references on this subject. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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