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The logic of the strongest and the weakest t-norms
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The logic of the strongest and the weakest t-norms Matteo Bianchi Department of Computer Science, Università degli Studi di Milano, Via Comelico 39/41, 20135, Milano, Italy Received 23 July 2014; received in revised form 14 January 2015; accepted 16 January 2015
Abstract It is well known that the strongest t-norm, that is the largest with respect to the pointwise order, is the minimum. In 2001, the logic MTL was introduced as the base of a framework of many-valued logics, and in 2002 it was shown that it is the logic of all left-continuous t-norms and their residua. Within this family of logics, the many-valued logic associated with the minimum t-norm is the Gödel one, whilst there is no logic associated to the drastic product t-norm. Indeed the drastic product is not left-continuous, and hence it does not have a residuum. However, in a recent paper the logic DP has been studied, by showing that the monoidal operation of every DP-chain is like the drastic product t-norm. In this paper we present the logic EMTL, whose algebraic variety is the smallest to contain the ones of Gödel- and DP-algebras. We show that the chains in this algebraic variety are exactly all the Gödel- and DP-chains, we classify and axiomatize all the subvarieties, and we show some limitative results concerning the amalgamation property. © 2015 Elsevier B.V. All rights reserved. Keywords: Many-valued logics; Gödel-logic; Drastic product logic; Drastic product t-norm; Minimum t-norm; Amalgamation property; Residuated lattices
1. Introduction and motivations Triangular norms, t-norms for short, were introduced by Menger in [25], and subsequently studied by Schweizer, Sklar and other people in the area of probabilistic metric spaces: see [30,24] for a historical overview. Also, many connections between logic and t-norms were pointed out, in particular after the introduction of fuzzy sets by Zadeh [34]: from then a lot of research has been done, concerning both the theory and the applications. At the end of nineties, in [19], Hájek introduced the basic logic BL, as the base of a large framework of many-valued logics, strictly connected with residuated lattices [17] and t-norms. In [8] it was shown that BL is the logic of all continuous t-norms and their residua, in the sense that every continuous t-norm and the associated residuum induce an algebraic structure, called standard BL-algebra, and BL is complete w.r.t. the class of standard BL-algebras. In [14] a weaker logic than BL, MTL, was introduced, and in [23] it was shown that MTL is the logic of all left-continuous t-norms and their residua. The acronym MTL stands for monoidal t-norm based logic: indeed MTL can be axiomatized as an extension E-mail address: [email protected]. URL: http://www.mat.unimi.it/users/mbianchi/. http://dx.doi.org/10.1016/j.fss.2015.01.013 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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of the Höhle’s monoidal logic ML [20,14] or as an extension of the Full Lambek calculus FL [17,14]. MTL is, among the substructural logics, the weakest many-valued logic to be t-norm based: indeed, a t-norm admits a residuum iff it is left-continuous [4]. In this framework of logics is also contained the well-known Gödel–Dummett logic, G: it was briefly introduced by Kurt Gödel in [18], and subsequently axiomatized by M. Dummett in [12]. This logic is complete w.r.t. the algebra induced by the minimum t-norm [19]. By using the terminology of [24, Definition 1.4] a t-norm ∗ is stronger (weaker) than a t-norm ∗ if, for every x, y ∈ [0, 1], x ∗ y ≥ x ∗ y (x ∗ y ≤ x ∗ y). Since the minimum is known to be the strongest t-norm [24], we can say that G is the logic of the strongest t-norm. The weakest t-norm (see [24]) is instead drastic product ∗D , defined as x ∗D y = 0, if x, y < 1, and x ∗D y = min{x, y} otherwise: however, there is no axiomatic extension of MTL that is ∗D -based, i.e. that is complete w.r.t. an MTL-algebra having ∗D as monoidal operation, and [0, 1] as carrier. This happens because ∗D is not left-continuous, and does not have a residuum. In [22] a modification to ∗D has been introduced, in the way to obtain a left-continuous t-norm: the author presented a family of parametrized isomorphic t-norms, called revised drastic product. In [31] the corresponding logic has been studied, and was called RDP. In [29,21] the logic S3 MTL was introduced, and in [2] it was renamed in DP. The reason for the new name is that, even if DP is not t-norm based, the monoidal operation ∗ of every DP-chain is such that x ∗ y = 0, if x, y < 1, and x ∗ y = min{x, y} otherwise, as it happens for ∗D . So, we can say that DP is the logic of the weakest t-norm. In this paper we introduce the logic EMTL, whose corresponding variety is the smallest to contain the ones of Gödel- and DP-algebras. In particular, we show that every EMTL-chain is a DP-chain or a Gödel chain, and hence we can say that EMTL is the logic of the strongest and the weakest t-norms. We now describe the structure of the paper. After some basic background in Section 2, in Section 3 we introduce EMTL, by showing that the corresponding variety EMTL is generated by all the Gödel- and DP-algebras: in particular, the class of EMTL-chains coincides with all the Gödel- and DP-chains. We also discuss the relation between EMTL and other varieties, by showing also that it cannot be axiomatized with equations with only a variable from MTL. In Section 4 we study which classes of chains generates EMTL, and we classify and axiomatize all its subvarieties. Finally, in Section 5 we show some positive and negative results concerning the amalgamation property, for EMTL and its subvarieties. 2. Preliminaries We assume that the reader is acquainted with many-valued logics as developed by P. Hájek: see [19,14,10] for details. In particular we focus on some axiomatic extensions of the monoidal t-norm based logic MTL, firstly introduced by Esteva and Godo in [14], and proved to be logic of all left-continuous t-norm and their residua in [23]. 2.1. Syntax The language of MTL is based over the connectives {∧, &, →, ⊥}: the formulas are built in the usual inductive way from these connectives, and a denumerable set VAR of variables. The set of formulas is called FORM. Useful derived connectives are the following: def
¬ϕ = ϕ → ⊥ def ϕ ∨ ψ = (ϕ → ψ) → ψ ∧ (ψ → ϕ) → ϕ def
= ¬⊥
(negation) (disjunction) (top)
MTL can be axiomatized with a Hilbert style calculus: for the reader’s convenience, we list the axioms of MTL: (ϕ → ψ) → (ψ → χ) → (ϕ → χ) (A1) (ϕ&ψ) → ϕ
(A2)
(ϕ&ψ) → (ψ&ϕ)
(A3)
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(ϕ ∧ ψ) → ϕ
(A4)
(ϕ ∧ ψ) → (ψ ∧ ϕ) ϕ&(ϕ → ψ) → (ψ ∧ ϕ) ϕ → (ψ → χ) → (ϕ&ψ) → χ (ϕ&ψ) → χ → ϕ → (ψ → χ) (ϕ → ψ) → χ → (ψ → ϕ) → χ → χ
(A5)
⊥→ϕ
(A6) (A7a) (A7b) (A8) (A9)
As inference rule we have modus ponens: ϕ ϕ→ψ (MP) ψ An axiomatic extension of MTL is a logic obtained by adding one or more axiom schemata to it. A theory is a set of formulas: the notion of proof and logical consequence are defined as in the classical case. In this paper we focus on the following axiomatic extensions of MTL (EMTL will be introduced in Section 3): C3 MTL, G, DP, GHP, WNM, RDP [7,2,14,1,19,31]. The first five are axiomatized as MTL plus, respectively, the following axioms: (ϕ&ϕ)&ϕ ↔ ϕ. (C3 ) (ϕ&ϕ) ↔ ϕ.
(ID)
¬(ϕ&ϕ) ∨ ϕ. ¬(ϕ&ϕ) ∨ ϕ → (ϕ&ϕ) . ¬(ϕ&ψ) ∨ (ϕ ∧ ψ) → (ϕ&ψ) .
(DP) (GHP) (WNM)
RDP is axiomatized as WNM plus: (ϕ → ¬ϕ) ∨ ¬¬ϕ.
(RDP)
2.2. Semantics The semantics related to MTL and its axiomatic extensions is given by particular algebraic structures, called MTLalgebras. This class of algebras forms a variety: see [5] for details about algebraic varieties, and any unexplained notion of universal algebra. Definition 1. An MTL-algebra A is an algebra A = A, ∗, ⇒, , , 0, 1 such that: 1. 2. 3. 4.
A, , , 0, 1 is a bounded lattice with minimum 0 and maximum 1.
A, ∗, 1 is a commutative monoid.
∗, ⇒ forms a residuated pair: z ∗ x ≤ y iff z ≤ x ⇒ y for all x, y, z ∈ A. The following axiom holds for all x, y ∈ A: (x ⇒ y) (y ⇒ x) = 1
(Prelinearity)
A totally ordered MTL-algebra is called MTL-chain. An MTL-algebra is called standard whenever its carrier is [0, 1]. In the rest of the paper the notation ∼ x will denote x ⇒ 0. Given an MTL-algebra, a negation fixpoint is an element x such that ∼ x = x. Remark 1. By [14,4] we have that an MTL-algebra is standard if and only if its monoidal operation ∗ is a leftcontinuous t-norm (see [24] for a monograph on t-norms). Moreover, it is easy to check that, if an MTL-algebra has a negation fixpoint, then it is unique.
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Let L be an axiomatic extension of MTL. It is known (see [29,9]) that L is algebraizable in the sense of [3], and that the equivalent algebraic semantics forms a subvariety of MTL-algebras, called L-algebras. We will denote by L such variety. On the other hand, each subvariety L of MTL is algebraizable, and we will denote L the corresponding axiomatic extension of MTL. In particular L is the extension of MTL via a set of axioms {ϕ}i∈I if and only if L is the subvariety of MTL-algebras satisfying {ϕ¯ = 1}i∈I , where ϕ¯ is obtained from ϕ by replacing each occurrence of &, →, ∧, ∨, ¬, ⊥, with ∗, ⇒, , , ∼, 0, 1, and every formula symbol occurring in ϕ with an individual variable. Given a class K of MTL-algebras, with V(K) we denote the variety generated by K. A class K of MTL-algebras is said to be generic for a variety V of MTL-algebras whenever V(K) = V. Concerning the varieties associated to the logics mentioned before we have that DP ⊂ RDP, G ⊂ RDP, and RDP ⊂ WNM ⊂ GHP ⊂ C3 MTL ⊂ MTL. In every variety of MTL-algebras the subdirectly irreducible algebras are totally ordered [10,9,14]: hence if two varieties of MTL-algebras V1 and V2 have the same class of chains it holds that V1 = V2 . This will be the key point to show that the variety of EMTL-algebras is the smallest to contain G and DP. We now give some results about the chains in GHP, G and DP: Lemma 1. (See [1].) Let A be a GHP-chain. For every x ∈ A, and if x > ∼ x, then x ∗ x = x, if x ≤ ∼ x, then x ∗ x = 0. As regards to Gödel, we have that: Lemma 2. (See [12,19].) In every Gödel chain the operations ∗ and ⇒ are the following ones, for every x, y ∈ A: 1 if x ≤ y, x ∗ y = min{x, y} x ⇒ y = (1) y otherwise, Concerning DP, we have that: Lemma 3. (See [29,2].) Every DP-chain A with more than two elements has a coatom c. Moreover, the operations ∗ and ⇒ of a such chain are the following ones, for every x, y ∈ A: 1 if x ≤ y, 0 if x, y < 1, x∗y = x ⇒ y = c if 1 > x > y, (2) min{x, y} otherwise. y if x = 1. Observe that the coatom c is a negation fixpoint, that is ∼c = c. As a consequence of the previous results we have that: Theorem 1. (See [12,19,2].) Let L ∈ {G, DP}. Then: • Given two L-chains A and B, if their lattice reducts are isomorphic, i.e. A, ≤A B, ≤B , then A B. • Given two L-chains A, B with |A| < |B|, then A embeds into B. 2.3. Completeness We now discuss the notion of completeness, that is a generalization of the one associated to classical logic. Let A be an MTL-algebra. An A-assignment over variables is a map e : VAR → A. Every such map can be extended in a unique way to an A-evaluation over formulas v : FORM → A, in an inductive way: see [10] for details. Given a formula ϕ, the notation A | ϕ indicates that v(ϕ) = 1, for every A-evaluation over formulas v. If K is a class of MTL-algebras, the notation K | ϕ indicates that B | ϕ for every B ∈ K.
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Definition 2. An axiomatic extension L of MTL is said to be complete w.r.t. a class K of L-algebras whenever, for every formula ϕ: L ϕ
iff
K | ϕ.
As pointed out in [10,9] an axiomatic extension L of MTL is complete w.r.t. a class K of L-algebras if and only if V(K) = L. The following result holds: Theorem 2. (See [14].) Let L be an axiomatic extension of MTL. Then L is complete w.r.t. the class of L-chains. Equivalently, L is generated by the class of its chains. It is important to point out that this theorem is not only a technical result, but it reflects a particular way of understanding a logic. In particular, we refer to the point of view presented in [19, page 2], about logics with a comparative notion of truth. That is, sentences can be compared according to their truth-values: clearly this presumes that the truth-values are comparable, and hence totally ordered. See [6] for a discussion about this topic. There are some well-established classes of chains with respect to one can be interested in studying completeness. For example the class of standard algebras. Definition 3. Given an axiomatic extension L of MTL, we say that L enjoys the standard completeness whenever L is complete w.r.t. the class of standard L-algebras. Many of the logics mentioned in the preceding sections enjoy the standard completeness: Theorem 3. (See [23,14,31,1,12,7].) Let L be one of MTL, C3 MTL, G, GHP, WNM, RDP. Then L enjoys the standard completeness. However, not all the axiomatic extensions of MTL enjoy this property. An example is given by DP, in which by Lemma 3 there are no standard DP-chains. However, we have a weaker notion of standard completeness: as we will see in a moment, this is an approximation of standard completeness. Theorem 4. (See [2].) For c ∈ (0, 1), let [0, 1]c be the DP-chain whose carrier is [0, c] ∪ {1}. Then for every 0 < c < 1, DP is complete w.r.t. [0, 1]c : equivalently, V([0, 1]c ) = DP. Clearly, more c is “near” to 1, and more [0, 1]c approximates a standard MTL-algebra. In this paper we focus our attention on some varieties of MTL-algebras, in particular G, DP, and the smallest containing both, EMTL, that will be introduced in the next section. We conclude the section with the following result, that describes which are the generic chains for G and DP, as well as their subvarieties. Remark 2. By Lemmas 2 and 3 and Theorem 1, for L ∈ {G, DP}, all the finite L-chains with the same cardinality are isomorphic. So, in the rest of the paper we will refer to the L-chain of h elements as the one with carrier 1 {0, h−1 , . . . , h−1 h−1 }, and the order given by the one of real numbers. Let L ∈ {G, DP}. With Lk , k ≥ 2, we denote the variety generated by the finite L-chain of k elements. Theorem 5. (See [12,13,2,9].) Let L ∈ {G, DP}. • Let K be a class of L-chains. If K contains an infinite L-chain or K contains infinitely many non-isomorphic finite L-chains, then V(K) = L.
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• Then the only proper subvarieties of L are the varieties Lk , with k ≥ 2. An L-chain belongs to Lk if and only if it has no more than k elements. • For k ≥ 2, Gk is axiomatized as MTL plus ( i
We conclude with some results concerning the join of varieties of MTL-algebras. Theorem 6. (See [17, Lemma 5.25], [33, Theorem 3.2], [32].) Let L be an axiomatic extension of MTL, and let L1 , L2 be two logics axiomatized by adding to L, respectively, the axioms ϕ1 , and ϕ2 , with no variables in common (if not, we can rename the variables). Then the meet of L1 and L2 (in the lattice of the axiomatic extensions of MTL), L1 ∧ L2 can be axiomatized as L plus ϕ1 ∨ ϕ2 . As we already seen before, every variety of MTL-algebras is generated by the class of its totally ordered members. Let L1 ∧ L2 be a logic satisfying the conditions of Theorem 6: one can ask if the class of the chains contained in the corresponding variety coincide with the union of L1 -chains and L2 -chains. As we will see in a moment, the answer is positive. Lemma 4. Let A be an MTL-chain, and ϕ, ψ be formulas. If A | ϕ ∨ ψ , then A | ϕ or A | ψ . Proof. Immediate from the fact that ∨ is interpreted as maximum, on every MTL-chain. 2 We can now state the previously mentioned result: Theorem 7. Let L be an axiomatic extension of MTL, and let L1, L2 be two logics axiomatized by adding to L, respectively, the axioms ϕ1 , ϕ2 , with no variables in common (if not, we can rename the variables). If M = L1 ∧ L2 , then the class of M-chains is given by the chains in L1 ∪ L2 . Proof. Clearly every chain in L1 ∪ L2 belongs to M. Assume by contradiction that A is a chain in M \ (L1 ∪ L2 ): / M. Hence the class of hence for every i ∈ {1, 2}, A | ϕi . By Lemma 4 we have that A | ϕ1 ∨ ϕ2 , and hence A ∈ M-chains is given by the chains in L1 ∪ L2 . 2 3. The logic EMTL Consider the following axiom: (ϕ → ϕ&ϕ) ∨ ¬(ψ&ψ) ∨ ψ
(E)
and call EMTL the logic obtained by adding (E) to MTL: note that (E) is the disjunction of (ID) and (DP). We show that this logic has the peculiarity of being the meet, in the lattice of axiomatic extension of MTL, of G and DP: as a consequence, we have that every EMTL-chain is a Gödel-chain or a DP-chain. The name EMTL takes his origin from this fact: the monoidal operation ∗ of every EMTL-chain is the minimum, the largest (w.r.t. the pointwise order) monoidal operation definable over an MTL-chain, or it is such that x ∗ y = 0 if x, y < 1, and x ∗ y = min{x, y} otherwise. In this last case ∗ is the smallest monoidal operation definable over an MTL-chain. For these reasons the acronym EMTL stands for extreme MTL. Theorem 8. Every EMTL-chain is a Gödel-chain or a DP-chain. As a consequence, EMTL = V(G ∪ DP). Proof. Immediate from Theorems 6 and 7, by observing that (E) is the disjunction of an instantiation of (DP) and (ID), with no variables in common. 2 Corollary 1. EMTL is the smallest variety containing G and DP. Observe that EMTL ⊆ RDP: in particular, this inclusion is strict, as the following example shows.
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G (DP)
(ID)
EMTL (E)
RDP (E)
(RDP)
WNM (DP)
(ID) (WNM)
GHP (GHP)
C3 MTL C3
MTL Fig. 1. The position of EMTL in the lattice of the axiomatic extensions of MTL (ordered by strength of the logic).
Example 3.1. Consider the MTL-chain A = {0, a, b, 1}, ∗, ⇒, min, max, 0, 1, with 0 < a < b < 1 and the operations ∗ and ⇒ are defined as follows. ∗ 0 a b 1
0 0 0 0 0
a 0 0 a a
b 0 a b b
1 0 a b 1
⇒ 0 a b 1
0 1 a 0 0
a 1 1 b a
b 1 1 1 b
1 1 1 1 1
It is easy to check that A is an RDP-chain, but not an EMTL-chain, as ∼(∼a a) = a < 1, ∼(a ∗ b) = a < 1, b < 1. We conclude the section with the following result: Theorem 9. The variety EMTL cannot be axiomatized with equations with only a variable from MTL. Proof. Let A be the RDP-chain of Example 3.1, and B, C be the subalgebras of A with carrier, respectively, {0, b, 1}, {0, a, 1}. Note that both belong to EMTL, as B ∈ G and C ∈ DP. By contradiction, suppose that T = {ti (x) = 1 | i ∈ I } is a set of equations in one variable axiomatizing EMTL: clearly, B and C satisfy all equations in T . Note however, that A = B ∪ C, whence A satisfies all equations in T , too. Hence A ∈ EMTL, in contrast with Example 3.1. 2 A graphical picture of these logics is given in Fig. 1. 4. Completeness, generic chains, and subvarieties of EMTL In this section we study which classes of chains generates EMTL, and we classify all its subvarieties. We begin with the following theorem: Theorem 10. • There is no generic EMTL-chain: that is, for every EMTL-chain A, V(A) ⊂ EMTL. • The standard completeness fails to hold for EMTL. • EMTL is locally finite.
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Proof. The first claim follows by Theorem 5 and the fact that every EMTL-chain is a Gödel- or a DP-chain. The second claim follows from the fact that the only standard algebra in EMTL is the standard Gödel algebra [0, 1]G , and V([0, 1]G ) = G, by Theorems 3 and 5. The third claim is due to the fact that EMTL ⊂ WNM, that is known to be locally finite (see for example [29]). 2 We now tackle the problem of study which classes of EMTL-chains generates the whole variety. Theorem 11. Let K be a class of EMTL-chains: then V(K) = EMTL if, • K contains at least an infinite Gödel-chain and an infinite DP-chain or • K contains infinitely many non-isomorphic finite Gödel chains and infinitely many non-isomorphic finite DPchains or • K contains at least an infinite Gödel-chain and infinitely many non-isomorphic finite DP-chains or • K contains at least an infinite DP-chain and infinitely many non-isomorphic finite Gödel-chains. Proof. Immediate by Theorems 5 and 8.
2
We now classify all the subvarieties of EMTL. As a consequence, we will see that the classes of EMTL-chains mentioned in the previous theorem are the only that generate EMTL. Theorem 12. The proper subvarieties of EMTL are the following ones: 1. G and its subvarieties. That is, by Theorem 5, G and Gk , for k ≥ 2. 2. DP and its subvarieties. That is, by Theorem 5, DP and DPk , for k ≥ 2. 3. The family of varieties EMTLh,k = V(Gh ∪ DPk ), with h, k > 2. EMTLh,k is generated by all the classes of EMTL-chains containing the Gödel chain of h elements, the DP-chain of k elements, and neither Gödel chains with more than h elements nor DP-chains with more than k elements. For h, k > 2, EMTLh,k , is axiomatized as EMTL plus: 2 (xi ⇒ xi+1 ) ∗ x ⇒ x (yi ⇒ yi+1 ) ∗ ∼ y 2 y = 1. (Eh,k )
i
i
4. The family of varieties EMTL∞,k = V(G ∪ DPk ), with k > 2. EMTL∞,k is generated by all the classes of EMTL-chains containing: At least an infinite Gödel chain or infinitely many non-isomorphic Gödel chains. The DP-chain of k elements, and no DP-chains with more than k elements. For k > 2, EMTL∞,k is axiomatized as EMTL plus: y ⇒ y2 (xi ⇒ xi+1 ) ∗ ∼ x 2 x = 1.
(E∞,k )
i
5. The family of varieties EMTLh,∞ = V(Gh ∪ DP), with h > 2. EMTLh,∞ is generated by all the classes of EMTL-chains containing: At least an infinite DP-chain or infinitely many non-isomorphic DP-chains. The Gödel chain of h elements, and no Gödel chains with more than h elements. For h > 2, EMTLh,∞ is axiomatized as EMTL plus: (Eh,∞ ) (xi ⇒ xi+1 ) ∗ x ⇒ x 2 ∼ y 2 y = 1.
i
Proof. 1–2 Immediate, by Theorem 8. 3 By Theorem 5, EMTLh,k = V(Gh ∪ DPk ): the axiomatization of this variety follows by Theorems 5 and 6. As a consequence of these facts and Theorems 5 and 7, we have that EMTLh,k is generated by all the classes of
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EMTL-chains containing the Gödel chain of h elements, the DP-chain of k elements, and neither Gödel chains with more than h elements nor DP-chains with more than k elements. 4 By Theorem 5, EMTL∞,k = V(G ∪ DPk ): the axiomatization of this variety follows by Theorems 6 and 5. As a consequence of these facts and Theorems 5 and 7, EMTL∞,k is generated by all the classes of EMTL-chains containing: • At least an infinite Gödel chain or infinitely many non-isomorphic Gödel chains. • The DP-chain of k elements, and no DP-chains with more than k elements. 5 By Theorem 5, EMTLh,∞ = V(Gh ∪ DP): the axiomatization of this variety follows by Theorems 5 and 6. As a consequence of these facts and Theorems 5 and 7, EMTLh,∞ is generated by all the classes of EMTL-chains containing: • At least an infinite DP-chain or infinitely many non-isomorphic DP-chains. • The Gödel chain of h elements, and no Gödel chains with more than h elements. Finally, consider a class K of EMTL-chains different to the ones mentioned in cases 3–5 of this theorem. Then K is one of the classes mentioned in Theorem 5 or in Theorem 11. In the first case we have that V(K) is one of L or Lk , with k ≥ 2 and L ∈ {G, DP}. In the second case V(K) = EMTL. We conclude that the only proper subvarieties of EMTL are the ones mentioned in Theorem 12. 2 Corollary 2. For k, h ≥ 2 the following hold. • The chains in EMTLh,k coincides with the ones in Gh ∪ DPk . • The chains in EMTLh,∞ coincides with the ones in Gh ∪ DP. • The chains in EMTL∞,k coincides with the ones in G ∪ DPk . 2
Proof. Immediate by Theorems 6, 7 and 12.
We finally arrive to the characterization of the classes of EMTL-chains that generates the whole variety. Corollary 3. A class K of EMTL-chains is generic for EMTL if and only if it is of the type specified in Theorem 11. Proof. Immediate by Theorems 5, 11, 12.
2
Concerning the relations between the subvarieties of EMTL, in terms of inclusion, we have that: Theorem 13. For h, k ≥ 3 it holds that: • • • •
Gh ∪ DPk ⊂ EMTLh,k . G ∪ DPh ⊂ EMTL∞,h , Gh ∪ DP ⊂ EMTLh,∞ . If r ≥ h and s ≥ k, then EMTLh,k ⊆ EMTLr,s . if r ≥ h and s ≥ k then EMTLh,k ⊂ EMTL∞,s and EMTLh,k ⊂ EMTLr,∞ .
Proof. Immediate from the previous results.
2
5. Amalgamation property In this section we will show some positive and negative results concerning the amalgamation property, for EMTL and its subvarieties. Definition 4. We say that a variety K of MTL-algebras has the amalgamation property (AP) if for every tuple (called i
j
V-formation) A, B, C, i, j , where A, B, C ∈ K and A − → B, A − → C, there is a tuple (called amalgam) D, h, k, h
k
→ D, C − → D, such that h ◦ i = k ◦ j . with D ∈ K, B −
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For the varieties of MTL-algebras there is a sufficient condition, to show the AP. Theorem 14. (See [27,26].) Let K be a variety of MTL-algebras and Klin be the class of its chains. If Klin enjoys the AP then the same holds for K. For EMTL we have a negative result, concerning the AP. Theorem 15. The amalgamation property fails to hold for EMTL. Proof. Let us denote with G3 , DP 3 , respectively, the Gödel chain and the DP-chain with carrier {0, 12 , 1}, with 0 < 12 < 1. By Theorem 8 we immediately see that there is no EMTL-chain in which both G3 , DP 3 can be embedded. We now show there is no EMTL-algebra in which G3 , DP 3 can be embedded: suppose by contradiction that such an algebra exists, and call it A. Clearly A cannot be totally ordered, but because DP 3 embeds into it, then A must have a negation fixpoint: this means that in the decomposition of A as subdirect product of subdirectly irreducible EMTL-algebras (that are totally ordered), we can only have DP-chains with at least three elements. Indeed, if at least a component of the subdirect product of A does not have a negation fixpoint, then also A cannot have it. Moreover, also G3 embeds into A, and hence this last one must have a non-trivial idempotent element: however the only way to define a non-trivial idempotent element, from a subdirect product of DP-chains, is by taking tuples of 0’s and 1’s, and the negation of such an element is not 0. Hence we cannot embed G3 into A. Hence for every V-formation of EMTL-algebras B, G3 , DP 3 , i, j there is no way to find an amalgam, in EMTL: this follows from the previous results, and the fact that B must coincide with the two elements boolean algebra. We conclude that the AP fails to hold for EMTL. 2 We now analyze the AP for the subvarieties of EMTL. We first recall that: Theorem 16. (See [16,27].) The only subvarieties of G for which the AP holds are G, G3 , and the varieties of boolean algebras. Concerning DP we have a similar result: Theorem 17. The only subvarieties of DP for which the AP holds are DP, DP3 , DP4 , and the varieties of boolean algebras. Proof. We first show that the AP holds for DP. By Theorem 14 it is enough to show the claim for the DP-chains. The proof is an adaptation of the one given in i
j
[16, Proposition 6.20] for G. Consider the tuple A, B, C, i, j , where A, B, C are DP-chains and A − → B, A − → C. If A has two elements then the claim is immediate: indeed, by [2, Theorem 3] and [28, Theorem 6] the class of DP-chains enjoys the joint embedding property (that is, for every pair of DP-chains there is a DP-chain in which both can be embedded), and hence we can find an amalgam. Assume now that |A| > 2: by Lemma 3 it follows that A, B, C must have a negation fixpoint. We can assume that A = B ∩ C, otherwise we can take two isomorphic copies of B and C that satisfy that condition, by renaming their elements. Note that the negation fixpoint of A, B, C belongs to B ∩ C. Set D = B ∪ C and define a binary relation ≤ over B ∪ C such that, for every x, y ∈ B ∪ C, x < y iff 1. 2. 3. 4.
x, y ∈ B and x
It is easy to see that ≤ is a partial order over B ∪ C: extend it to a total order ≤ over B ∪ C. Define two operations ∗, ⇒ over D as in Eq. (2), and call D the corresponding MTL-algebra: by Lemma 3 it follows that D is a DP-chain.
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Construct the maps h : B → D and k : C → D in the way that h(b) = b and k(c) = c, for every b ∈ B, c ∈ C: h
k
→ D, C − → D and h ◦ i = k ◦ j . a direct inspection shows that they extend to a pair of homomorphisms h, k such that B − Hence DP enjoys the AP. Observe that DP3 coincides with the variety generated by the three element MV-chain, for which the AP holds (see [11, Theorem 13]). Consider now the case of DP4 : by Theorem 14 it is enough to show the claim for the DP4 -chains. If we denote by DP k the DP-chain (up to isomorphisms) of k elements, we can observe that the only non-trivial chains (up to isomorphisms) in DP4 are 2, DP 3 and DP 4 . A direct inspection shows that, for every V-formation of DP4 -chains, DP 4 (equipped with two suitable embeddings) is an amalgam: hence DP4 enjoys the AP. Finally, consider DPn , with n > 4, and take the V-formation A, B, C, i, j , where A has {0 < a1 < · · · < an−4 < f < 1} as a lattice reduct, B has {0 < b1 < · · · < bn−3 < f < 1} as a lattice reduct, and C has {0 < c1 < · · · < cn−3 < f < 1} as a lattice reduct. The embeddings i, j are defined as follows: • For s ∈ {0, f, 1}, i(s) = s and j (s) = s. • For s ∈ {1, . . . , n − 4}, i(as ) = bs+1 and j (as ) = cs . Suppose by contradiction that an amalgam D, h, k exists, in DPn : this means that, for every a ∈ A, h(i(a)) = h
k
→ D and C − → D. Hence there are r, s ∈ D such that 0 < r = h(b1 ) < h(b2 ) = k(c1 ) < · · · < h(bn−3 ) = k(j (a)), B − k(cn−4 ) < s = k(cn−3 ) < h(f ) = k(f ) < 1. Let us call F the DP-chain having exactly these elements as carrier: we have that |F| = n + 1, and since F is a subalgebra of D, then F ∈ DPn . This, however, is a contradiction, because DPn is generated by the n-elements DP-chain, and hence by Theorem 5 every chain in DPn cannot have more than n elements. Hence the AP fails to hold for DPn . 2 Moving back to EMTL, we have that: Theorem 18. For h, k ≥ 3, the AP fails to hold for EMTLh,k , EMTLh,∞ , EMTL∞,k . Proof. The chains G3 and DP 3 are contained in all these varieties, and hence the same proof of Theorem 15 can be applied. 2 Summing up, we have that: Theorem 19. The only subvarieties of EMTL in which the AP holds are G, G3 , DP, DP3 , DP4 and the variety of boolean algebras. Proof. Immediate by Theorems 12, 15, 16, 17, 18.
2
References [1] S. Aguzzoli, M. Bianchi, On some questions concerning the axiomatisation of WNM-algebras and their subvarieties, Fuzzy Sets Syst. (2014), http://dx.doi.org/10.1016/j.fss.2014.07.007. [2] S. Aguzzoli, M. Bianchi, D. Valota, A note on drastic product logic, in: Information Processing and Management of Uncertainty— Proceedings of 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Part II, IPMU 2014, Montpellier, France, July 15–19, 2014, in: Communications in Computer and Information Science, vol. 443, Springer, 2014, pp. 365–374, preprint on: arXiv:1406.7166. [3] W. Blok, D. Pigozzi, Algebraizable logics, Mem. Am. Math. Soc. 77 (396) (1989), available on: http://ow.ly/rV2GS. [4] D.Boixader, F.Esteva, L.Godo, On the continuity of t-norms on bounded chains, in: Proceedings of the 8th IFSA World Congress, IFSA’99, Taipei, Taiwan, August 1999, pp. 476–479. [5] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York, 1981, an updated and revised electronic edition is available on: http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html. [6] L. Bˇehounek, P. Cintula, Fuzzy logics as the logics of chains, Fuzzy Sets Syst. 157 (5) (2006) 604–610, http://dx.doi.org/10.1016/j.fss. 2005.10.005. [7] A. Ciabattoni, F. Esteva, L. Godo, T-norm based logics with n-contraction, Neural Netw. World 12 (5) (2002) 441–452.
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The logic of the strongest and the weakest t-norms Matteo Bianchi Department of Computer Science, Università degli Studi di Milano, Via Comelico 39/41, 20135, Milano, Italy Received 23 July 2014; received in revised form 14 January 2015; accepted 16 January 2015
Abstract It is well known that the strongest t-norm, that is the largest with respect to the pointwise order, is the minimum. In 2001, the logic MTL was introduced as the base of a framework of many-valued logics, and in 2002 it was shown that it is the logic of all left-continuous t-norms and their residua. Within this family of logics, the many-valued logic associated with the minimum t-norm is the Gödel one, whilst there is no logic associated to the drastic product t-norm. Indeed the drastic product is not left-continuous, and hence it does not have a residuum. However, in a recent paper the logic DP has been studied, by showing that the monoidal operation of every DP-chain is like the drastic product t-norm. In this paper we present the logic EMTL, whose algebraic variety is the smallest to contain the ones of Gödel- and DP-algebras. We show that the chains in this algebraic variety are exactly all the Gödel- and DP-chains, we classify and axiomatize all the subvarieties, and we show some limitative results concerning the amalgamation property. © 2015 Elsevier B.V. All rights reserved. Keywords: Many-valued logics; Gödel-logic; Drastic product logic; Drastic product t-norm; Minimum t-norm; Amalgamation property; Residuated lattices
1. Introduction and motivations Triangular norms, t-norms for short, were introduced by Menger in [25], and subsequently studied by Schweizer, Sklar and other people in the area of probabilistic metric spaces: see [30,24] for a historical overview. Also, many connections between logic and t-norms were pointed out, in particular after the introduction of fuzzy sets by Zadeh [34]: from then a lot of research has been done, concerning both the theory and the applications. At the end of nineties, in [19], Hájek introduced the basic logic BL, as the base of a large framework of many-valued logics, strictly connected with residuated lattices [17] and t-norms. In [8] it was shown that BL is the logic of all continuous t-norms and their residua, in the sense that every continuous t-norm and the associated residuum induce an algebraic structure, called standard BL-algebra, and BL is complete w.r.t. the class of standard BL-algebras. In [14] a weaker logic than BL, MTL, was introduced, and in [23] it was shown that MTL is the logic of all left-continuous t-norms and their residua. The acronym MTL stands for monoidal t-norm based logic: indeed MTL can be axiomatized as an extension E-mail address: [email protected]. URL: http://www.mat.unimi.it/users/mbianchi/. http://dx.doi.org/10.1016/j.fss.2015.01.013 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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of the Höhle’s monoidal logic ML [20,14] or as an extension of the Full Lambek calculus FL [17,14]. MTL is, among the substructural logics, the weakest many-valued logic to be t-norm based: indeed, a t-norm admits a residuum iff it is left-continuous [4]. In this framework of logics is also contained the well-known Gödel–Dummett logic, G: it was briefly introduced by Kurt Gödel in [18], and subsequently axiomatized by M. Dummett in [12]. This logic is complete w.r.t. the algebra induced by the minimum t-norm [19]. By using the terminology of [24, Definition 1.4] a t-norm ∗ is stronger (weaker) than a t-norm ∗ if, for every x, y ∈ [0, 1], x ∗ y ≥ x ∗ y (x ∗ y ≤ x ∗ y). Since the minimum is known to be the strongest t-norm [24], we can say that G is the logic of the strongest t-norm. The weakest t-norm (see [24]) is instead drastic product ∗D , defined as x ∗D y = 0, if x, y < 1, and x ∗D y = min{x, y} otherwise: however, there is no axiomatic extension of MTL that is ∗D -based, i.e. that is complete w.r.t. an MTL-algebra having ∗D as monoidal operation, and [0, 1] as carrier. This happens because ∗D is not left-continuous, and does not have a residuum. In [22] a modification to ∗D has been introduced, in the way to obtain a left-continuous t-norm: the author presented a family of parametrized isomorphic t-norms, called revised drastic product. In [31] the corresponding logic has been studied, and was called RDP. In [29,21] the logic S3 MTL was introduced, and in [2] it was renamed in DP. The reason for the new name is that, even if DP is not t-norm based, the monoidal operation ∗ of every DP-chain is such that x ∗ y = 0, if x, y < 1, and x ∗ y = min{x, y} otherwise, as it happens for ∗D . So, we can say that DP is the logic of the weakest t-norm. In this paper we introduce the logic EMTL, whose corresponding variety is the smallest to contain the ones of Gödel- and DP-algebras. In particular, we show that every EMTL-chain is a DP-chain or a Gödel chain, and hence we can say that EMTL is the logic of the strongest and the weakest t-norms. We now describe the structure of the paper. After some basic background in Section 2, in Section 3 we introduce EMTL, by showing that the corresponding variety EMTL is generated by all the Gödel- and DP-algebras: in particular, the class of EMTL-chains coincides with all the Gödel- and DP-chains. We also discuss the relation between EMTL and other varieties, by showing also that it cannot be axiomatized with equations with only a variable from MTL. In Section 4 we study which classes of chains generates EMTL, and we classify and axiomatize all its subvarieties. Finally, in Section 5 we show some positive and negative results concerning the amalgamation property, for EMTL and its subvarieties. 2. Preliminaries We assume that the reader is acquainted with many-valued logics as developed by P. Hájek: see [19,14,10] for details. In particular we focus on some axiomatic extensions of the monoidal t-norm based logic MTL, firstly introduced by Esteva and Godo in [14], and proved to be logic of all left-continuous t-norm and their residua in [23]. 2.1. Syntax The language of MTL is based over the connectives {∧, &, →, ⊥}: the formulas are built in the usual inductive way from these connectives, and a denumerable set VAR of variables. The set of formulas is called FORM. Useful derived connectives are the following: def
¬ϕ = ϕ → ⊥ def ϕ ∨ ψ = (ϕ → ψ) → ψ ∧ (ψ → ϕ) → ϕ def
= ¬⊥
(negation) (disjunction) (top)
MTL can be axiomatized with a Hilbert style calculus: for the reader’s convenience, we list the axioms of MTL: (ϕ → ψ) → (ψ → χ) → (ϕ → χ) (A1) (ϕ&ψ) → ϕ
(A2)
(ϕ&ψ) → (ψ&ϕ)
(A3)
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(ϕ ∧ ψ) → ϕ
(A4)
(ϕ ∧ ψ) → (ψ ∧ ϕ) ϕ&(ϕ → ψ) → (ψ ∧ ϕ) ϕ → (ψ → χ) → (ϕ&ψ) → χ (ϕ&ψ) → χ → ϕ → (ψ → χ) (ϕ → ψ) → χ → (ψ → ϕ) → χ → χ
(A5)
⊥→ϕ
(A6) (A7a) (A7b) (A8) (A9)
As inference rule we have modus ponens: ϕ ϕ→ψ (MP) ψ An axiomatic extension of MTL is a logic obtained by adding one or more axiom schemata to it. A theory is a set of formulas: the notion of proof and logical consequence are defined as in the classical case. In this paper we focus on the following axiomatic extensions of MTL (EMTL will be introduced in Section 3): C3 MTL, G, DP, GHP, WNM, RDP [7,2,14,1,19,31]. The first five are axiomatized as MTL plus, respectively, the following axioms: (ϕ&ϕ)&ϕ ↔ ϕ. (C3 ) (ϕ&ϕ) ↔ ϕ.
(ID)
¬(ϕ&ϕ) ∨ ϕ. ¬(ϕ&ϕ) ∨ ϕ → (ϕ&ϕ) . ¬(ϕ&ψ) ∨ (ϕ ∧ ψ) → (ϕ&ψ) .
(DP) (GHP) (WNM)
RDP is axiomatized as WNM plus: (ϕ → ¬ϕ) ∨ ¬¬ϕ.
(RDP)
2.2. Semantics The semantics related to MTL and its axiomatic extensions is given by particular algebraic structures, called MTLalgebras. This class of algebras forms a variety: see [5] for details about algebraic varieties, and any unexplained notion of universal algebra. Definition 1. An MTL-algebra A is an algebra A = A, ∗, ⇒, , , 0, 1 such that: 1. 2. 3. 4.
A, , , 0, 1 is a bounded lattice with minimum 0 and maximum 1.
A, ∗, 1 is a commutative monoid.
∗, ⇒ forms a residuated pair: z ∗ x ≤ y iff z ≤ x ⇒ y for all x, y, z ∈ A. The following axiom holds for all x, y ∈ A: (x ⇒ y) (y ⇒ x) = 1
(Prelinearity)
A totally ordered MTL-algebra is called MTL-chain. An MTL-algebra is called standard whenever its carrier is [0, 1]. In the rest of the paper the notation ∼ x will denote x ⇒ 0. Given an MTL-algebra, a negation fixpoint is an element x such that ∼ x = x. Remark 1. By [14,4] we have that an MTL-algebra is standard if and only if its monoidal operation ∗ is a leftcontinuous t-norm (see [24] for a monograph on t-norms). Moreover, it is easy to check that, if an MTL-algebra has a negation fixpoint, then it is unique.
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Let L be an axiomatic extension of MTL. It is known (see [29,9]) that L is algebraizable in the sense of [3], and that the equivalent algebraic semantics forms a subvariety of MTL-algebras, called L-algebras. We will denote by L such variety. On the other hand, each subvariety L of MTL is algebraizable, and we will denote L the corresponding axiomatic extension of MTL. In particular L is the extension of MTL via a set of axioms {ϕ}i∈I if and only if L is the subvariety of MTL-algebras satisfying {ϕ¯ = 1}i∈I , where ϕ¯ is obtained from ϕ by replacing each occurrence of &, →, ∧, ∨, ¬, ⊥, with ∗, ⇒, , , ∼, 0, 1, and every formula symbol occurring in ϕ with an individual variable. Given a class K of MTL-algebras, with V(K) we denote the variety generated by K. A class K of MTL-algebras is said to be generic for a variety V of MTL-algebras whenever V(K) = V. Concerning the varieties associated to the logics mentioned before we have that DP ⊂ RDP, G ⊂ RDP, and RDP ⊂ WNM ⊂ GHP ⊂ C3 MTL ⊂ MTL. In every variety of MTL-algebras the subdirectly irreducible algebras are totally ordered [10,9,14]: hence if two varieties of MTL-algebras V1 and V2 have the same class of chains it holds that V1 = V2 . This will be the key point to show that the variety of EMTL-algebras is the smallest to contain G and DP. We now give some results about the chains in GHP, G and DP: Lemma 1. (See [1].) Let A be a GHP-chain. For every x ∈ A, and if x > ∼ x, then x ∗ x = x, if x ≤ ∼ x, then x ∗ x = 0. As regards to Gödel, we have that: Lemma 2. (See [12,19].) In every Gödel chain the operations ∗ and ⇒ are the following ones, for every x, y ∈ A: 1 if x ≤ y, x ∗ y = min{x, y} x ⇒ y = (1) y otherwise, Concerning DP, we have that: Lemma 3. (See [29,2].) Every DP-chain A with more than two elements has a coatom c. Moreover, the operations ∗ and ⇒ of a such chain are the following ones, for every x, y ∈ A: 1 if x ≤ y, 0 if x, y < 1, x∗y = x ⇒ y = c if 1 > x > y, (2) min{x, y} otherwise. y if x = 1. Observe that the coatom c is a negation fixpoint, that is ∼c = c. As a consequence of the previous results we have that: Theorem 1. (See [12,19,2].) Let L ∈ {G, DP}. Then: • Given two L-chains A and B, if their lattice reducts are isomorphic, i.e. A, ≤A B, ≤B , then A B. • Given two L-chains A, B with |A| < |B|, then A embeds into B. 2.3. Completeness We now discuss the notion of completeness, that is a generalization of the one associated to classical logic. Let A be an MTL-algebra. An A-assignment over variables is a map e : VAR → A. Every such map can be extended in a unique way to an A-evaluation over formulas v : FORM → A, in an inductive way: see [10] for details. Given a formula ϕ, the notation A | ϕ indicates that v(ϕ) = 1, for every A-evaluation over formulas v. If K is a class of MTL-algebras, the notation K | ϕ indicates that B | ϕ for every B ∈ K.
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Definition 2. An axiomatic extension L of MTL is said to be complete w.r.t. a class K of L-algebras whenever, for every formula ϕ: L ϕ
iff
K | ϕ.
As pointed out in [10,9] an axiomatic extension L of MTL is complete w.r.t. a class K of L-algebras if and only if V(K) = L. The following result holds: Theorem 2. (See [14].) Let L be an axiomatic extension of MTL. Then L is complete w.r.t. the class of L-chains. Equivalently, L is generated by the class of its chains. It is important to point out that this theorem is not only a technical result, but it reflects a particular way of understanding a logic. In particular, we refer to the point of view presented in [19, page 2], about logics with a comparative notion of truth. That is, sentences can be compared according to their truth-values: clearly this presumes that the truth-values are comparable, and hence totally ordered. See [6] for a discussion about this topic. There are some well-established classes of chains with respect to one can be interested in studying completeness. For example the class of standard algebras. Definition 3. Given an axiomatic extension L of MTL, we say that L enjoys the standard completeness whenever L is complete w.r.t. the class of standard L-algebras. Many of the logics mentioned in the preceding sections enjoy the standard completeness: Theorem 3. (See [23,14,31,1,12,7].) Let L be one of MTL, C3 MTL, G, GHP, WNM, RDP. Then L enjoys the standard completeness. However, not all the axiomatic extensions of MTL enjoy this property. An example is given by DP, in which by Lemma 3 there are no standard DP-chains. However, we have a weaker notion of standard completeness: as we will see in a moment, this is an approximation of standard completeness. Theorem 4. (See [2].) For c ∈ (0, 1), let [0, 1]c be the DP-chain whose carrier is [0, c] ∪ {1}. Then for every 0 < c < 1, DP is complete w.r.t. [0, 1]c : equivalently, V([0, 1]c ) = DP. Clearly, more c is “near” to 1, and more [0, 1]c approximates a standard MTL-algebra. In this paper we focus our attention on some varieties of MTL-algebras, in particular G, DP, and the smallest containing both, EMTL, that will be introduced in the next section. We conclude the section with the following result, that describes which are the generic chains for G and DP, as well as their subvarieties. Remark 2. By Lemmas 2 and 3 and Theorem 1, for L ∈ {G, DP}, all the finite L-chains with the same cardinality are isomorphic. So, in the rest of the paper we will refer to the L-chain of h elements as the one with carrier 1 {0, h−1 , . . . , h−1 h−1 }, and the order given by the one of real numbers. Let L ∈ {G, DP}. With Lk , k ≥ 2, we denote the variety generated by the finite L-chain of k elements. Theorem 5. (See [12,13,2,9].) Let L ∈ {G, DP}. • Let K be a class of L-chains. If K contains an infinite L-chain or K contains infinitely many non-isomorphic finite L-chains, then V(K) = L.
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• Then the only proper subvarieties of L are the varieties Lk , with k ≥ 2. An L-chain belongs to Lk if and only if it has no more than k elements. • For k ≥ 2, Gk is axiomatized as MTL plus ( i
We conclude with some results concerning the join of varieties of MTL-algebras. Theorem 6. (See [17, Lemma 5.25], [33, Theorem 3.2], [32].) Let L be an axiomatic extension of MTL, and let L1 , L2 be two logics axiomatized by adding to L, respectively, the axioms ϕ1 , and ϕ2 , with no variables in common (if not, we can rename the variables). Then the meet of L1 and L2 (in the lattice of the axiomatic extensions of MTL), L1 ∧ L2 can be axiomatized as L plus ϕ1 ∨ ϕ2 . As we already seen before, every variety of MTL-algebras is generated by the class of its totally ordered members. Let L1 ∧ L2 be a logic satisfying the conditions of Theorem 6: one can ask if the class of the chains contained in the corresponding variety coincide with the union of L1 -chains and L2 -chains. As we will see in a moment, the answer is positive. Lemma 4. Let A be an MTL-chain, and ϕ, ψ be formulas. If A | ϕ ∨ ψ , then A | ϕ or A | ψ . Proof. Immediate from the fact that ∨ is interpreted as maximum, on every MTL-chain. 2 We can now state the previously mentioned result: Theorem 7. Let L be an axiomatic extension of MTL, and let L1, L2 be two logics axiomatized by adding to L, respectively, the axioms ϕ1 , ϕ2 , with no variables in common (if not, we can rename the variables). If M = L1 ∧ L2 , then the class of M-chains is given by the chains in L1 ∪ L2 . Proof. Clearly every chain in L1 ∪ L2 belongs to M. Assume by contradiction that A is a chain in M \ (L1 ∪ L2 ): / M. Hence the class of hence for every i ∈ {1, 2}, A | ϕi . By Lemma 4 we have that A | ϕ1 ∨ ϕ2 , and hence A ∈ M-chains is given by the chains in L1 ∪ L2 . 2 3. The logic EMTL Consider the following axiom: (ϕ → ϕ&ϕ) ∨ ¬(ψ&ψ) ∨ ψ
(E)
and call EMTL the logic obtained by adding (E) to MTL: note that (E) is the disjunction of (ID) and (DP). We show that this logic has the peculiarity of being the meet, in the lattice of axiomatic extension of MTL, of G and DP: as a consequence, we have that every EMTL-chain is a Gödel-chain or a DP-chain. The name EMTL takes his origin from this fact: the monoidal operation ∗ of every EMTL-chain is the minimum, the largest (w.r.t. the pointwise order) monoidal operation definable over an MTL-chain, or it is such that x ∗ y = 0 if x, y < 1, and x ∗ y = min{x, y} otherwise. In this last case ∗ is the smallest monoidal operation definable over an MTL-chain. For these reasons the acronym EMTL stands for extreme MTL. Theorem 8. Every EMTL-chain is a Gödel-chain or a DP-chain. As a consequence, EMTL = V(G ∪ DP). Proof. Immediate from Theorems 6 and 7, by observing that (E) is the disjunction of an instantiation of (DP) and (ID), with no variables in common. 2 Corollary 1. EMTL is the smallest variety containing G and DP. Observe that EMTL ⊆ RDP: in particular, this inclusion is strict, as the following example shows.
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G (DP)
(ID)
EMTL (E)
RDP (E)
(RDP)
WNM (DP)
(ID) (WNM)
GHP (GHP)
C3 MTL C3
MTL Fig. 1. The position of EMTL in the lattice of the axiomatic extensions of MTL (ordered by strength of the logic).
Example 3.1. Consider the MTL-chain A = {0, a, b, 1}, ∗, ⇒, min, max, 0, 1, with 0 < a < b < 1 and the operations ∗ and ⇒ are defined as follows. ∗ 0 a b 1
0 0 0 0 0
a 0 0 a a
b 0 a b b
1 0 a b 1
⇒ 0 a b 1
0 1 a 0 0
a 1 1 b a
b 1 1 1 b
1 1 1 1 1
It is easy to check that A is an RDP-chain, but not an EMTL-chain, as ∼(∼a a) = a < 1, ∼(a ∗ b) = a < 1, b < 1. We conclude the section with the following result: Theorem 9. The variety EMTL cannot be axiomatized with equations with only a variable from MTL. Proof. Let A be the RDP-chain of Example 3.1, and B, C be the subalgebras of A with carrier, respectively, {0, b, 1}, {0, a, 1}. Note that both belong to EMTL, as B ∈ G and C ∈ DP. By contradiction, suppose that T = {ti (x) = 1 | i ∈ I } is a set of equations in one variable axiomatizing EMTL: clearly, B and C satisfy all equations in T . Note however, that A = B ∪ C, whence A satisfies all equations in T , too. Hence A ∈ EMTL, in contrast with Example 3.1. 2 A graphical picture of these logics is given in Fig. 1. 4. Completeness, generic chains, and subvarieties of EMTL In this section we study which classes of chains generates EMTL, and we classify all its subvarieties. We begin with the following theorem: Theorem 10. • There is no generic EMTL-chain: that is, for every EMTL-chain A, V(A) ⊂ EMTL. • The standard completeness fails to hold for EMTL. • EMTL is locally finite.
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Proof. The first claim follows by Theorem 5 and the fact that every EMTL-chain is a Gödel- or a DP-chain. The second claim follows from the fact that the only standard algebra in EMTL is the standard Gödel algebra [0, 1]G , and V([0, 1]G ) = G, by Theorems 3 and 5. The third claim is due to the fact that EMTL ⊂ WNM, that is known to be locally finite (see for example [29]). 2 We now tackle the problem of study which classes of EMTL-chains generates the whole variety. Theorem 11. Let K be a class of EMTL-chains: then V(K) = EMTL if, • K contains at least an infinite Gödel-chain and an infinite DP-chain or • K contains infinitely many non-isomorphic finite Gödel chains and infinitely many non-isomorphic finite DPchains or • K contains at least an infinite Gödel-chain and infinitely many non-isomorphic finite DP-chains or • K contains at least an infinite DP-chain and infinitely many non-isomorphic finite Gödel-chains. Proof. Immediate by Theorems 5 and 8.
2
We now classify all the subvarieties of EMTL. As a consequence, we will see that the classes of EMTL-chains mentioned in the previous theorem are the only that generate EMTL. Theorem 12. The proper subvarieties of EMTL are the following ones: 1. G and its subvarieties. That is, by Theorem 5, G and Gk , for k ≥ 2. 2. DP and its subvarieties. That is, by Theorem 5, DP and DPk , for k ≥ 2. 3. The family of varieties EMTLh,k = V(Gh ∪ DPk ), with h, k > 2. EMTLh,k is generated by all the classes of EMTL-chains containing the Gödel chain of h elements, the DP-chain of k elements, and neither Gödel chains with more than h elements nor DP-chains with more than k elements. For h, k > 2, EMTLh,k , is axiomatized as EMTL plus: 2 (xi ⇒ xi+1 ) ∗ x ⇒ x (yi ⇒ yi+1 ) ∗ ∼ y 2 y = 1. (Eh,k )
i
i
4. The family of varieties EMTL∞,k = V(G ∪ DPk ), with k > 2. EMTL∞,k is generated by all the classes of EMTL-chains containing: At least an infinite Gödel chain or infinitely many non-isomorphic Gödel chains. The DP-chain of k elements, and no DP-chains with more than k elements. For k > 2, EMTL∞,k is axiomatized as EMTL plus: y ⇒ y2 (xi ⇒ xi+1 ) ∗ ∼ x 2 x = 1.
(E∞,k )
i
5. The family of varieties EMTLh,∞ = V(Gh ∪ DP), with h > 2. EMTLh,∞ is generated by all the classes of EMTL-chains containing: At least an infinite DP-chain or infinitely many non-isomorphic DP-chains. The Gödel chain of h elements, and no Gödel chains with more than h elements. For h > 2, EMTLh,∞ is axiomatized as EMTL plus: (Eh,∞ ) (xi ⇒ xi+1 ) ∗ x ⇒ x 2 ∼ y 2 y = 1.
i
Proof. 1–2 Immediate, by Theorem 8. 3 By Theorem 5, EMTLh,k = V(Gh ∪ DPk ): the axiomatization of this variety follows by Theorems 5 and 6. As a consequence of these facts and Theorems 5 and 7, we have that EMTLh,k is generated by all the classes of
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EMTL-chains containing the Gödel chain of h elements, the DP-chain of k elements, and neither Gödel chains with more than h elements nor DP-chains with more than k elements. 4 By Theorem 5, EMTL∞,k = V(G ∪ DPk ): the axiomatization of this variety follows by Theorems 6 and 5. As a consequence of these facts and Theorems 5 and 7, EMTL∞,k is generated by all the classes of EMTL-chains containing: • At least an infinite Gödel chain or infinitely many non-isomorphic Gödel chains. • The DP-chain of k elements, and no DP-chains with more than k elements. 5 By Theorem 5, EMTLh,∞ = V(Gh ∪ DP): the axiomatization of this variety follows by Theorems 5 and 6. As a consequence of these facts and Theorems 5 and 7, EMTLh,∞ is generated by all the classes of EMTL-chains containing: • At least an infinite DP-chain or infinitely many non-isomorphic DP-chains. • The Gödel chain of h elements, and no Gödel chains with more than h elements. Finally, consider a class K of EMTL-chains different to the ones mentioned in cases 3–5 of this theorem. Then K is one of the classes mentioned in Theorem 5 or in Theorem 11. In the first case we have that V(K) is one of L or Lk , with k ≥ 2 and L ∈ {G, DP}. In the second case V(K) = EMTL. We conclude that the only proper subvarieties of EMTL are the ones mentioned in Theorem 12. 2 Corollary 2. For k, h ≥ 2 the following hold. • The chains in EMTLh,k coincides with the ones in Gh ∪ DPk . • The chains in EMTLh,∞ coincides with the ones in Gh ∪ DP. • The chains in EMTL∞,k coincides with the ones in G ∪ DPk . 2
Proof. Immediate by Theorems 6, 7 and 12.
We finally arrive to the characterization of the classes of EMTL-chains that generates the whole variety. Corollary 3. A class K of EMTL-chains is generic for EMTL if and only if it is of the type specified in Theorem 11. Proof. Immediate by Theorems 5, 11, 12.
2
Concerning the relations between the subvarieties of EMTL, in terms of inclusion, we have that: Theorem 13. For h, k ≥ 3 it holds that: • • • •
Gh ∪ DPk ⊂ EMTLh,k . G ∪ DPh ⊂ EMTL∞,h , Gh ∪ DP ⊂ EMTLh,∞ . If r ≥ h and s ≥ k, then EMTLh,k ⊆ EMTLr,s . if r ≥ h and s ≥ k then EMTLh,k ⊂ EMTL∞,s and EMTLh,k ⊂ EMTLr,∞ .
Proof. Immediate from the previous results.
2
5. Amalgamation property In this section we will show some positive and negative results concerning the amalgamation property, for EMTL and its subvarieties. Definition 4. We say that a variety K of MTL-algebras has the amalgamation property (AP) if for every tuple (called i
j
V-formation) A, B, C, i, j , where A, B, C ∈ K and A − → B, A − → C, there is a tuple (called amalgam) D, h, k, h
k
→ D, C − → D, such that h ◦ i = k ◦ j . with D ∈ K, B −
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For the varieties of MTL-algebras there is a sufficient condition, to show the AP. Theorem 14. (See [27,26].) Let K be a variety of MTL-algebras and Klin be the class of its chains. If Klin enjoys the AP then the same holds for K. For EMTL we have a negative result, concerning the AP. Theorem 15. The amalgamation property fails to hold for EMTL. Proof. Let us denote with G3 , DP 3 , respectively, the Gödel chain and the DP-chain with carrier {0, 12 , 1}, with 0 < 12 < 1. By Theorem 8 we immediately see that there is no EMTL-chain in which both G3 , DP 3 can be embedded. We now show there is no EMTL-algebra in which G3 , DP 3 can be embedded: suppose by contradiction that such an algebra exists, and call it A. Clearly A cannot be totally ordered, but because DP 3 embeds into it, then A must have a negation fixpoint: this means that in the decomposition of A as subdirect product of subdirectly irreducible EMTL-algebras (that are totally ordered), we can only have DP-chains with at least three elements. Indeed, if at least a component of the subdirect product of A does not have a negation fixpoint, then also A cannot have it. Moreover, also G3 embeds into A, and hence this last one must have a non-trivial idempotent element: however the only way to define a non-trivial idempotent element, from a subdirect product of DP-chains, is by taking tuples of 0’s and 1’s, and the negation of such an element is not 0. Hence we cannot embed G3 into A. Hence for every V-formation of EMTL-algebras B, G3 , DP 3 , i, j there is no way to find an amalgam, in EMTL: this follows from the previous results, and the fact that B must coincide with the two elements boolean algebra. We conclude that the AP fails to hold for EMTL. 2 We now analyze the AP for the subvarieties of EMTL. We first recall that: Theorem 16. (See [16,27].) The only subvarieties of G for which the AP holds are G, G3 , and the varieties of boolean algebras. Concerning DP we have a similar result: Theorem 17. The only subvarieties of DP for which the AP holds are DP, DP3 , DP4 , and the varieties of boolean algebras. Proof. We first show that the AP holds for DP. By Theorem 14 it is enough to show the claim for the DP-chains. The proof is an adaptation of the one given in i
j
[16, Proposition 6.20] for G. Consider the tuple A, B, C, i, j , where A, B, C are DP-chains and A − → B, A − → C. If A has two elements then the claim is immediate: indeed, by [2, Theorem 3] and [28, Theorem 6] the class of DP-chains enjoys the joint embedding property (that is, for every pair of DP-chains there is a DP-chain in which both can be embedded), and hence we can find an amalgam. Assume now that |A| > 2: by Lemma 3 it follows that A, B, C must have a negation fixpoint. We can assume that A = B ∩ C, otherwise we can take two isomorphic copies of B and C that satisfy that condition, by renaming their elements. Note that the negation fixpoint of A, B, C belongs to B ∩ C. Set D = B ∪ C and define a binary relation ≤ over B ∪ C such that, for every x, y ∈ B ∪ C, x < y iff 1. 2. 3. 4.
x, y ∈ B and x
It is easy to see that ≤ is a partial order over B ∪ C: extend it to a total order ≤ over B ∪ C. Define two operations ∗, ⇒ over D as in Eq. (2), and call D the corresponding MTL-algebra: by Lemma 3 it follows that D is a DP-chain.
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Construct the maps h : B → D and k : C → D in the way that h(b) = b and k(c) = c, for every b ∈ B, c ∈ C: h
k
→ D, C − → D and h ◦ i = k ◦ j . a direct inspection shows that they extend to a pair of homomorphisms h, k such that B − Hence DP enjoys the AP. Observe that DP3 coincides with the variety generated by the three element MV-chain, for which the AP holds (see [11, Theorem 13]). Consider now the case of DP4 : by Theorem 14 it is enough to show the claim for the DP4 -chains. If we denote by DP k the DP-chain (up to isomorphisms) of k elements, we can observe that the only non-trivial chains (up to isomorphisms) in DP4 are 2, DP 3 and DP 4 . A direct inspection shows that, for every V-formation of DP4 -chains, DP 4 (equipped with two suitable embeddings) is an amalgam: hence DP4 enjoys the AP. Finally, consider DPn , with n > 4, and take the V-formation A, B, C, i, j , where A has {0 < a1 < · · · < an−4 < f < 1} as a lattice reduct, B has {0 < b1 < · · · < bn−3 < f < 1} as a lattice reduct, and C has {0 < c1 < · · · < cn−3 < f < 1} as a lattice reduct. The embeddings i, j are defined as follows: • For s ∈ {0, f, 1}, i(s) = s and j (s) = s. • For s ∈ {1, . . . , n − 4}, i(as ) = bs+1 and j (as ) = cs . Suppose by contradiction that an amalgam D, h, k exists, in DPn : this means that, for every a ∈ A, h(i(a)) = h
k
→ D and C − → D. Hence there are r, s ∈ D such that 0 < r = h(b1 ) < h(b2 ) = k(c1 ) < · · · < h(bn−3 ) = k(j (a)), B − k(cn−4 ) < s = k(cn−3 ) < h(f ) = k(f ) < 1. Let us call F the DP-chain having exactly these elements as carrier: we have that |F| = n + 1, and since F is a subalgebra of D, then F ∈ DPn . This, however, is a contradiction, because DPn is generated by the n-elements DP-chain, and hence by Theorem 5 every chain in DPn cannot have more than n elements. Hence the AP fails to hold for DPn . 2 Moving back to EMTL, we have that: Theorem 18. For h, k ≥ 3, the AP fails to hold for EMTLh,k , EMTLh,∞ , EMTL∞,k . Proof. The chains G3 and DP 3 are contained in all these varieties, and hence the same proof of Theorem 15 can be applied. 2 Summing up, we have that: Theorem 19. The only subvarieties of EMTL in which the AP holds are G, G3 , DP, DP3 , DP4 and the variety of boolean algebras. Proof. Immediate by Theorems 12, 15, 16, 17, 18.
2
References [1] S. Aguzzoli, M. Bianchi, On some questions concerning the axiomatisation of WNM-algebras and their subvarieties, Fuzzy Sets Syst. (2014), http://dx.doi.org/10.1016/j.fss.2014.07.007. [2] S. Aguzzoli, M. Bianchi, D. Valota, A note on drastic product logic, in: Information Processing and Management of Uncertainty— Proceedings of 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Part II, IPMU 2014, Montpellier, France, July 15–19, 2014, in: Communications in Computer and Information Science, vol. 443, Springer, 2014, pp. 365–374, preprint on: arXiv:1406.7166. [3] W. Blok, D. Pigozzi, Algebraizable logics, Mem. Am. Math. Soc. 77 (396) (1989), available on: http://ow.ly/rV2GS. [4] D.Boixader, F.Esteva, L.Godo, On the continuity of t-norms on bounded chains, in: Proceedings of the 8th IFSA World Congress, IFSA’99, Taipei, Taiwan, August 1999, pp. 476–479. [5] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York, 1981, an updated and revised electronic edition is available on: http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html. [6] L. Bˇehounek, P. Cintula, Fuzzy logics as the logics of chains, Fuzzy Sets Syst. 157 (5) (2006) 604–610, http://dx.doi.org/10.1016/j.fss. 2005.10.005. [7] A. Ciabattoni, F. Esteva, L. Godo, T-norm based logics with n-contraction, Neural Netw. World 12 (5) (2002) 441–452.
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