On the direct decomposability of pseudo-t-norms, t-norms and implication operators on product lattices

Fuzzy Sets and Systems 158 (2007) 2494 – 2503 www.elsevier.com/locate/fss

On the direct decomposability of pseudo-t-norms, t-norms and implication operators on product lattices夡 Zhudeng Wanga, b,∗ , Jin-xuan Fanga a Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China b Department of Mathematics, Yancheng Teachers College, Jiangsu 224002, People’s Republic of China

Received 5 June 2006; received in revised form 22 June 2007; accepted 22 June 2007 Available online 26 March 2007

Abstract In this paper, we discuss the direct decomposability of pseudo-t-norms and t-norms on a product lattice, characterize the implication operator on a product lattice that is a direct product of two implication operators and study the direct decomposability of Simplications, R-implications, n-reciprocal R-implications and n-reciprocal QL-implications on a product lattice. © 2007 Elsevier B.V. All rights reserved. Keywords: Connective; t-Norm; Pseudo-t-norm; Implication; Direct decomposition

1. Introduction In fuzzy logic, connectives AND, OR and NOT are usually modelled by t-norms, t-conorms, and strong negations on [0, 1], respectively [31]. In recent papers many authors studied the properties of these logical operators on [0, 1] (e.g., see [1,7,8,10,11,21,24,25,32]). In [10], Fodor introduced the notion of weak t-norms on [0, 1] and studied the relations between weak t-norms and fuzzy implications on [0, 1]. Moreover, some authors introduced the notion of t-norms on a complete lattice or a poset, discussed the relation between the t-norms and the implications (see [5,6,15,22,23]). Noting that general QL-implications on [0, 1] (see [8,11]) cannot be induced by weak t-norms (or t-norms) on [0, 1], we introduced the concepts of pseudo-t-norms and implication operators on a complete Brouwerian lattice L in [29], discussed the relations between the set of all infinitely ∨-distributive pseudo-t-norms on L and the set of all infinitely ∧-distributive implications on L in detail and pointed out that three fundamental classes of fuzzy implications on [0, 1] (i.e., S-implications, R-implications and QL-implications on [0, 1]) (see [8,11]) and Yager’s implications (see [32]) on [0, 1] are all infinitely ∧-distributive implications that can be induced by infinitely ∨-distributive pseudo-t-norms on L when L = [0, 1]. In [30], we studied their direct products and direct decompositions. In [28], we laid bare the formulas for calculating the pseudo-t-norms and implication operators generated by binary operations.

夡 This project is supported by the National Natural Science Foundation of China (No. 10671094) and Science Foundation of Yancheng Teachers College. ∗ Corresponding author. E-mail address: [email protected] (Z. Wang).

0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.06.011

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The concept of the direct product of t-norms on a product lattice was introduced by De Baets and Mesiar [2]. They characterized ∨-distributive t-norms on a product lattice that are direct product of t-norms. In [18], Jenei and De Baets presented a method for constructing t-norms on a product lattice which are not direct product of two t-norms and formulated three open problems. In [19,20], Karacal and Khadjiev discussed two problems of them. This paper is a continuation of [27–30]. In Section 2, we briefly recall some definitions and results about t-norms, pseudo-t-norms and implication operators on a complete Brouwerian lattice. In Section 3, based on De Baets and Mesiar [2], Jenei and De Baets [18], Karacal and Khadjiev [20] and Karacal [19], we discuss direct decompositions of pseudo-t-norms and t-norms. In Section 4, we characterize the implication operator on a product lattice that is a direct product of two implication operators and study the direct decomposability of S-implications, R-implications, n-reciprocal R-implications and n-reciprocal QL-implications on a product lattice. 2. Preliminaries Let L = (L, ∧, ∨) be a lattice. For every y, z ∈ L, the relative pseudocomplement of y with respect to z, provided it exists, is the greatest element x such that x ∧ y z; it is denoted by y ⇒ z (i.e., y ⇒ z = max{x ∈ L|x ∧ y z}). L is said to be relatively pseudocomplemented provided the relative pseudocomplement y ⇒ z exists for every y, z ∈ L. A Heyting algebra (see [4]) is a relatively pseudocomplemented lattice with 0, i.e., a bounded one. If L is a relatively pseudocomplemented lattice, then ⇒ can be viewed as a binary operation on L and there exists the greatest element, 1, of the lattice: 1 = x ⇒ x, for all x ∈ L. Consequently, a Heyting algebra is an algebra L= (L, ∧, ∨, ⇒, 0, 1), where (L, ∧, ∨, 0, 1) is a bounded lattice with greatest element and minimum element and the binary operation ⇒ on L verifies: for all x, y, z ∈ L, x y ⇒ z if and only if x ∧ y z. Gödel algebras [12] (or linear Heyting algebras or L-algebras [14]) are Heyting algebras verifying the condition (x ⇒ y) ∨ (y ⇒ x) = 1 and the Gödel t-norm and its associated residuum (Gödel implication) on [0, 1] are  1 if x y, x G y = min(x, y) = x ∧ y, x →G y = y if x > y. Lukasiewicz t-norm, product t-norm and Gödel t-norm (and their associated residua) correspond to the most significant fuzzy logics: Lukasiewicz logic, Product logic and Gödel logic, respectively. The MV algebras, the Product algebras and the Gödel algebras constitute the algebraic models for these three types of logics. The class of BL algebras contains the MV algebras, the Product algebras and the Gödel algebras. The standard Gödel algebra is the BL algebra ([0, 1], min, max, G , →G , 0, 1), determined by the above Gödel t-norm. Moreover, a proper Heyting algebra (i.e., which is not a Gödel algebra) is not linearly ordered. A Heyting algebra L= (L, ∧, ∨, ⇒, 0, 1) is complete when (L, ∧, ∨, 0, 1) is a complete lattice. A (complete) Brouwerian lattice is the dual of a (complete) Heyting algebra. By Theorem 15 in [3], a complete lattice (L, ∧, ∨, 0, 1) is a Heyting algebra if and only if it satisfies the identity a ∧ (∨j ∈J bj ) = ∨j ∈J (a ∧ bj ), i.e., the join operation is completely distributive on meet operation in L. Throughout this paper, unless otherwise stated, L, L1 and L2 always represent any given complete Brouwerian lattices with maximal element 1 and minimal element 0; J stands for any index set. The direct product L1 × L2 of L1 and L2 is the set of all couples (a, b) with a ∈ L1 and b ∈ L2 , partially ordered by the rule that (a1 , b1 ) (a2 , b2 ) if and only if a1 a2 in L1 and b1 b2 in L2 . It is easy to see that the direct product L1 × L2 is also a complete Brouwerian lattice (see [3]) and (a1 , b1 ) ∨ (a2 , b2 ) = (a1 ∨ a2 , b1 ∨ b2 ), (a1 , b1 ) ∧ (a2 , b2 ) = (a1 ∧ a2 , b1 ∧ b2 ), where (a1 , b1 ) and (a2 , b2 ) are any two elements of L1 × L2 . Moreover, for any (a, b) ∈ L1 × L2 , we define the projections by l((a, b)) = a, r((a, b)) = b. Below, we briefly recall some definitions and results about t-norms, pseudo-t-norms and implications operators on L.

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Definition 2.1 (De Baets and Mesiar [2], Drossos [5], Dubois and Prade [7], Ma and Wu [22], Wang [26]). A binary operation T (or S) on L is called a t-norm (or t-conorm) if it satisfies the following conditions: (1) (2) (3) (4)

T (T (a, b), c) = T (a, T (b, c)), a, b, c ∈ L (associative law); T (a, b) = T (b, a), a, b ∈ L (commutative law); b c ⇒ T (a, b)T (a, c), a, b, c ∈ L (monotonicity); T (a, 1) = a (or S(a, 0) = a), a ∈ L (boundary condition).

T-norms and t-conorms on L are, respectively, called triangular norms and triangular conorms on L, and t-norms and t-conorms on [0, 1] are, respectively, called triangular norms and triangular conorms on [0, 1] (see Klement et al. [21]). Definition 2.2 (Wang and Yu [29]). (1) A binary operation T on L is called a pseudo-t-norm if it satisfies the following conditions: (T 1) T (1, a) = a and T (0, a) = 0, a ∈ L, (T 2) b c ⇒ T (a, b)T (a, c), a, b, c ∈ L. Especially, the pseudo-t-norm T is said to be infinitely ∨-distributive if 

(T∨ ) a, bj ∈ L; j ∈ J ⇒ T (a,

bj ) =

j ∈J



T (a, bj ).

j ∈J

(2) An implication operator (or an implication) I on L is a binary operation on L, satisfying two conditions: (I 1) I (1, b) = b, I (0, b) = 1, b ∈ L; (I 2) b c ⇒ I (a, b) I (a, c), a, b, c ∈ L. An implication operator I on L is called infinitely ∧-distributive if it satisfies the following condition:   (I∧ ) a, bj ∈ L; j ∈ J ⇒ I (a, bj ) = I (a, bj ). j ∈J

j ∈J

Denote by T (L) and I (L) the set of all infinitely ∨-distributive pseudo-t-norms on L and the set of all infinitely ∧-distributive implications on L, respectively. Clearly, if T is an infinitely ∨-distributive t-norm on L (see [26]), then T ∈ T (L). Another notion “pseudo-t-norm” was introduced, first on [0, 1] (see [9]), then, more generally, on a poset (L, , 1) (see [16,17]) by eliminating the axiom of commutativity from the definition of a t-norm. Such a pseudo-t-norm T (i.e., a non-commutative t-norm) has a pair of associated implication, (→L , →R ), verifying the non-commutative version of the residuation rule. That pseudo-t-norm and its pair of associated implications added to (L, , 1) determine the structures called porims (partially ordered residuated integral monoids) and also the pseudo-BCK algebras. The pseudo-t-norm in Definition 2.2 is non-commutative and non-associative and satisfies only (T 1) and (T 2). Thus, any t-norm on [0, 1] (on (L, , 1)) is a pseudo-t-norm from [9] on [0, 1] (on (L, , 1), respectively), any pseudot-norm from [9] on [0, 1] is a Fodor’s weak-t-norm (see [10]) on [0, 1] and any Fodor’s weak-t-norm on [0, 1] is a pseudo-t-norm in Definition 2.2. Han and Li [13] pointed out Definition 3.1 and Examples 3.1 and 4.1 in [29] are incorrect. In [27], we corrected these errors. In fact, noting that the least upper bound of the empty set is 0 and the largest lower bound of the empty set is 1, we have that ⎛ ⎞   T (a, 0) = T ⎝a, bj ⎠ = T (a, bj ) = 0 ∀a ∈ L, ⎛ I (a, 1) = I ⎝a,

j ∈J

 j ∈J

⎞

bj ⎠ =

j ∈J



I (a, bj ) = 1

j ∈J

when T ∈ T (L), I ∈ I (L) and J = ∅.

∀a ∈ L

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Definition 2.3 (Wang and Yu [29]). Let A ∈ LL×L . Define I (A), T (A) ∈ LL×L as follows: I (A)(a, b) = ∨{u ∈ L|A(a, u) b}

∀a, b ∈ L,

T (A)(a, b) = ∧{u ∈ L|A(a, u) b}

∀a, b ∈ L.

Below, we list some properties for a pseudo-t-norm and for its associated implication. Theorem 2.1 (Wang and Yu [29]). (1) If T ∈ T (L), then T (a, I (T )(a, 0)) = 0∀a ∈ L. (2) If T ∈ T (L), then T (a, c) b ⇔ c I (T )(a, b)∀a, b, c ∈ L. (3) If I ∈ I (L), then I (a, b)c ⇔ T (I )(a, c)b∀a, b, c ∈ L, Theorem 2.2 (Wang and Yu [29]). (1) If T is a pseudo-t-norm on L, then T ∈ T (L) if and only if for any a, b ∈ L, I (T )(a, b) is the largest element of set {u ∈ L|T (a, u) b}. (2) If I is an implication on L, then I ∈ I (L) if and only if for any a, b ∈ L, T (I )(a, b) is the smallest element of set {u ∈ L|I (a, u) b}. Theorem 2.3 (Wang and Yu [29]). (1) If T ∈ T (L), then I (T ) ∈ I (L) and T (I (T )) = T . (2) If I ∈ I (L), then T (I ) ∈ T (L) and I (T (I )) = I . (3) (T (L), ) is anti-order isomorphic to (I (L), ). Theorem 2.4 (Wang and Yu [29]). Let  be the anti-order isomorphism in Theorem 2.3. Then, (1) (T1 ∨ T2 ) = (T1 ) ∧ (T2 )∀T1 , T2 ∈ T (L), (2) −1 (I1 ∧ I2 ) = −1 (I1 ) ∨ −1 (I2 )∀I1 , I2 ∈ I (L). Theorem 2.5 (Wang and Yu [29]). Let T be a pseudo-t-norm on L and I = I (T ). Then, the following statements are equivalent: (1) T (a, c)b ⇔ c I (a, b)∀a, b, c ∈ L. (2) T satisfies the condition (T∨ ). Moreover, by the proof of Theorem 4.3 in [29], we see that I (T ) is an implication when T is a pseudo-t-norm on L and T (A) is a pseudo-t-norm when T is an implication on L. 1 ×L1 2 ×L2 Definition 2.4 (De Baets and Mesiar [2], Wang and Yu [30]). Let A1 ∈ LL and A2 ∈ LL . A binary operation 1 2 A on L1 × L2 , defined by

A((a1 , b1 ), (a2 , b2 )) := (A1 (a1 , a2 ), A2 (b1 , b2 ))

∀(a1 , b1 ), (a2 , b2 ) ∈ L1 × L2 ,

is called the direct product (or Cartesian product) of A1 and A2 , and written as A = A1 × A2 . 3. Direct decomposability of pseudo-t-norms and t-norms In this section, based on De Baets and Mesiar [2], Jenei and De Baets [18] and Karacal and Khadjiev [20], we discuss direct decompositions of pseudo-t-norms and t-norms on a product lattice. In [2], De Baets and Mesiar characterized ∨-distributive t-norms on a product lattice that are the direct product of two t-norms. In [30], we examined direct decompositions of pseudo-t-norms. Moreover, we have the following theorem. Theorem 3.1. Let T be a t-norm (pseudo-t-norm) on L1 × L2 . Then, T is a direct product of two t-norms (pseudo-tnorms) if and only if T satisfies the following conditions: (T 3) l(T ((a, b), (c, d))) = l(T ((a, 0), (c, 0)))∀a, c ∈ L1 , b, d ∈ L2 ; (T 4) r(T ((a, b), (c, d))) = r(T ((0, b), (0, d)))∀a, c ∈ L1 , b, d ∈ L2 .

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Proof. If there exist two t-norms (pseudo-t-norms) T1 on L1 and T2 on L2 such that T = T1 × T2 , then, for any a, c ∈ L1 , b, d ∈ L2 , l(T ((a, b), (c, d))) = l(T1 (a, c), T2 (b, d)) = T1 (a, c) = l(T ((a, 0), (c, 0))), r(T ((a, b), (c, d))) = r(T1 (a, c), T2 (b, d)) = T2 (b, d) = r(T ((0, b), (0, d))), and T1 (a, c) = l(T ((a, 0), (c, 0))),

T2 (b, d) = r(T ((0, b), (0, d))).

Conversely, suppose that T satisfies conditions (T 3) and (T 4). For any a, c ∈ L1 , b, d ∈ L2 , let T1 (a, c) = l(T ((a, 0), (c, 0))),

T2 (b, d) = r(T ((0, b), (0, d))).

When T is a t-norm on L1 × L2 , we have that T1 (a, c) = l(T ((a, 0), (c, 0))) = l(T ((c, 0), (a, 0))) = T1 (c, a)

∀a, c ∈ L1 ;

c1  c2 , a, c1 , c2 ∈ L1 ⇒ T ((a, 0), (c1 , 0))T ((a, 0), (c2 , 0)) ⇒ T1 (a, c1 ) = l(T ((a, 0), (c1 , 0)))l(T ((a, 0), (c2 , 0))) = T1 (a, c2 ); T1 (a, T1 (b, c)) = l(T ((a, 0), (T1 (b, c), 0))) = l(T ((a, 0), (l(T ((b, 0), (c, 0))), 0))) = l(T ((a, 0), (l(T ((b, 0), (c, 0))), r(T ((b, 0), (c, 0)))))) = l(T ((a, 0), T ((b, 0), (c, 0)))) = l(T (T ((a, 0), (b, 0)), (c, 0))) = T1 (T1 (a, b), c) ∀a, b, c ∈ L1 ; T1 (1, a) = l(T ((1, 0), (a, 0))) = l(T ((1, 1), (a, 0))) = l(a, 0) = a

∀a ∈ L1 .

Thus, T1 is a t-norm on L1 . Similarly, we can show that T2 is a t-norm on L2 . On the other hand, for any a, c ∈ L1 , b, d ∈ L2 , we have that l(T ((a, b), (c, d))) = l(T ((a, 0), (c, 0))) = T1 (a, c), r(T ((a, b), (c, d))) = r(T ((0, b), (0, d))) = T2 (b, d), and hence, T ((a, b), (c, d)) = (l(T ((a, b), (c, d))), r(T ((a, b), (c, d)))) = (T1 (a, c), T2 (b, d)) = (T1 × T2 )((a, b), (c, d)), i.e., T = T1 × T2 . When T is a pseudo-t-norm on L1 × L2 , we see that T1 (0, a) = l(T ((0, 0), (a, 0))) = l(0, 0) = 0

∀a ∈ L1 .

Thus, T1 is a pseudo-t-norm on L1 . Similarly, T2 is a pseudo-t-norm on L2 and T = T1 × T2 .



By the proof of Theorem 3.1, we can see that the conditions (T 3) and (T 4) can be, respectively, replaced by the following conditions: (T 3) l(T ((a, b), (c, d))) = l(T ((a, 1), (c, 1))) ∀a, c ∈ L1 , b, d ∈ L2 ; (T 4) r(T ((a, b), (c, d))) = r(T ((1, b), (1, d))) ∀a, c ∈ L1 , b, d ∈ L2 . Moreover, if T ∈ T (L1 × L2 ) and T satisfies conditions (T 3) and (T 4) in Theorem 3.1, then it follows from the proof of Theorem 3.4 in [30] that T1 ∈ T (L1 ), T2 ∈ T (L2 ). Example 3.1. The binary operation ⊗ on [0, 1]2 defined by (a1 , a2 ) ⊗ (b1 , b2 ) = (a1 ∧ b1 , [(a1 ∧ a2 ) ∨ (b1 ∧ b2 )] ∧ a2 ∧ b2 )

∀(a1 , a2 ), (b1 , b2 ) ∈ [0, 1]2

is a continuous t-norm (see Jenei and De Baets [18]) on [0, 1]2 which is not a direct product of two t-norms on [0, 1].

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In fact, when a, b ∈ (0, 1], we see that 0 = r[(0, a) ⊗ (0, b)]  = r[(1, a) ⊗ (1, b)] = a ∧ b. Thus, it follows from Theorem 3.1 that ⊗ is not a direct product of two t-norms on [0, 1]. Example 3.2. Let L1 = {0, 1} and L2 ⎧ x, ⎪ ⎪ ⎨ (0, 1), T (x, y) = T (y, x) = (1, 0), ⎪ ⎪ ⎩ (0, 0),

= {0, 21 , 1}. We define a commutative binary operation T on L1 × L2 as follows: y = (1, 1), x = y = (0, 1), x, y ∈ {(1, 0), (1, 21 )}, otherwise,

where x, y ∈ L1 × L2 . It is easy to verify that T is a t-norm (see Karacal and Khadjiev [20]) on L1 × L2 . But, 0 = r[T ((0, 21 ), (0, 1))]  = r[T ((1, 21 ), (1, 1))] = 21 . Thus, it follows from Theorem 3.1 that T is not a direct product of two t-norms on L1 and L2 . Example 3.3. The binary operation ⊗ on [0, 1]2 defined by  (0, 0) if (a, b), (c, d)  = (1, 1) and a ∧ c  21 , T ((a, b), (c, d)) = (a ∧ c, b ∧ d) otherwise is a t-norm on [0, 1]2 (see [19]). But, r(T ((1, 1), (1, 1))) = r((1, 1))  = r((0, 0)) = r(T ((0, 1), (0, 1))). Thus, T is not a direct product of two t-norms on [0, 1] by Theorem 3.1. In [30], we present a sufficient condition of an infinitely ∨-distributive pseudo-t-norm on a product lattice that can be decomposed into a direct product of two infinitely ∨-distributive pseudo-t-norms. Below, we give out a necessary and sufficient condition of a pseudo-t-norm with the identity on a product lattice that can be decomposed into a direct product of two pseudo-t-norms. Theorem 3.2. A pseudo-t-norm T with the identity (1, 1) on L1 × L2 is a direct product of two pseudo-t-norms if and only if it satisfies the following condition: (T 5) T ((a, b), (c, d)) = T ((a, 0), (c, 0)) ∨ T ((0, b), (0, d))

∀a, c ∈ L1 , b, d ∈ L2 .

Proof. If there exist two pseudo-t-norms T1 on L1 and T2 on L2 such that T = T1 × T2 , then, for any a, c ∈ L1 , b, d ∈ L2 , T ((a, 0), (c, 0)) ∨ T ((0, b), (0, d)) = (T1 (a, c), T2 (0, 0)) ∨ (T1 (0, 0), T2 (b, d)) = (T1 (a, c), 0)) ∨ (0, T2 (b, d)) = (T1 (a, c), T2 (b, d)) = T ((a, b), (c, d)). Conversely, if T satisfies the condition (T 5), then r(T ((a, b), (c, d))) = r(T ((a, 0), (c, 0))) ∨ r(T ((0, b), (0, d)))

∀a, c ∈ L1 , b, d ∈ L2 .

Noting that r(T ((a, 0), (c, 0))) r(T ((a, 0), (1, 1))) = r((a, 0)) = 0, we see that r(T ((a, b), (c, d))) = r(T ((0, b), (0, d)))

∀a, c ∈ L1 , b, d ∈ L2 .

Similarly, we have that l(T ((a, b), (c, d))) = l(T ((a, 0), (c, 0)))

∀a, c ∈ L1 , b, d ∈ L2 .

Thus, it follows from Theorem 3.1 that T is a direct product of two pseudo-t-norms.



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In particular, a t-norm T on L1 × L2 is a direct product of two t-norms if and only if it satisfies the condition (T 5) (see [20]). The following example illustrates that the condition in Theorem 3.2 that T is a pseudo-t-norm with the identity (1, 1) is indispensable. Example 3.4. We define a binary operation T on [0, 1]2 as follows: ⎧ (c, d), (a, b) = (1, 1), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (0, 0), (a, b) = (0, 0), T ((a, b), (c, d)) = (0, d), (a, b) = (0, 1), ⎪ ⎪ ⎪ (c, 0), (a, b) = (1, 0), ⎪ ⎪ ⎩ (1, 1), otherwise. It is easy to verify that T is a pseudo-t-norm on [0, 1]2 and satisfies the condition (T 5). But, 1 = l[T ((0, 13 ), (0, 1))]  = l[T ((0, 0), (0, 0))] = 0, and hence it follows from Theorem 3.1 that T is not a direct product of two pseudo-t-norms. Noting that T ((a, b), (1, 1)) = (1, 1)  = (a, b)

∀a, b ∈ (0, 1),

we can see that T is not a pseudo-t-norm with the identity (1, 1) on [0, 1]2 . 4. Direct decomposability of implication operators In [19], Karacal studied the direct decomposability of S-implication operators on a product lattice. In this section, based on Karacal [19], we study the implication operator on a product lattice that is a direct product of two implication operators and discuss the direct decomposability of S-implications, R-implications, n-reciprocal R-implications and n-reciprocal QL-implications on a product lattice. Firstly, we characterize the implication operator on a product lattice that is a direct product of two implication operators. Theorem 4.1. Let I be an implication operator on L1 × L2 . Then, I is a direct product of two implication operators if and only if I satisfies the following conditions: (I 3) l(I ((a, b), (c, d))) = l(I ((a, 1), (c, 1))) ∀a, c ∈ L1 , b, d ∈ L2 ; (I 4) r(I ((a, b), (c, d))) = r(I ((1, b), (1, d))) ∀a, c ∈ L1 , b, d ∈ L2 . Proof. Necessity: Refer to the proof of Theorem 3.1. Sufficiency: Assume that I satisfies the conditions (I 3) and (I 4). Let I1 (a, c) = l(I ((a, 1), (c, 1))),

I2 (b, d) = r(I ((1, b), (1, d)))

∀a, c ∈ L1 , b, d ∈ L2 .

If I is an implication operator on L1 × L2 , then c1 c2 , a, c1 , c2 ∈ L1 ⇒ I ((a, 1), (c1 , 1))I ((a, 1), (c2 , 1)) ⇒ I1 (a, c1 ) = l(I ((a, 1), (c1 , 1)))l(I ((a, 1), (c2 , 1))) = I1 (a, c2 ); I1 (1, c) = l(I ((1, 1), (c, 1))) = l(c, 1) = c

∀c ∈ L1 ;

I1 (0, c) = l(I ((0, 1), (c, 1))) = l(I ((0, 0), (c, 1))) = l(1, 1) = 1

∀c ∈ L1 .

Thus, I1 is an implication operator on L1 . Similarly, we can show that I2 is an implication operator on L2 and l = l1 × l2 .  By the proof of Theorems 3.1 and 4.1, we see that the conditions (I 3) and (I 4) can be, respectively, replaced by the following conditions: (I 3) l(I ((a, b), (c, d))) = l(I ((a, 0), (c, 0))) ∀a, c ∈ L1 , b, d ∈ L2 ; (I 4) r(I ((a, b), (c, d))) = r(I ((0, b), (0, d))) ∀a, c ∈ L1 , b, d ∈ L2 .

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Moreover, by Theorem 4.1, we have the following theorem. Theorem 4.2. If I is an implication operator on L1 × L2 and satisfies the condition (I 5) (a1 , b1 )(a2 , b2 ) ⇒ I ((a1 , b1 ), (c, d)) I ((a2 , b2 ), (c, d)) ∀a1 , a2 , c ∈ L1 , b1 , b2 , d ∈ L2 , then I is a direct product of two implication operators if and only if it satisfies the condition: (I 6) I ((a, b), (c, d)) = I ((a, 1), (c, 1)) ∧ I ((1, b), (1, d))

∀a, c ∈ L1 , b, d ∈ L2 .

Proof. If there exist two implication operators I1 on L1 and I2 on L2 such that I = I1 × I2 , then, for any a, c ∈ L1 , b, d ∈ L2 , I ((a, 1), (c, 1)) ∧ I ((1, b), (1, d)) = (I1 (a, c), I2 (1, 1)) ∧ (I1 (1, 1), I2 (b, d)) = (I1 (a, c), 1)) ∧ (1, I2 (b, d)) = (I1 (a, c), I2 (b, d)) = I ((a, b), (c, d)). Conversely, if I satisfies the conditions (I 5) and (I 6), then r(I ((a, b), (c, d))) = r(I ((a, 1), (c, 1))) ∧ r(I ((1, b), (1, d)))

∀a, c ∈ L1 , b, d ∈ L2 .

Noting that r(I ((a, 1), (c, 1)))r(I ((1, 1), (c, 1))) = r((c, 1)) = 1, we see that r(I ((a, b), (c, d))) = r(I ((1, b), (1, d)))

∀a, c ∈ L1 , b, d ∈ L2 .

Similarly, we have that l(I ((a, b), (c, d))) = l(I ((a, 1), (c, 1)))

∀a, c ∈ L1 , b, d ∈ L2 .

Thus, it follows from Theorem 4.1 that I is a direct product of two implication operators.



Below, by Theorem 4.2, we study the direct decomposability of S-implications, R-implications, n-reciprocal R-implications and n-reciprocal QL-implications on a product lattice. Example 4.1. S-implication (see [8,11]) on L1 × L2 is given by IS ((a, b), (c, d)) = S(n(a, b), (c, d)) ∀(a, b), (c, d) ∈ L1 × L2 , where S is a t-conorm and n a strong negation on L1 × L2 . It is easy to see that IS satisfies the condition (I 5). Thus, IS is a direct product of two implication operators if and only if it satisfies the condition (I 6). When IS = I1 × I2 , I1 = l(S(n(a, 1), (c, 1)) is an S-implication operator on L1 and I2 = r(S(n(1, b), (1, d)) is an S-implication operator on L2 (see [19, Corollary 2]). Example 4.2. R-implication (see [8,11]) on L1 × L2 is defined by using the concept of residuation as follows: IR ((a, b), (c, d)) = ∨{(x, y)|T ((a, b), (x, y))(c, d)} ∀(a, b), (c, d) ∈ L1 × L2 , where T is a t-norm on L1 × L2 . We can see that IR satisfies the condition (I 5) and hence IR is a direct product of two implication operators if and only if it satisfies the condition (I 6). When T is an infinitely ∨-distributive t-norm (i.e., left-continuous t-norm (see [22])) on L1 × L2 and IR = I1 × I2 , it follows from Theorem 3.1 in [30] that T = T (I1 × I2 ) = T (I1 ) × T (I2 ) is a direct product of two infinitely ∨-distributive t-norms, and I1 = I (T (I1 )) and I2 = I (T (I2 )) are two infinitely ∧-distributive R-implication operators.

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Example 4.3. n-Reciprocal R-implications (see [11]) on L1 × L2 are of the form IRn ((a, b), (c, d)) = ∨{(x, y)|T (n(c, d), (x, y))n(a, b)} ∀(a, b), (c, d) ∈ L1 × L2 , where T is a t-norm and n a strong negation on L1 × L2 . It is easy to verify that IRn satisfies the condition (I 5) and hence IRn is a direct product of two implication operators if and only if it satisfies the condition (I 6). Clearly, IRn ((a, b), (c, d)) = IR (n(c, d), n(a, b)). When n is the direct product of a strong negation n1 on L1 and a strong negation n2 on L2 (see [19]), we have that IRn ((a, b), (c, d)) = IR ((n1 (c), n2 (d)), (n1 (a), n2 (b))). Moreover, if T is an infinitely ∨-distributive t-norm on L1 × L2 and IR = I1 × I2 , then it follows from Example 4.2 that I1 and I2 are two infinitely ∧-distributive R-implication operators and IRn ((a, b), (c, d)) = (I1 (n1 (c), n1 (a)), I2 (n2 (d), n2 (b))) = (I1n1 (a, c), I2n2 (b, d)), i.e., IRn is the direct product of two n-reciprocal R-implications I1n1 on L1 and I2n2 on L2 . Example 4.4. n-Reciprocal QL-implications (see [11]) on L1 × L2 are introduced by n ((a, b), (c, d)) = S((c, d), T (n(a, b), n(c, d))) ∀(a, b), (c, d) ∈ L1 × L2 , IQL

where T is a t-norm, S a t-conorm and n a strong negation on L1 × L2 . n satisfies the condition (I 5) and hence I n is a direct product of two implication Similarly, we can show that IQL QL operators if and only if it satisfies the condition (I 6). When S, T, and n are, respectively, the direct products of two t-conorms S1 and S2 , two t-norms T1 and T2 , and two strong negations n1 and n2 , n ((a, b), (c, d)) = S((c, d), T (n(a, b), n(c, d))) IQL = (S1 (c, T1 (n1 (a), n1 (c))), S2 (d, T2 (n2 (b), n2 (d)))) n1 n2 = (IQL (a, c), IQL (b, d)). n is the direct product of two n-reciprocal QL-implications I n1 on L and I n2 on L . Thus, IQL 1 2 QL QL Finally, we illustrate that the condition (I 5) in Theorem 4.2 is indispensable by the following example.

Example 4.5. We define a binary operation I on [0, 1]2 as follows: ⎧ (c, d), (a, b) = (1, 1), ⎪ ⎪ ⎪ ⎪ ⎨ (1, 1), (a, b) = (0, 0), I ((a, b), (c, d)) = (1, d), (a, b) = (0, 1), ⎪ ⎪ (c, 1), (a, b) = (1, 0), ⎪ ⎪ ⎩ (0, 0), otherwise. It can be shown that I is an implication on [0, 1]2 and satisfies the condition (I 6). But, 0 = l[I ((1, 21 ), (1, 0))]  = l[I ((1, 1), (1, 0))] = 1, and hence it follows from Theorem 4.1 that I is not a direct product of two implication operators [0, 1]. Noting that (0, 0) = I ((0, 21 ), (1, 1)) < I ((0, 1), (1, 1))] = (1, 1), we see that I does not satisfy the condition (I 5).

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5. Conclusion In this paper we have dealt with direct decompositions of pseudo-t-norms, t-norms and implication operators on a product lattice and studied the direct decomposability of S-implications, R-implications, n-reciprocal R-implications and n-reciprocal QL-implications on a product lattice. For QL-implications on a product lattice, whether some results similar to Examples 4.1-4.4 are true or not, this could be a topic in further investigations. Acknowledgements This work was done during the author’s visit to Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences. The author wishes to thank Dr. Liming Ge for his help during the preparation of this paper; Area Editor and the anonymous referees for their valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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