A new structure for uninorms on bounded lattices

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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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A new structure for uninorms on bounded lattices

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Yexing Dan a,b , Bao Qing Hu a,b,∗

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a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China b Computational Science, Hubei Key Laboratory, Wuhan University, Wuhan 430072, People’s Republic of China

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Received 13 September 2018; received in revised form 5 January 2019; accepted 3 February 2019

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Abstract

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This article investigates the construction of uninorms on bounded lattices. First, by using the fact that triangular norms (t-norms) and triangular conorms (t-conorms) on an arbitrary bounded lattice always exist, we introduce a new construction of uninorms on arbitrary bounded lattices with the neutral element. Then we illustrate how our new construction method is different from some existing methods for the construction of uninorms on bounded lattices. Some illustrative examples for the construction of uninorms on bounded lattices are provided. © 2019 Elsevier B.V. All rights reserved.

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Keywords: Uninorms; t-Norms; t-Conorms; Bounded lattices

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1. Introduction Uninorms [2,11,19,21–23,26,28,32,33] with the neutral element e on the unit interval, as an important generalization of triangular norms (t-norms) and triangular conorms (t-conorms) [18,22], play an important role in lots of fields, such as in fuzzy logic [1], fuzzy set theory, and fuzzy system modeling [34]. In contrast to t-norms and t-conorms, uninorms allow the neutral element e to lie anywhere in the unit interval. In particular, a uninorm U is a t-norm T when e = 1 and a t-conorm S when e = 0. For uninorms with the neutral element e different from 1 (the top element) and 0 (the bottom element) on the unit interval, the construction of those functions is an important task not only from the theoretical point of view, such as Umin and Umax [11], but also for their applications. On the other hand, t-norms and t-conorms on the unit interval, as an extension of the conjunction and disjunction in classical two-valued logic, respectively, have been extended to bounded lattices in the literature (see, e.g., [10,20,24]). In addition, uninorms on bounded lattices have been investigated in many studies (see, e.g., [7,14,16,25,27,29–31]), especially in work related to the construction of uninorms on bounded lattices. Uninorms on bounded lattices were introduced by Karaçal and Mesiar [15] in 2015. They introduced the existence of uninorms with the neutral element e ∈ L \ {0L , 1L } on an arbitrary bounded lattice L. They obtained the greatest uninorm and the smallest uninorm with the neutral element e ∈ L \ {0L , 1L }.

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* Corresponding author at: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China.

E-mail addresses: [email protected] (Y. Dan), [email protected] (B.Q. Hu). https://doi.org/10.1016/j.fss.2019.02.001 0165-0114/© 2019 Elsevier B.V. All rights reserved.

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In 2016, Çaylı et al. [9] proposed a new class of uninorms on bounded lattices. They showed the existence of idempotent uninorms with the neutral element e ∈ L \ {0L , 1L } on bounded lattices. In addition, they discussed uninorms with neutral element e that is incomparable with U (0, 1) and provided an example of an idempotent uninorm that is neither disjunctive nor conjunctive. In 2017, Çaylı and Karaçal [8] provided some new methods for the construction of uninorms on bounded lattices. Compared with the previous work, they presented new methods for constructing uninorms on an arbitrary bounded lattice L with some additional constraints on the neutral element e ∈ L \ {0L , 1L }. Some illustrative examples for the construction of uninorms on bounded lattices were provided. In 2018, based on [8], Çaylı [5] proposed new construction approaches for uninorms on bounded lattices without the need for some additional constraints on the neutral element e ∈ L \ {0L , 1L } and studied the main characteristics of uninorms on bounded lattices. Also, the relationship between the uninorms the author proposed and previous uninorms was discussed. The construction of uninorms on bounded lattices by some other approaches can be found in the literature [4,6]. In this article, we propose new construction approaches for defining uninorms on an arbitrary bounded lattice with the neutral element. In addition, we exemplify the differences between our new construction approaches and the previous approaches. When considering the specific structures of uninorms on bounded lattices, we mainly illustrate the relationship between the uninorms on bounded lattices we present in this article and the uninorms proposed by Karaçal and Mesiar [15], Bodjanova and Kalina [4], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5] as Theorems 3.1, 3.2, 3.3, 3.4 and 3.5, and 3.6, respectively. The relationship between the uninorms we present in Theorems 3.7 and 3.14 and other uninorms such as those proposed by Çaylı [6] can be discussed in an analogous way; we leave this task to interested readers. Moreover, readers can refer to the literature [5,6] for more comparabilities. The article is organized as follows. In Section 2 we recall some basic notions of bounded lattices and some properties related to them. In Section 3 we propose a new construction of uninorms with underlying t-norms and t-conorms on bounded lattices. In addition, we provide some examples to illustrate the differences between the new construction of uninorms on bounded lattices proposed by us and some existing uninorms. In Section 4 we provide concluding remarks. 2. Preliminaries

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In this section, we recall some basic concepts of bounded lattices and some properties related to them, which will be used in the sequel. A bounded lattice (L, ≤, 0L , 1L ) [12] is a lattice that has top element 1L and bottom element 0L . In the following, unless stated otherwise, we denote L as a bounded lattice.

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Definition 2.1. (See Birkhoff [3].) Let a, b ∈ L. We use the notation a  b to denote that a and b are incomparable. In the following, Ia denotes the family of all incomparable elements with a; that is, Ia = {x ∈ L | x  a}. Let a, b ∈ L with a ≤ b. Then a subinterval [a, b] of L can be defined as follows [3]: [a, b] = {x ∈ L | a ≤ x ≤ b}.

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(a, b] = {x ∈ L | a < x ≤ b},

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[a, b) = {x ∈ L | a ≤ x < b}

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(a, b) = {x ∈ L | a < x < b}.

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Definition 2.2. (See Birkhoff [3].) An element a of L is called an atom if a > 0L and there is no x ∈ L such that 0L < x < a. An element a of L is called a coatom if a < 1L and there is no x ∈ L such that a < x < 1L .

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Definition 2.3. (See Karaçal and Mesiar [15].) An operation U : L2 −→ L is called a uninorm on L if for all x, y, z ∈ L it satisfies the following conditions:

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(1) (2) (3) (4)

U (x, y) = U (y, x). U (x, U (y, z)) = U (U (x, y), z). U (x, y) ≤ U (x, z) whenever y ≤ z. There exists some element e ∈ L, called the neutral element, such that U (e, x) = x for all x ∈ L.

Definition 2.4. (See Çaylı et al. [9].) Let U be a uninorm on L with the neutral element e ∈ L. Then U is called an idempotent uninorm if U (x, x) = x for all x ∈ L. A uninorm U on L is called conjunctive if U (0L , 1L ) = 0L and disjunctive if U (0L , 1L ) = 1L . In particular, the uninorm U defined in Definition 2.3 is a t-norm T when e = 1L and is a t-conorm S when e = 0L . In the following, we show an example for some t-norms and t-conorms on L, which will be used in the sequel.

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Example 2.5. (See Karaçal and Mesiar [15].) Some basic t-norms and t-conorms on L are listed here: (1) The greatest t-norm T∧ : T∧ (x, y) = x ∧ y. (2) The smallest t-conorm S∨ : S∨ (x, y) = x ∨ y. (3) The smallest t-norm TW :  x ∧ y if 1L ∈ {x, y}, TW (x, y) = otherwise. 0L

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(4) The greatest t-conorm SW :  x ∨ y if 0L ∈ {x, y}, SW (x, y) = otherwise. 1L Readers can refer to [13,17] for more details of t-norms and t-conorms on bounded lattices. In recent years, the construction of uninorms on bounded lattices has been a popular research area. The main purpose of this article is to construct new uninorms on an arbitrary bounded lattice L. We recall some existing discussions for uninorms on bounded lattices, which will be used in the sequel. Proposition 2.6. (See Karaçal and Mesiar [15].) Let U be a uninorm on L with the neutral element e ∈ L \ {0L , 1L }. Then the following statements hold: (1) (2) (3) (4) (5)

x ∧ y ≤ U (x, y) ≤ x ∨ y for all (x, y) ∈ [0L , e] × [e, 1L ] ∪ [e, 1L ] × [0L , e]. U (x, y) ≤ x for all (x, y) ∈ L × [0L , e]. U (x, y) ≤ y for all (x, y) ∈ [0L , e] × L. x ≤ U (x, y) for all (x, y) ∈ L × [e, 1L ]. y ≤ U (x, y) for all (x, y) ∈ [e, 1L ] × L.

Proposition 2.7. (See Karaçal and Mesiar [15].) Let U be a uninorm with the neutral element e on L. Then the following statements hold:

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(1) T ∗ = U |[0L ,e]2 : [0L , e]2 −→ [0L , e] is a t-norm on [0L , e]. (2) S ∗ = U |[e,1L ]2 : [e, 1L ]2 −→ [e, 1L ] is a t-conorm on [e, 1L ]. Proposition 2.8. (See Çaylı et al. [9].) Let U be an idempotent uninorm on L with the neutral element e ∈ L \{0L , 1L }. Then the following statements hold:

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(1) If (x, y) ∈ [0L , e]2 , U (x, y) = x ∧ y. (2) If (x, y) ∈ [e, 1L ]2 , U (x, y) = x ∨ y.

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We note from Proposition 2.8 that the only idempotent t-norm on [0L , e] is T∧ and the only idempotent t-conorm on [e, 1L ] is S∨ .

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3. Construction of uninorms with underlying t-norms and t-conorms on bounded lattices

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Similarly to the investigation of uninorms on a unit interval, the notion of uninorms has been extended to bounded lattices and the existence of uninorms on bounded lattices was introduced by Karaçal and Mesiar [15]. Bodjanova and Kalina [4], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5] introduced new methods for constructing uninorms on bounded lattices. Their results can be used to enrich the class of uninorms on bounded lattices and to analyze their structure. In this section, considering the existence of a t-norm on [0L , e]2 and a t-conorm on [e, 1L ]2 , we propose a new construction of uninorms on an arbitrary bounded lattice with the neutral element. We discuss the relationship between the uninorms on bounded lattices presented in Theorems 3.7 and 3.14 and the uninorms proposed by Karaçal and Mesiar [15], Bodjanova and Kalina [4], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5] as Theorems 3.1, 3.2, 3.3, 3.4 and 3.5, and 3.6, respectively. Some examples are provided to demonstrate that our new construction approach is different from the proposals of Karaçal and Mesiar [15], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5]. First, we describe the existing construction methods proposed by Karaçal and Mesiar [15], Bodjanova and Kalina [4], Çaylı et al. [9], Çaylı and Karaçal [8], and Çaylı [5].

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Theorem 3.1. (See Karaçal and Mesiar [15].) Let e ∈ L \ {0L , 1L }. If Te is a t-norm on [0L and Se is a t-conorm on [e, 1L ]2 , then the functions Ut1 : L2 −→ L and Us1 : L2 −→ L defined as follows are uninorms on L: , e]2

⎧ ⎪ Te (x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x ∨ y Ut1 (x, y) = y ⎪ ⎪ ⎪x ⎪ ⎪ ⎪ ⎩1 L

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if (x, y) ∈ [e, 1L ]2 , if (x, y) ∈ [0L , e) × [e, 1L ] ∪ [e, 1L ] × [0L , e), if (x, y) ∈ [e, 1L ] × Ie , if (x, y) ∈ Ie × [e, 1L ], otherwise.

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Theorem 3.2. (See Bodjanova and Kalina [4].) Assume that T : L2 −→ L and S : L2 −→ L are a t-norm and a t-conorm, respectively. Let e ∈ L be an element distinct from both 0L and 1L . Let us denote by Ud : L2 −→ L and Uc : L2 −→ L the following:

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⎧ ⎪ T (x, y) if x < e, y < e, ⎪ ⎪ ⎪ ⎪ ⎪ if x ≤ e, y ≮ e, ⎨y Ud (x, y) = x if x ≮ e, y ≤ e, ⎪ ⎪ ⎪ S(x, y) if x > e, y > e, ⎪ ⎪ ⎪ ⎩S(x ∨ e, y ∨ e) otherwise

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⎧ ⎪ Se (x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x ∧ y Us1 (x, y) = y ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎩0 L

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if (x, y) ∈ [0L , e]2 , if (x, y) ∈ [0L , e] × (e, 1L ] ∪ (e, 1L ] × [0L , e], if (x, y) ∈ [0L , e] × Ie , if (x, y) ∈ Ie × [0L , e], otherwise

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⎧ ⎪ S(x, y) if x > e, y > e, ⎪ ⎪ ⎪ ⎪ ⎪ if x ≥ e, y ≯ e, ⎨y Uc (x, y) = x if x ≯ e, y ≥ e, ⎪ ⎪ ⎪ T (x, y) if x < e, y < e, ⎪ ⎪ ⎪ ⎩T (x ∧ e, y ∧ e) otherwise.

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Then Ud is a disjunctive uninorm and Uc is a conjunctive uninorm with the neutral element e, respectively.

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Theorem 3.3. (See Çaylı et al. [9].) Let e ∈ L \ {0L , 1L }. If Te is a t-norm on [0L , e]2 and Se is a t-conorm on [e, 1L ]2 , then the functions Ut2 : L2 −→ L and Us2 : L2 −→ L defined as follows are uninorms on L: ⎧ Te (x, y) if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎨y if (x, y) ∈ [0L , e] × Ie , Ut2 (x, y) = ⎪ x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎩ x∨y otherwise and

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⎧ Se (x, y) ⎪ ⎪ ⎪ ⎨y Us2 (x, y) = ⎪ x ⎪ ⎪ ⎩ x ∧y

if (x, y) ∈ [e, 1L ]2 , if (x, y) ∈ [e, 1L ] × Ie , if (x, y) ∈ Ie × [e, 1L ], otherwise.

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Theorem 3.4. (See Çaylı and Karaçal [8].) Let e ∈ L \ {0L , 1L }. Suppose that either x ∨ y > e for all x  e and y  e or x ∨ y  e for all x  e and y  e. If Te is a t-norm on [0L , e]2 , then the function Ut3 : L2 −→ L defined as follows is a uninorm on L with the neutral element e: ⎧ ⎪ Te (x, y) if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [0L , e] × [e, 1L ] ∪ [e, 1L ] × [0L , e] ∪ Ie × Ie , ⎨x ∨ y Ut3 (x, y) = y if (x, y) ∈ [0L , e] × Ie , ⎪ ⎪ ⎪ x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎪ ⎩1 otherwise. L

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Theorem 3.5. (See Çaylı and Karaçal [8].) Let e ∈ L \ {0L , 1L }. Suppose that either x ∧ y < e for all x  e and y  e or x ∧ y  e for all x  e and y  e. If Se is a t-conorm on [e, 1L ]2 , then the function Us3 : L2 −→ L defined as follows is a uninorm on L with the neutral element e: ⎧ ⎪ Se (x, y) if (x, y) ∈ [e, 1L ]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [0L , e] × [e, 1L ] ∪ [e, 1L ] × [0L , e] ∪ Ie × Ie , ⎨x ∧ y Us3 (x, y) = y if (x, y) ∈ [e, 1L ] × Ie , ⎪ ⎪ ⎪ x if (x, y) ∈ Ie × [e, 1L ], ⎪ ⎪ ⎪ ⎩0 otherwise. L

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Theorem 3.6. (See Çaylı [5].) Let e ∈ L \ {0L , 1L }. If Te is a t-norm on [0L , e]2 and Se is a t-conorm on [e, 1L ]2 , then the functions UeT : L2 −→ L and UeS : L2 −→ L are uninorms on L with the neutral element e, where ⎧ Te (x, y) if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎪ ⎪ x ∨y if (x, y) ∈ [0L , e] × [e, 1L ] ∪ [e, 1L ] × [0L , e], ⎪ ⎪ ⎪ ⎨y if (x, y) ∈ [0L , e] × Ie , UeT (x, y) = ⎪ x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎪ ⎪ ⎪x ∨ y ∨ e if (x, y) ∈ Ie × Ie , ⎪ ⎪ ⎩ otherwise 1L

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if (x, y) ∈ [e, 1L ]2 , if (x, y) ∈ [0L , e] × [e, 1L ] ∪ [e, 1L ] × [0L , e], if (x, y) ∈ [e, 1L ] × Ie , if (x, y) ∈ Ie × [e, 1L ], if (x, y) ∈ Ie × Ie , otherwise.

We now introduce a new construction of uninorms on an arbitrary bounded lattice L with the neutral element. More precisely, we obtain two classes of uninorms with underlying t-norms and t-conorms on an arbitrary bounded lattice L. In particular, compared with the uninorms given in Theorem 3.3, for all x, y ∈ Ie , we try to replace the corresponding value x ∧ y (or x ∨ y) with x ∧ y ∧ e (or x ∨ y ∨ e) in L to construct new uninorms on an arbitrary bounded lattice L based on t-norms and t-conorms, which can be an extension of the class of Umin (or Umax ) on the unit interval. First, we introduce one class of uninorm U(T ,e) given in the following theorem. Theorem 3.7. Let e ∈ L \ {0L , 1L }. Then the function U(T ,e) : L2 −→ L defined as follows is a uninorm on L with the neutral element e, where Te is a t-norm on [0L , e]2 : ⎧ Te (x, y) if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎨y if (x, y) ∈ [0L , e] × Ie , U(T ,e) (x, y) = ⎪ x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎩ x ∨ y ∨ e otherwise.

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Proof. It should be noticed from Definition 2.3 that the commutativity and the neutral element are easily checked. The proof can be divided into two parts related to the monotonicity and associativity as follows: (I) For monotonicity, we show that if x ≤ y, then U(T ,e) (x, z) ≤ U(T ,e) (y, z) for all z ∈ L. The proof can be split into all possible cases.

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1.1.2. z ∈ (e, 1L ] or z ∈ Ie , U(T ,e) (x, z) = z = U(T ,e) (y, z).

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1.2. y ∈ (e, 1L ].

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1.2.1. z ∈ [0L , e], U(T ,e) (x, z) = Te (x, z) ≤ x ∧ z ≤ y ∧ z ≤ y = U(T ,e) (y, z). 1.2.2. z ∈ (e, 1L ] or z ∈ Ie , U(T ,e) (x, z) = z ≤ y ∨ z ∨ e = U(T ,e) (y, z).

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U(T ,e) (x, z) = Te (x, z) ≤ Te (y, z) = U(T ,e) (y, z).

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U(T ,e) (x, z) = x ≤ y = U(T ,e) (y, z).

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U(T ,e) (x, z) = x ∨ z ∨ e ≤ y ∨ z ∨ e = U(T ,e) (y, z). 3.2. y ∈ Ie .

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U(T ,e) (x, z) = x ≤ y = U(T ,e) (y, z).

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U(T ,e) (x, z) = x ∨ z ∨ e ≤ y ∨ z ∨ e = U(T ,e) (y, z).

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(II) For associativity, we need to check that U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (U(T ,e) (x, y), z) for all x, y, z ∈ L. The proof can be split into all possible cases.

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1. Let x ∈ [0L , e].

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, Te (y, z)) = Te (x, Te (y, z)) = Te (Te (x, y), z) = U(T ,e) (Te (x, y), z)

48

= U(T ,e) (U(T ,e) (x, y), z).

49

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3.2.2. z ∈ (e, 1L ] or z ∈ Ie ,

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3.1.2. z ∈ (e, 1L ] or z ∈ Ie ,

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3.1. y ∈ (e, 1L ).

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3. Let x ∈ Ie . Then y ∈ (e, 1L ) or y ∈ Ie .

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U(T ,e) (x, z) = x ∨ z ∨ e ≤ y ∨ z ∨ e = U(T ,e) (y, z).

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U(T ,e) (x, z) = x ≤ y = U(T ,e) (y, z).

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2.1. z ∈ [0L , e],

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2. Let x ∈ (e, 1L ]. Then y ∈ (e, 1L ].

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U(T ,e) (x, z) = z ≤ y ∨ z ∨ e = U(T ,e) (y, z).

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U(T ,e) (x, z) = Te (x, z) ≤ x ∧ z ≤ y ∧ z ≤ y = U(T ,e) (y, z).

3 4

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1.1.2. z ∈ (e, 1L ] or z ∈ Ie , U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, z) = z = U(T ,e) (Te (x, y), z) = U(T ,e) (U(T ,e) (x, y), z).

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1.2. y ∈ (e, 1L ].

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1.2.1. z ∈ [0L , e],

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y) = y = U(T ,e) (y, z) = U(T ,e) (U(T ,e) (x, y), z).

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1.2.2. z ∈ (e, 1L ] or z ∈ Ie ,

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1.3. y ∈ Ie .

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1.3.1. z ∈ [0L , e],

12 13

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y) = y = U(T ,e) (y, z) = U(T ,e) (U(T ,e) (x, y), z).

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z)) = y ∨ z = U(T ,e) (y, z) = U(T ,e) (U(T ,e) (x, y), z).

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z ∨ e) = y ∨ z ∨ e

21 22

= U(T ,e) (y, z)

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= U(T ,e) (U(T ,e) (x, y), z). 2. Let x ∈ (e, 1L ]. 2.1. y ∈ [0L , e].

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2.1.1. z ∈ [0L , e],

29 30

U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, Te (y, z)) = x = U(T ,e) (x, z) = U(T ,e) (U(T ,e) (x, y), z).

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, z) = x ∨ z = U(T ,e) (x, z) = U(T ,e) (U(T ,e) (x, y), z). 2.2. y ∈ (e, 1L ].

36 37

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2.2.1. z ∈ [0L , e],

38 39

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y) = x ∨ y = U(T ,e) (x ∨ y, z) = U(T ,e) (U(T ,e) (x, y), z).

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z) = x ∨ y ∨ z

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= U(T ,e) (x ∨ y, z)

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= U(T ,e) (U(T ,e) (x, y), z).

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2.3. y ∈ Ie .

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2.2.2. z ∈ (e, 1L ] or z ∈ Ie ,

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1.3.3. z ∈ Ie ,

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1.3.2. z ∈ (e, 1L ],

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z) = y ∨ z = U(T ,e) (y, z) = U(T ,e) (U(T ,e) (x, y), z).

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2.3.1. z ∈ [0L , e], U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y) = x ∨ y = U(T ,e) (x ∨ y, z) = U(T ,e) (U(T ,e) (x, y), z).

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2.3.2. z ∈ (e, 1L ],

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= U(T ,e) (x ∨ y, z)

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= U(T ,e) (U(T ,e) (x, y), z).

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z) = x ∨ y ∨ z

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2.3.3. z ∈ Ie ,

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= U(T ,e) (x ∨ y, z)

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= U(T ,e) (U(T ,e) (x, y), z).

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3. Let x ∈ Ie .

U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, Te (y, z)) = x = U(T ,e) (x, z) = U(T ,e) (U(T ,e) (x, y), z). 3.1.2. z ∈ (e, 1L ], 3.1.3. z ∈ Ie ,

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3.2.1. z ∈ [0L , e], 3.2.2. z ∈ (e, 1L ] or z ∈ Ie ,

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= U(T ,e) (x ∨ y, z)

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= U(T ,e) (U(T ,e) (x, y), z).

36 37

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3.3.1. z ∈ [0L , e],

49 50 51 52

34 35 36 37

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y) = x ∨ y ∨ e = U(T ,e) (x ∨ y ∨ e, z) = U(T ,e) (U(T ,e) (x, y), z).

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3.3. y ∈ Ie .

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z) = x ∨ y ∨ z

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y) = x ∨ y = U(T ,e) (x ∨ y, z) = U(T ,e) (U(T ,e) (x, y), z).

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3.2. y ∈ (e, 1L ].

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, z) = x ∨ z ∨ e = U(T ,e) (x, z) = U(T ,e) (U(T ,e) (x, y), z).

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, z) = x ∨ z = U(T ,e) (x, z) = U(T ,e) (U(T ,e) (x, y), z).

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3.1.1. z ∈ [0L , e],

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3.1. y ∈ [0L , e].

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z ∨ e) = x ∨ y ∨ z ∨ e = x ∨ y ∨ z

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3.3.2. z ∈ (e, 1L ],

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z) = x ∨ y ∨ z =x ∨y ∨e∨z = U(T ,e) (x ∨ y ∨ e, z) = U(T ,e) (U(T ,e) (x, y), z).

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[m3SC+; v1.295; Prn:5/02/2019; 14:55] P.10 (1-18)

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Fig. 1. The uninorm U(T ,e) on L.

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3.3.3. z ∈ Ie ,

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U(T ,e) (x, U(T ,e) (y, z)) = U(T ,e) (x, y ∨ z ∨ e) = x ∨ y ∨ z ∨ e

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= U(T ,e) (x ∨ y ∨ e, z)

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= U(T ,e) (U(T ,e) (x, y), z).

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Remark 3.8. The uninorm U(T ,e) ⎧ ⎪ Te (x, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨y U(T ,e) (x, y) = x ⎪ ⎪ ⎪ x ∨y ∨e ⎪ ⎪ ⎪ ⎩x ∨ y

presented in Theorem 3.7 can be described as the following equivalent structure: if (x, y) ∈ [0L , e]2 , if (x, y) ∈ [0L , e] × (e, 1L ] ∪ [0L , e] × Ie , if (x, y) ∈ (e, 1L ] × [0L , e] ∪ Ie × [0L , e], if (x, y) ∈ Ie × Ie , otherwise.

The structure of U(T ,e) on L is shown in Fig. 1. Remark 3.9. Let U(T ,e) be a uninorm in Theorem 3.7.

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22 23 24 25 26 27 28 29 30 31 32 33

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(1) The uninorm U(T ,e) is disjunctive; that is, U(T ,e) (0L , 1L ) = 1L . (2) In general, the uninorm U(T ,e) does not show the existence of idempotent uninorms on bounded lattices. More precisely, if x ∨ y is incomparable with e for all x, y ∈ Ie , then U(T ,e) (x, x) = x ∨ e = x. Moreover, if x ∨ y is comparable with e for all x, y ∈ Ie , then also (x ∨ y) > e and hence U(T ,e) = Ut2 for all x, y ∈ L. The uninorm Ut2 defined in Theorem 3.3 can be used to show the existence of idempotent uninorms on bounded lattices. (3) The uninorm U(T ,e) is equivalent to the following equation: ⎧ ⎪ if (x, y) ∈ [0L , e]2 , ⎨Te (x, y) U(T ,e) (x, y) = φ(x) ∨ φ(y) ∨ e if (x, y) ∈ Ie × Ie , ⎪ ⎩ φ(x) ∨ φ(y) otherwise, where φ : L −→ L is a mapping given by  0L if x ∈ [0L , e], φ(x) = x otherwise. (4) When we put S = S∨ in Theorem 3.2, the uninorm U(T ,e) should be a special case of the uninorm Ud introduced in Theorem 3.2. However, there is a little difference between the uninorm U(T ,e) and the uninorm Ud from the point of view of their specific structures. Precisely, we take the value of U(T ,e) (x, y) as y for all (x, y) ∈ [0L , e] × (e, 1L ], while

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Y. Dan, B.Q. Hu / Fuzzy Sets and Systems ••• (••••) •••–•••

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Bodjanova and Kalina [4] in Theorem 3.2 take the value of Ud (x, y) as y for all (x, y) ∈ [0L , e] × [e, 1L ]. That is to say, for all x ∈ [0L , e], Ud (x, e) = e instead of x. Thus, e is not the neutral element of Ud in Theorem 3.2 and Ud is not a uninorm. If we make a little revision for Ud in Theorem 3.2 as follows, then Ud is a uninorm, and when we put S = S∨ in Theorem 3.2, the uninorm U(T ,e) is a special case of the uninorm Ud : ⎧ ⎪ T (x, y) if x ≤ e, y ≤ e, ⎪ ⎪ ⎪ ⎪ ⎪ y if x ≤ e, y  e, ⎨ Ud (x, y) = x if x  e, y ≤ e, ⎪ ⎪ ⎪S(x, y) if x > e, y > e, ⎪ ⎪ ⎪ ⎩S(x ∨ e, y ∨ e) otherwise. Corollary 3.10. If we take Te = T∧ on [0L , e]2 in Theorem 3.7, then we obtain the following uninorm on L: ⎧ x ∧y if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎨y if (x, y) ∈ [0L , e] × Ie , U(T ,e) (x, y) = ⎪ x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎩ x ∨ y ∨ e otherwise.

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Corollary 3.11. If e ∈ L \ {0L } is a coatom of L, then, according to Theorem 3.7, we obtain the following uninorm on L, where Te is a t-norm on [0L , e]2 : ⎧ ⎪ Te (x, y) if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [0L , e] × Ie , ⎨y U(T ,e) (x, y) = x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎪ if (x, y) ∈ [0L , e] × {1L } ∪ {1L } × [0L , e], 1L ⎪ ⎪ ⎪ ⎩x ∨ y ∨ e otherwise. Corollary 3.12. If e ∈ L \ {0L } is a coatom of L and we put Te = T∧ on [0L , e]2 in Theorem 3.7, then we obtain the following uninorm on L: ⎧ ⎪ x ∧y if (x, y) ∈ [0L , e]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [0L , e] × Ie , ⎨y U(T ,e) (x, y) = x if (x, y) ∈ Ie × [0L , e], ⎪ ⎪ ⎪ if (x, y) ∈ [0L , e] × {1L } ∪ {1L } × [0L , e], 1 ⎪ L ⎪ ⎪ ⎩x ∨ y ∨ e otherwise.

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In the following, we discuss the relationship between the uninorm U(T ,e) defined in Theorem 3.7 and the uninorms Ut1 , Ut2 , Ut3 , and UeT proposed in Theorems 3.1, 3.3, 3.4, and 3.6, respectively. In addition, some illustrative examples for those uninorms are provided. First, we introduce the partially ordered relations on the set consisting of uninorms on L. We use the notation U to denote the set of all uninorms on L. Two uninorms U1 and U2 on L are said to be pointwise ordered (i.e., U1 ≤ U2 ) if U1 (x, y) ≤ U2 (x, y) for all x, y ∈ L. Thus, U is a partially ordered set with pointwise order [6,15]. We denote U (e) as the family of all uninorms on L with the neutral element e ∈ L. Similarly, each U (e) is a partially ordered set too. The following conclusions can be obtained immediately by the uninorms U(T ,e) , Ut1 , Ut2 , Ut3 , and UeT defined in Theorems 3.7, 3.1, 3.3, 3.4, and 3.6, respectively:

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40 41 42 43 44 45 46 47 48 49 50

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• Ut2 ≤ U(T ,e) ≤ Ut1 . • U(T ,e) ≤ UeT .

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[m3SC+; v1.295; Prn:5/02/2019; 14:55] P.12 (1-18)

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Fig. 2. The lattice L1 .

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Table 1 The uninorm Ut1 on L1 . Ut1 0L1 x y 0L1 0L1 x y x x 1L1 1L1 y y 1L1 1L1 z z 1L1 1L1 e 0L1 x y k k 1L1 1L1 1L1 1L1 1L1 1L1

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z z 1L1 1L1 1L1 z 1L1 1L1

e 0L1 x y z e k 1L1

k k 1L1 1L1 1L1 k 1L1 1L1

1L1 1L1 1L1 1L1 1L1 1L1 1L1 1L1

Table 2 The uninorm Ut2 on L1 . Ut2 0L1 x y 0L1 0L1 x y x x x z y y z y z z z z e 0L1 x y k k k k 1L1 1L1 1L1 1L1

26 27 28 29 30 31 32 33 34

39 40 41

44 45

e 0L1 x y z e k 1L1

k k k k k k k 1L1

1L1 1L1 1L1 1L1 1L1 1L1 1L1 1L1

48 49 50

• Ut3 ≤ UeT ≤ Ut1 . • If L is a chain (emphasis on [0, 1]), then U(T ,e) = Ut2 ≤ Ut1 and Ut1 = UeT = Ut3 . The uninorm U(T ,e) with the neutral element e does not have to coincide with the uninorms Ut1 , Ut2 , Ut3 , and UeT with the neutral element e on bounded lattices in general. We illustrate this argument by an example as follows.

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Example 3.13. Consider the lattice (L1 = {0L1 , x, y, z, e, k, 1L1 }, ≤, 0L1 , 1L1 ) described in Fig. 2. By using of the construction approaches in Theorems 3.1, 3.3, 3.4, 3.6, and 3.7, taking the t-norm Te = T∧ on [0L1 , e]2 , we define the uninorms Ut1 , Ut2 , Ut3 , UeT , and U(T ,e) by Tables 1, 2, 3, 4, and 5, respectively.

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• • • •

U(T ,e) = Ut1 U(T ,e) = Ut2 U(T ,e) = Ut3 U(T ,e) = UeT

since U(T ,e) (x, x) = k < 1L1 = Ut1 (x, x). since U(T ,e) (x, x) = k > x = Ut2 (x, x). since U(T ,e) (k, k) = k < 1L1 = Ut3 (k, k). since U(T ,e) (x, k) = k < 1L1 = UeT (x, k).

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z z z z z z k 1L1

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We introduce another class of uninorm U(S,e) given in the following theorem.

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Y. Dan, B.Q. Hu / Fuzzy Sets and Systems ••• (••••) •••–•••

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Table 3 The uninorm Ut3 on L1 . Ut3 0L1 x y 0L1 0L1 x y x x x z y y z y z z z z e 0L1 x y k k 1L1 1L1 1L1 1L1 1L1 1L1

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z z z z z z 1L1 1L1

e 0L1 x y z e k 1L1

k k 1L1 1L1 1L1 k 1L1 1L1

1L1 1L1 1L1 1L1 1L1 1L1 1L1 1L1

12

UeT

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0L1 x y z e k 1L1

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0L1 0L1 x y z 0L1 k 1L1

x x k k k x 1L1 1L1

y y k k k y 1L1 1L1

z z k k k z 1L1 1L1

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e 0L1 x y z e k 1L1

k k 1L1 1L1 1L1 k 1L1 1L1

1L1 1L1 1L1 1L1 1L1 1L1 1L1 1L1

Table 5 The uninorm U(T ,e) on L1 . U(T ,e) 0L1 x y 0L1 0L1 x y x x k k y y k k z z k k e 0L x y k k k k 1L1 1L1 1L1 1L1

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13 14 15 16 17 18 19

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z z k k k z k 1L1

e 0L1 x y z e k 1L1

k k k k k k k 1L1

1L1 1L1 1L1 1L1 1L1 1L1 1L1 1L1

Theorem 3.14. Let e ∈ L \ {0L , 1L }. Then the function U(S,e) : L2 −→ L defined as follows is a uninorm on L with the neutral element e, where Se is a t-conorm on [e, 1L ]2 : ⎧ Se (x, y) if (x, y) ∈ [e, 1L ]2 , ⎪ ⎪ ⎪ ⎨y if (x, y) ∈ [e, 1L ] × Ie , U(S,e) (x, y) = ⎪ x if (x, y) ∈ Ie × [e, 1L ], ⎪ ⎪ ⎩ x ∧ y ∧ e otherwise. Proof. It can be proved with Theorem 3.7 in an analogous way.

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Remark 3.15. The uninorm U(S,e) defined in Theorem 3.14 can be described as the following equivalent structure: ⎧ ⎪ Se (x, y) if (x, y) ∈ [e, 1L ]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [e, 1L ] × [0L , e) ∪ [e, 1L ] × Ie , ⎨y U(S,e) (x, y) = x if (x, y) ∈ [0L , e) × [e, 1L ] ∪ Ie × [e, 1L ], ⎪ ⎪ ⎪ x ∧ y ∧ e if (x, y) ∈ Ie × Ie , ⎪ ⎪ ⎪ ⎩x ∧ y otherwise.

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The structure of U(S,e) on L is shown in Fig. 3.

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Table 4 The uninorm UeT on L1 .

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Remark 3.16. Analogously to Remark 3.9, we give some points as follows. Let U(S,e) be a uninorm in Theorem 3.14.

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Fig. 3. The uninorm U(S,e) on L.

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(1) The uninorm U(S,e) is conjunctive; that is, U(S,e) (0L , 1L ) = 0L . (2) In general, the uninorm U(S,e) does not illustrate the existence of idempotent uninorms on bounded lattices. More precisely, if x ∧ y is incomparable with e for all x, y ∈ Ie , then U(S,e) (x, x) = x ∧ e = x. Moreover, if x ∧ y is comparable with e for all x, y ∈ Ie , then also (x ∧ y) < e and hence U(S,e) = Us2 for all x, y ∈ L. The uninorm Us2 constructed in Theorem 3.3 can be used to show the existence of idempotent uninorms on bounded lattices. (3) The uninorm U(S,e) is equivalent to the following equation: ⎧ ⎪ if (x, y) ∈ [e, 1L ]2 , ⎨Se (x, y) U(S,e) (x, y) = ϕ(x) ∧ ϕ(y) ∧ e if (x, y) ∈ Ie × Ie , ⎪ ⎩ ϕ(x) ∧ ϕ(y) otherwise, where ϕ : L −→ L is a mapping given by  1L if x ∈ [e, 1L ], ϕ(x) = x otherwise. (4) When we put T = T∧ in Theorem 3.2, the uninorm U(S,e) should be a special case of the uninorm Uc introduced in Theorem 3.2. However, there is a little difference between the uninorm U(S,e) and the uninorm Uc from the point of view of their specific structures. Precisely, we take the value of U(S,e) (x, y) as y for all (x, y) ∈ [e, 1L ] × [0L , e), while Bodjanova and Kalina [4] in Theorem 3.2 take the value of Uc (x, y) as y for all (x, y) ∈ [e, 1L ] × [0L , e]. That is to say, for all x ∈ [e, 1L ], Uc (x, e) = e instead of x. Thus, e is not the neutral element of Uc in Theorem 3.2 and Uc is not a uninorm. If we make a little revision for Uc in Theorem 3.2 as follows, then Uc is a uninorm, and when we put T = T∧ in Theorem 3.2, the uninorm U(S,e) is a special case of the uninorm Uc : ⎧ ⎪ S(x, y) if x ≥ e, y ≥ e, ⎪ ⎪ ⎪ ⎪ ⎪ y if x ≥ e, y  e, ⎨ Uc (x, y) = x if x  e, y ≥ e, ⎪ ⎪ ⎪ T (x, y) if x < e, y < e, ⎪ ⎪ ⎪ ⎩T (x ∧ e, y ∧ e) otherwise. ]2

Corollary 3.17. If we put Se = S∨ on [e, 1L in Theorem 3.14, then we obtain the following uninorm on L: ⎧ x ∨y if (x, y) ∈ [e, 1L ]2 , ⎪ ⎪ ⎪ ⎨y if (x, y) ∈ [e, 1L ] × Ie , U(S,e) (x, y) = ⎪ x if (x, y) ∈ Ie × [e, 1L ], ⎪ ⎪ ⎩ x ∧ y ∧ e otherwise.

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Fig. 4. The lattice L2 .

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Corollary 3.18. If e ∈ L \ {1L } is an atom of L, then, according to Theorem 3.14, we obtain the following uninorm on L, where Se is a t-conorm on [e, 1L ]2 : ⎧ ⎪ Se (x, y) if (x, y) ∈ [e, 1L ]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [e, 1L ] × Ie , ⎨y U(S,e) (x, y) = x if (x, y) ∈ Ie × [e, 1L ], ⎪ ⎪ ⎪ if (x, y) ∈ {0L } × [e, 1L ] ∪ [e, 1L ] × {0L }, 0L ⎪ ⎪ ⎪ ⎩x ∧ y ∧ e otherwise.

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Corollary 3.19. If e ∈ L \ {1L } is an atom of L and we put Se = S∨ on [e, 1L following uninorm on L: ⎧ ⎪ x ∨y if (x, y) ∈ [e, 1L ]2 , ⎪ ⎪ ⎪ ⎪ ⎪ if (x, y) ∈ [e, 1L ] × Ie , ⎨y U(S,e) (x, y) = x if (x, y) ∈ Ie × [e, 1L ], ⎪ ⎪ ⎪ if (x, y) ∈ {0L } × [e, 1L ] ∪ [e, 1L ] × {0L }, 0L ⎪ ⎪ ⎪ ⎩x ∧ y ∧ e otherwise.

]2

in Theorem 3.14, then we obtain the

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Similarly to the discussion of the relationship among the uninorms U(T ,e) , Ut1 , Ut2 , Ut3 , and UeT with pointwise order [6,15], we investigate the relationship between the uninorm U(S,e) defined in Theorem 3.14 and the uninorms Us1 , Us2 , Us3 , and UeS proposed in Theorems 3.1, 3.3, 3.5, and 3.6, respectively. In addition, we provide some illustrative examples for those uninorms.

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• • • •

Us1 ≤ U(S,e) ≤ Us2 . UeS ≤ U(S,e) . Us1 ≤ UeS ≤ Us3 . If L is a chain (emphasis on [0, 1]), then Us1 ≤ Us2 = U(S,e) and Us1 = UeS = Us3 .

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The uninorm U(S,e) with the neutral element e does not have to coincide with the uninorms Us1 , Us2 , Us3 , and UeS with the neutral element e on bounded lattices in general. We demonstrate this argument by an example as follows.

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Example 3.20. Consider the lattice (L2 = {0L2 , x, y, z, e, k, 1L2 }, ≤, 0L2 , 1L2 ) described in Fig. 4. By using of the construction approaches in Theorems 3.1, 3.3, 3.5, 3.6, and 3.14, putting the t-conorm Se = S∨ on [e, 1L2 ]2 , we define the uninorms Us1 , Us2 , Us3 , UeS , and U(S,e) by Tables 6, 7, 8, 9, and 10, respectively.

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• U(S,e) = Us1 since U(S,e) (y, y) = k > 0L2 = Us1 (y, y). • U(S,e) = Us2 since U(S,e) (y, y) = k < y = Us2 (y, y).

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Table 6 The uninorm Us1 on L2 . Us1 0L2 k z 0L2 0L2 0L2 0L2 k 0L2 0L2 0L2 0L2 0L2 0L2 z 0L2 0L2 0L2 x y 0L2 0L2 0L2 0L2 k z e 1L2 0L2 k z

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x 0L2 0L2 0L2 0L2 0L2 x x

y 0L2 0L2 0L2 0L2 0L2 y y

e 0L2 k z x y e 1L2

1L2 0L2 k z x y 1L2 1L2

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Table 7 The uninorm Us2 on L2 . Us2 0L2 k z 0L2 0L2 0L2 0L2 k 0L2 k k 0L2 k z z 0L2 k z x y 0L2 k z 0L2 k z e 1L2 0L2 k z

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Table 8 The uninorm Us3 on L2 . Us3 0L2 k z 0L2 0L2 0L2 0L2 k 0L2 0L2 0L2 0L2 0L2 z z 0L2 0L2 z x y 0L2 0L2 z 0L2 k z e 1L2 0L2 k z

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y 0L2 k z z y y y

e 0L2 k z x y e 1L2

1L2 0L2 k z x y 1L2 1L2

UeS 0L2 k z x y e 1L2

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x 0L2 0L2 z x z x x

y 0L2 0L2 z z y y y

e 0L2 k z x y e 1L2

1L2 0L2 k z x y 1L2 1L2

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0L2 0L2 0L2 0L2 0L2 0L2 0L2 0L2

k 0L2 0L2 0L2 0L2 0L2 k k

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z 0L2 0L2 k k k z z

x 0L2 0L2 k k k x x

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• U(S,e) = Us3 since U(S,e) (k, k) = k > 0L2 = Us3 (k, k). • U(S,e) = UeS since U(S,e) (x, k) = k > 0L2 = UeS (x, k).

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Table 9 The uninorm UeS on L2 .

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Uninorms on bounded lattices have been extensively studied similarly to research into uninorms on the unit interval. In particular, the construction of uninorms related to algebraic structures on bounded lattices is still an active research area. In this article, from the mathematical point of view, we have further investigated the topic of uninorms on bounded lattices with the neutral element e ∈ L \ {0L , 1L }. Precisely, we propose a new structure of uninorms on bounded lattices with the neutral element e ∈ L \ {0L , 1L }. The main results of this article are as follows:

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Table 10 The uninorm U(S,e) on L2 . U(S,e) 0L2 k z 0L2 0L2 0L2 0L2 k 0L2 k k 0L2 k k z 0L2 k k x y 0L2 k k 0L2 k z e 1L2 0L2 k z

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x 0L2 k k k k x x

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e 0L2 k z x y e 1L2

1L2 0L2 k z x y 1L2 1L2

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Readers can refer to Remarks 3.9 and 3.16, respectively, for more on comparabilities. • In addition, to well understand the relationship between the uninorms U(T ,e) and U(S,e) and some existing uninorms on bounded lattices, we provided some corresponding examples: Examples 3.13 and 3.20. • Moreover, in general, the uninorms U(T ,e) and U(S,e) defined in Theorems 3.7 and 3.14, respectively, are not idempotent. We expect that our results can be applied in many-valued logics (with truth value range L) and in several branches of mathematics and information sciences dealing with algebraic structures equipped with product-like operations [15]. For future work, we plan to consider the following issues: (1) Similarly to the construction of uninorms on bounded lattices by means of t-norms and t-conorms [5], we will continue to investigate the construction of uninorms on bounded lattices by use of both t-norms and t-conorms, and will study the corresponding properties. (2) We will consider the construction of uninorms on bounded lattices by using the method of ordinal sums. Acknowledgements The authors express their sincere thanks to the editors and anonymous reviewers for their most valuable comments and suggestions for improving this article greatly. The work described in this article was supported by grants from the National Natural Science Foundation of China (grant nos. 11571010 and 61179038).

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References

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[1] [2] [3] [4]

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(1) if x ∨ y is incomparable with e for all x, y ∈ Ie , then U(T ,e) ≥ Ut2 ; if x ∨ y is comparable with e for all x, y ∈ Ie , then also (x ∨ y) > e and hence U(T ,e) = Ut2 for all x, y ∈ L. (2) If x ∧ y is incomparable with e for all x, y ∈ Ie , then U(S,e) ≤ Us2 ; if x ∧ y is comparable with e for all x, y ∈ Ie , then also (x ∨ y) < e and hence U(S,e) = Us2 for all x, y ∈ L.

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• Based on a bounded lattice L, first, we introduced two new classes of uninorms U(T ,e) and U(S,e) with the neutral element e ∈ L \ {0L , 1L }, respectively. Then we obtained that the uninorms U(T ,e) and U(S,e) given in Theorems 3.7 and 3.14, respectively, are different from some existing uninorms on bounded lattices. For instance,

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