Distributivity equation between uninorms and Mayor’s aggregation operators 1

Abstract

Distributivity equation has been widely studied involving different classes of logical connectives or aggregation operators, such as implications, uninorms, t-operators and their generalizations. In this paper, we follow on these works by investigating the distributivity for uninorms and Mayor’s aggregation operators.

Keywords

Aggregation operators uninorms Mayor’s aggregation operators distributivity equation

1 Introduction

The distributivity property was posed many years ago (cf. Acz $\overset{´}{e} l$ [1], pp. 318-319) as an important property in many situations and it has gained more and more attention of researchers from both theory and application fields. Some studies have stressed the solutions of distributivity equations for t-norms and t-conorms [2 , 7], some for fuzzy implications and aggregations (such as t-norms, t-conorms, uninorms, nullnorms and so on, for instance, see [4-6 , 23]), and some for uninorms and t-operators or their generalizations (see, for instance, cf. [16 , 27-30]).

Recently, Jočić and Štajner-Papuga [13] investigated the distributivity between Mayor’s operators and some aggregation operators that involving the relaxed nullnorms and relaxed uninorms as well as aggregation operators without neutral and absorbing element. Qin and Wang [21] characterized solutions of distributivity equations between semi-t-operators and Mayor’s aggregation operators, which generalized results of distributivity between Mayor’s aggregation operators and semi-nullnorms [13]. In this paper, we will focus on the distributivity of uninorms and Mayor’s aggregation operators.

The paper is organized as follows. In Section 2 we recall some concepts and results of fuzzy connectives and aggregation operations that we will use along this paper. In Section 3, we explore the distributivity condition of Mayor’s aggregation operator over uninorms. In Section 4, we conversely discuss the equation that uninorms are distributive over Mayor’s aggregation operators. Finally, Section 5 includes some conclusions.

2 Preliminaries

In this section, we recall some concepts and results that will be used in the sequel.

Definition 2.1. [9 , 31] A binary operation U on [0, 1] ² is called a uninorm if it is associative, commutative, increasing in each place and has a neutral element e ∈ [0, 1].

A uninorm is called conjunctive if U (0, 1) =0 and disjunctive when U (0, 1) =1. Any uninorm U always satisfies U (0, 1) ∈ {0, 1}. Evidently, t-norms (or t-conorms, respectively) are uninorms with neutral element e = 1 (or e = 0, respectively). We denote the family of all uninorms with the neutral element e ∈ [0, 1] by $U$ . Furthermore, we list out the most studied four classes of uninorms which will be used in this paper:

$U_{e}^{min}$ and $U_{e}^{max}$ , those given by minimum and maximum respectively in [0, e [×] e, 1] ∪] e, 1] × [0, e [.

Idempotent uninorms, which satisfy U (x, x) = x for all x ∈ [0, 1].

Representable uninorms, those that have an additive (or multiplicative) generator.

Uninorms that are continuous in the open square ] 0, 1 [².

Theorem 2.2. [22, 31] Let e ∈ [0, 1]. Operations $U_{e}^{min} = {\begin{matrix} max & in [e,1]2, \\ min & otherwise; \end{matrix}$ and $U_{e}^{max} = {\begin{matrix} min & in [0,e]2, \\ max & otherwise . \end{matrix}$ are unique idempotent uninorms in $U_{e}^{min}$ and $U_{e}^{max}$ , respectively.

Theorem 2.3. [19 , 28] Let e ∈]0, 1 [. Then $U \in U_{e}$ is an idempotent uninorm if and only if there exists a decreasing function g : [0, 1] → [0, 1] with fixed point e, symmetric with respect to the identity function, such that U is given as follows: $U (x, y) = {\begin{matrix} min (x, y), \\ if y < g(x) or y=g(x) and x < g2 (x), \\ max (x, y), \\ if y > g(x) or y=g(x) and x > g2 (x), \\ max (x, y) or min (x, y), \\ if y=g(x) or y=g(x) and x=g2 (x) . \end{matrix}$ being commutative on the set of points (x, g (x)) such that x = g² (x).

The class of all idempotent uninorms with the neutral element e will be denoted by $U_{ide}^{e}$ .

Definition 2.4. [9, 31] Consider e ∈]0, 1 [. A binary operation U : [0, 1] ² → [0, 1] is a representable uninorm if and only if there exists a continuous strictly increasing function h : [0, 1] → [- ∞, + ∞] with h (0) = - ∞, h (e) =0 and h (1) =+ ∞ such that $U (x, y) = h^{- 1} (h (x) + h (y))$ for all (x, y) ∈ [0, 1] ² \ {(0, 1), (1, 0)} and U (0, 1) = U (1, 0) ∈ {0, 1}. The function h is usually called an additive generator of U.

It is easy to derive the multiplicative generator of representative uninorm U. Indeed, if h is an additive generator of U, then the function θ : [0, 1] → [0, + ∞] defined by $θ (x) = exp h (x)$ is a multiplicative generator of U, i.e., $U (x, y) = θ^{- 1} (θ (x) \cdot θ (y))$ for all x, y ∈ [0, 1].

We denote the class of representable uninorms with neutral element e by $U$ .

Theorem 2.5. [12, 28] Suppose $U \in U_{e}$ is continuous in ] 0, 1 [² with e ∈]0, 1 [. Then either one of the following cases is satisfied:

(i) There exist μ ∈ [0, e [, λ ∈ [0, μ], two continuous t-norms T₁ and T₂ and a representable uninorm R such that U can be represented as $U (x, y) = {\begin{matrix} λ T_{1} (\frac{x}{λ}, \frac{y}{λ}), \\ if x,y∈ [0,λ], \\ λ + (μ - λ) T_{2} (\frac{x - λ}{μ - λ}, \frac{y - λ}{μ - λ}), \\ if x,y∈ [λ,μ], \\ μ + (1 - μ) R (\frac{x - μ}{1 - μ}, \frac{y - μ}{1 - μ}), \\ if x,y∈ [μ,1[, \\ 1, \\ if min (x,y)∈]λ,1] and max (x,y)=1, \\ 1 or λ, \\ if (x,y)∈ { (λ,1),(1,λ)}, \\ min (x, y), \\ otherwise; \end{matrix}$ (1)

(ii) There exist ν ∈] e, 1], ω ∈ [ν, 1], two continuous t-conorms S₁ and S₂ and a representable uninorm R such that U can be represented as $U (x, y) = {\begin{matrix} ν + (ω - ν) S_{1} (\frac{x - ν}{ω - ν}, \frac{y - ν}{ω - ν}), \\ if x,y∈ [ν,ω], \\ ω + (1 - ω) S_{2} (\frac{x - ω}{1 - ω}, \frac{y - ω}{1 - ω}), \\ if x,y∈ [ω,1], \\ ν R (\frac{x}{ν}, \frac{y}{ν}), \\ if x,y∈]0,ν] \\ 0, \\ if max (x,y)∈ [and min (x,y)=0, \\ 0 or ω, \\ if (x,y)∈ {(ω 0), (0,ω)}, \\ max (x, y), \\ otherwise . \end{matrix}$ (2)

The class of all uninorms that are continuous in ] 0, 1 [² with the neutral element e will be denoted by $U_{e}^{cos}$ . A uninorm as in form (i) will be denoted by $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ and a uninorm as in form (ii) will be denoted by $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ . Evidently, the class of representable uninorms is a special case of the class of uninorms that are continuous in ] 0, 1 [².

Definition 2.6. [11, 16] A t-seminorm (or semi-t-norm, semicopula) T is an increasing operation T : [0, 1] ² → [0, 1] with neutral element 1.

A t-semiconorm is an increasing operation S : [0, 1] ² → [0, 1] with neutral element 0.

It is evident that a t-norm is a commutative and associative t-seminorm, a t-conorm is a commutative and associative t-semiconorm.

Definition 2.7. [13 , 21] A GM aggregation operator F is a commutative binary aggregation operator that satisfy the following boundary conditions for all x ∈ [0, 1]:

F (0, x) = F (0, 1) x and F (x, 1) = (1 - F (0, 1)) x + F (0, 1).

Let $GM$ denote the family of all GM aggregation operators. The following properties of the GM aggregation operators are essential for the further characterizations.

Theorem 2.8. [13 , 21] Let F be a GM aggregation operator. Then, the following holds:

(i) F is associative if and only if F is a t-norm or t-conorm;

(ii) F = min or F = max if and only if F (0, 1) =0 or F (0, 1) =1 and F (x, x) = x for all x ∈ [0, 1];

(iii) F is idempotent if and only if min ≤ F ≤ max.

Definition 2.9. [1] Let F, G : [0, 1] ² → [0, 1]. We say that F is distributive over G, if $F (x, G (y, z)) = G (F (x, y), F (x, z))$ (3) for all x, y, z ∈ [0, 1].

Lemma 2.10. [8, 21] Let F : X² → X have the right (left) neutral element e in a subset ∅ ≠ Y ⊂ X (i.e., ∀_x∈Y, F (x, e) = x (F (e, x) = x)). If the operation F is distributive over another operation G : X² → X that satisfies G (e, e) = e, then G is idempotent in Y.

Lemma 2.11. [8, 21] Every increasing operation F : [0, 1] ² → [0, 1] is distributive over max and min.

3 Distributivity equation for Mayor’s aggregation operators over uninorms

Lemma 3.1. Let $F \in GM$ and $U \in U_{e}$ . If F is distributive over U, then U is idempotent.

Proof. Assume that F (0, 1) = k. For any t ∈ [0, 1], we consider the following two cases:

If t ∈ [0, k], then there exists x ∈ [0, 1] such that t = kx. Hence, $\begin{matrix} t & = & kx = F (x, 0) = F (x, U (0, 0)) \\ = & U (F (x, 0), F (x, 0)) = U (kx, kx) \\ = & U (t, t) . \end{matrix}$

If t ∈ [k, 1], then there exists x ∈ [0, 1] such that t = k + (1 - k) x. Hence, $\begin{matrix} t & = & k + (1 - k) x = F (x, 1) \\ = & F (x, U (1, 1)) = U (F (x, 1), F (x, 1)) \\ = & U (t, t) . \end{matrix}$

Therefore, U is idempotent. □

From Lemma 2.11 and Lemma 3.1, we get the following results:

Corollary 3.2. Let $F \in GM$ . Then F is distributive over a t-norm T if and only if T = min.

Corollary 3.3. Let $F \in GM$ . Then F is distributive over a t-conorm S if and only if S = max.

Since the distributivity condition of GM aggregation operation over a t-norm (respectively t-conorm) have been investigated, in the following, we only consider the distributivity of GM aggregation operators over uninorms with neutral element e ∈]0, 1 [.

Lemma 3.4. Let $F \in GM$ such that F (0, 1) = k and $U \in U_{e}$ with e ∈]0, 1 [. If F is distributive over U, then k ∈ {0, 1}.

Proof. Firstly, we assume that U is a conjunctive uninorm. The distributivity implies that $\begin{matrix} k & = & F (1, 0) = F (1, U (0, 1)) \\ = & U (F (1, 0), F (1, 1)) \\ = & U (k, 1) . \end{matrix}$

If k ≥ e, then we have k = U (k, 1) ≥ U (e, 1) =1. If 0 ≤ k < e, then we have $\begin{matrix} k & = & F (0, 1) = F (0, U (e, 1)) \\ = & U (F (0, e), F (0, 1)) \\ = & U (ke, k) = ke . \end{matrix}$

The last step is due to that uninorm U is idempotent. Thus, combining that e ∈]0, 1 [, we get k = 0.

The case that U is a disjunctive uninorm can be similarly proved. □

Proposition 3.5. Let $F \in GM$ such that F (0, 1) =0 and $U \in U_{e}$ with e ∈]0, 1 [. F is distributive over U if and only if $U = U_{e}^{min}$ and F = (〈0, e, T₁〉, 〈e, 1, T₂〉), where T₁ and T₂ are commutative t-seminorms.

Proof. Note that F (x, 1) = x when F (0, 1) =0. Assume that F is distributive over U, then we get $\begin{matrix} e & = & F (e, 1) = F (e, U (1, e)) \\ = & U (F (e, 1), F (e, e)) = U (e, F (e, e)) \\ = & F (e, e) . \end{matrix}$

Furthermore, for any x ∈ [0, e], it follows from F (x, e) ≤ F (x, 1) = x that $\begin{matrix} x & = & F (x, 1) = F (x, U (e, 1)) \\ = & U (F (x, e), F (x, 1)) = U (F (x, e), x) \\ = & F (x, e) \land x = F (x, e) . \end{matrix}$

The last step but one is due to that U is idempotent. Thus, for any 0 ≤ x ≤ e ≤ y ≤ 1, from the monotonicity of F, we get x = F (x, e) ≤ F (x, y) ≤ F (x, 1) = x which indicates that F (x, y) = x = x ∧ y. In addition, for any x ∈] e, 1], we have e = F (e, e) ≤ F (x, e) ≤ F (1, e) = e, i.e., F (x, e) = e. Using the commutativity of F, we get F is an ordinal sum of two commutative t-seminorms T₁ and T₁.

Finally, we prove that $U = U_{e}^{min}$ . For any 0 ≤ x < e < y ≤ 1, we have $F (e, U (x, y)) = U (F (e, x), F (e, y)) = U (x, e) = x .$ If U (x, y) > e for some x ∈ [0, e [ and y ∈] e, 1], then x = F (e, U (x, y)) = e which is contradict with that x < e. Hence, for any 0 ≤ x < e < y ≤ 1, we have U (x, y) ≤ e. Thus, $U (x, y) = F (e, U (x, y)) = x = x \land y .$ Similarly, we have $U (x, y) = y = x \land y$ when 0 ≤ y < e < x ≤ 1. Therefore we have $U = U_{e}^{min}$ since U is idempotent.

Conversely, it is straightforward to prove that, when $U = U_{e}^{min}$ and F = (〈0, e, T₁〉, 〈e, 1, T₂〉), they satisfy Eq. (3) □

Corollary 3.6. Let $F \in GM$ and $U \in U_{e}$ with e ∈]0, 1 [, F is distributive over U. If $U = U_{e}^{max}$ , then F (0, 1) =1.

Proof. Assume that F (0, 1) ≠1, then it follows from Lemma 3.4 that F (0, 1) =0. Therefore, by Proposition 3.5, we get $U = U_{e}^{min}$ which contradicts with $U = U_{e}^{max}$ .

Analogously, we obtain the following results.

Proposition 3.7. Let $F \in GM$ such that F (0, 1) =1 and $U \in U_{e}$ with e ∈]0, 1 [. F is distributive over U if and only if $U = U_{e}^{max}$ and F = (〈0, e, S₁〉, 〈e, 1, S₂〉), where S₁ and S₂ are commutative t-semiconorms.

Corollary 3.8. Let $F \in GM$ and $U \in U_{e}$ with e ∈]0, 1 [, F is distributive over U. If $U = U_{e}^{min}$ , then F (0, 1) =0.

Summarizing the above results, we can characterize the solution of distributivity equation for Mayor’s aggregation operators over uninorms.

Theorem 3.9. Let $F \in GM$ and $U \in U_{e}$ . F is distributive over U if and only if one of the following results holds:

(i) U = min.

(ii) U = max.

(iii) $U = U_{e}^{min}$ and F = (〈0, e, T₁〉, 〈e, 1, T₂〉), where T₁ and T₂ are commutative t-seminorms.

(iv) $U = U_{e}^{max}$ and F = (〈0, e, S₁〉, 〈e, 1, S₂〉), where S₁ and S₂ are commutative t-semiconorms.

Remark 3.10. As we know that a semi-uninorm [14] is a non-decreasing operation U : [0, 1] ² → [0, 1] with neutral element e, i.e., by omitting commutativity and associativity from the definition of uninorms we get semi-uninorms. In all of the above results, commutativity and associativity are not needed. Therefore, Theorem 3.8 is not only suitable for uninorms but also for semi-uninorms. The differences between Theorem 3.9 and Theorem 23 in [13] are as follows:

(i) Theorem 23 (or Theorem 25) in [13] only discussed distributivity for Mayor’s aggregation operators over semi-uninorms in $N \max$ (or $N \min$ , respectively), those given by maximum (or minimum, respectively) in [0, e [×] e, 1] ∪] e, 1] × [0, e [. However, here we investigate distributivity for Mayor’s aggregation operators over any uninorms (or semi-uninorms).

(ii) The necessary condition presented by [13] in Theorem 23 is different from the sufficient and necessary condition in Theorem 3.9. Example 24 (in which F satisfies F (0, 1) =0 and $U = U_{e}^{max}$ ) in [13] illustrated that the sufficiency of Theorem 23 does not hold. In fact, Proposition 3.5 and Corollary 3.6 in this paper can explain this example properly, that is, the cases F (0, 1) =0 and $U = U_{e}^{max}$ could not exist simultaneously.

Example 3.11. (Example 24 in [13]) Let $F \in GM$ with F (0, 1) =0 and $U = U_{e}^{max}$ with 0 < e < 1. Evidently the distributivity of F over U does not hold, since the couple (F, U) does not satisfy any one of (i)-(iv) in Theorem 3.9.

Example 3.12. Let $U (x, y) = {\begin{matrix} max (x, y) & if (x,y)∈ [0.25,1]2, \\ min (x, y) & otherwise; \end{matrix}$ and $F (x, y) = {\begin{matrix} 4 xy, \\ if (x,y)∈ [0,0.25]2, \\ 0.25 + max (x + y - 1.25, 0), \\ if (x,y)∈ [0.25,1]2, \\ min (x, y), \\ otherwise . \end{matrix}$ Then $U = U_{0.25}^{min}$ and F = (〈0, 0.25, T_P〉, 〈0.25, 1, T_L〉). From Theorem 3.9, it follows that F is distributive over U.

4 Distributivity equation for uninorms over Mayor’s aggregation operators

In this section, we investigate the distributivity for uninorms over Mayor’s aggregation operators. Firstly, we consider the distributivity for t-norm and t-conorms over Mayor’s aggregation operators. Then we discuss the case of uninorms with neutral element e ∈]0, 1 [.

4.1 Distributivity for continuous t-norm T over $F \in GM$

In this part we will discuss the distributivity for continuous t-norm T over $F \in GM$ . Firstly, we need the following concept and lemmas:

Definition 4.1. [10] Let α ∈]0, 1 [. A binary operation T : [0, 1] ² → [0, 1] is said to be α-migrative if we have $T (α x, y) = T (x, α y)$ for all x, y ∈ [0, 1].

Lemma 4.2. [10] Let T be a continuous t-norm. If T is α-migrative, then T is strict.

Lemma 4.3. [14] A function T : [0, 1] ² → [0, 1] is a strict t-norm if and only if it is isomorphic to the product T_P, i.e., there exists an automorphism φ : [0, 1] → [0, 1] such that T (x, y) = φ^-1 (φ (x) · φ (y)) for all x, y ∈ [0, 1].

Lemma 4.4. Let T be a continuous t-norm and F a GM aggregation operator. If T is distributive on F, then F is idempotent.

Proof. It is directly from lemma 2.10. □

According to Lemma 4.4 and Theorem 2.8 (ii), we get the following result:

Corollary 4.5. Let T be a continuous t-norm and F a GM aggregation operator that satisfies F (0, 1) =0 (F (0, 1) =1, respectively). T is distributive on F if and only if F = min (F = max, respectively).

Theorem 4.6. Let $F \in GM$ with F (0, 1) = k ∈]0, 1 [ and T be a continuous t-norm. T is distributive over F if and only if T and F has the following forms: $T (x, y) = φ^{- 1} (φ (x) \cdot φ (y))$ (4) and $F (x, y) = {\begin{array}{l} φ^{- 1} [φ (x) \cdot φ (k + (1 - k) φ^{- 1} (\frac{φ (y)}{φ (x)}))], \\ if y \leq x and (x, y) \neq (0, 0), \\ φ^{- 1} [φ (y) \cdot φ (k + (1 - k) φ^{- 1} (\frac{φ (x)}{φ (y)}))], \\ if x < y, \\ 0, \\ if (x, y) = (0, 0), \end{array}$ (5) where φ : [0, 1] → [0, 1] is an automorphism of the unit interval [0, 1].

Proof. Assume that continuous t-norm T is distributive over GM aggregation operator F. For any x, y ∈ [0, 1], we have $\begin{matrix} T (x, ky) & = & T (x, F (0, y)) = F (T (x, 0), T (x, y)) \\ = & F (0, T (x, y)) \\ = & kT (x, y) . \end{matrix}$ Similarly, we have T (kx, y) = kT (x, y). By Definition 4.1, T is k-migrative. Therefore, it follows from Lemmas 4.2 and 4.3 that T has the form $T (x, y) = φ^{- 1} (φ (x) \cdot φ (y)),$ where φ : [0, 1] → [0, 1] is an automorphism of the unit interval [0, 1]. Thus, for any x, y, z ∈ [0, 1], the condition that T is distributive over F equivalents to $\begin{array}{l} φ^{- 1} (φ (x) \cdot φ (F (y, z))) \\ = F (φ^{- 1} (φ (x) \cdot φ (y)), φ^{- 1} (φ (x) \cdot φ (z))) \end{array}$ Therefore, it holds that $\begin{array}{l} φ (x) \cdot φ (F (y, z)) \\ = φ (F (φ^{- 1} (φ (x) \cdot φ (y)), φ^{- 1} (φ (x) \cdot φ (z)))) . \end{array}$

Let φ (x) = u, φ (y) = v and φ (z) = w, we get $u \cdot φ (F (φ^{- 1} (v), φ^{- 1} (w)) = φ (F (φ^{- 1} (uv), φ^{- 1} (uw)) .$ Define $F_{φ^{- 1}} (v, w) = φ (F (φ^{- 1} (v), φ^{- 1} (w)), (v, w) \in [0, 1]^{2},$ we have $u \cdot F_{φ^{- 1}} (v, w) = F_{φ^{- 1}} (uv, uw) .$ (6) Therefore, for any v ≤ u and (u, v) ≠ (0, 0), it follows from (6) that

$\begin{matrix} F_{φ^{- 1}} (u, v) & = & F_{φ^{- 1}} (u \cdot 1, u \cdot \frac{v}{u}) \\ = & u \cdot F_{φ^{- 1}} (1, \frac{v}{u}) \\ = & u \cdot φ (F (φ^{- 1} (1), φ^{- 1} (\frac{v}{u}))) \\ = & u \cdot φ (k + (1 - k) φ^{- 1} (\frac{v}{u})) . \end{matrix}$ Analogously, we get $F_{φ^{- 1}} (u, v) = v \cdot φ (k + (1 - k) φ^{- 1} (\frac{u}{v}))$ when v > u. Thus, by the definition of F_{φ
^-1}, we have F has the form (5).

Conversely, assume that there exists an automorphism φ : [0, 1] → [0, 1] such that T and F are give in (4) and (5), respectively. Now we prove that T is distributive over F.

Firstly, suppose that y ≤ z, and (y, z) ≠ (0, 0), then for any x ∈ [0, 1], we have $\begin{matrix} T (x, F (y, z)) = φ^{- 1} (φ (x) \cdot φ (F (y, z))) \\ = & φ^{- 1} (φ (x) \cdot φ (z) \cdot φ (k + (1 - k) φ^{- 1} (\frac{φ (y)}{φ (z)}))) . \end{matrix}$ Hence, we have $\begin{matrix} F (T (x, y), T (x, z)) \\ = & φ^{- 1} [φ (T (x, z)) \cdot φ (k + (1 - k) φ^{- 1} (\frac{φ (T (x, y))}{φ (T (x, z))}))] \\ = & φ^{- 1} [φ (x) \cdot φ (z) \cdot φ (k + (1 - k) φ^{- 1} (\frac{φ (x) \cdot φ (y)}{φ (x) \cdot φ (z)}))] \\ = & T (x, F (y, z)) . \end{matrix}$

Analogously, we can prove that T is distributive over F, if 0 ≤ z < y ≤ 1.

Finally, if (y, z) = (0, 0), then for any x ∈ [0, 1], we have $T (x, F (0, 0)) = T (x, 0) = 0 = F (T (x, 0), T (x, 0)) .$ From all the above, we get that T is distributive over F. □

Example 4.7. Let T (x, y) = xy and $F (x, y) = \frac{1}{2} (x + y)$ for all x, y ∈ [0, 1]. Then φ (x) = x, x ∈ [0, 1] and $k = F (0, 1) = \frac{1}{2}$ . It follows from Theorem 4.6 that T is distributive over F.

4.2 Distributivity of continuous t-conorm S over

F \in GM

We have discussed distributivity of continuous t-norm T over GM aggregation operator F. However, distributivity of continuous t-conorm S over GM aggregation operator F is similar. Therefore, here we only list the results without proofs.

Definition 4.8. [10] Let α ∈]0, 1 [. A binary operation S : [0, 1] ² → [0, 1] is said to be α-migrative if we have $S (x, α + (1 - α) y) = S (α + (1 - α) x, y) .$

Lemma 4.9. [10] Let S be a continuous t-conorm. If S is α-migrative, then S is strict.

Lemma 4.10. [14] A function S : [0, 1] ² → [0, 1] is a strict t-conorm if and only if there exists a continuous strictly decreasing function ψ : [0, 1] → [0, 1] with ψ (0) =1 and ψ (1) =0 such that S (x, y) = ψ^-1 (ψ (x) · ψ (y)) for all x, y ∈ [0, 1].

Lemma 4.11. Let S be a continuous t-conorm and F a GM aggregation operator. If S is distributive over F, then F is idempotent.

Corollary 4.12. Let S be a continuous t-conorm and F a GM aggregation operator that satisfies F (0, 1) =0 (F (0, 1) =1, respectively). S is distributive over F if and only if F = min (F = max, respectively).

Theorem 4.13. Let $F \in GM$ with F (0, 1) = k ∈]0, 1 [ and S be a continuous t-conorm. S is distributive over F if and only if S and F has the following forms: $S (x, y) = ψ^{- 1} (ψ (x) \cdot ψ (y))$ and $F (x, y) = {\begin{matrix} ψ^{- 1} [ψ (x) \cdot ψ (k \cdot ψ^{- 1} (\frac{ψ (y)}{ψ (x)}))], \\ if y≥ x and (x,y)≠ (1,1), \\ ψ^{- 1} [ψ (y) \cdot ψ (k \cdot ψ^{- 1} (\frac{ψ (x)}{ψ (y)}))], \\ if x> y, \\ 1, \\ if (x,y)=(1,1), \end{matrix}$ where ψ : [0, 1] → [0, 1] is a continuous strictly decreasing function with ψ (0) =1 and ψ (1) =0.

We have considered distributivity of continuous t-norm T and t-conorm S over GM aggregation operators F. In what follows we will discuss distributivity of uninorm U with neutral element e ∈]0, 1 [ over GM aggregation operators F.

4.3 Distributivity of $U \in U_{e}^{min}$ ( $U \in U_{e}^{max}$ ) over $F \in GM$

Lemma 4.14. [13] Let $F \in GM$ with F (0, 1) = k and $U \in U_{e}^{min}$ with e ∈]0, 1 [. If U is distributive over F, then k ∈ {0, 1}.

From Lemma 4.14, we only need to distinguish two cases: (i) F (0, 1) =1, (ii) F (0, 1) =0. Note that a GM aggregation operator F with F (0, 1) =1 becomes a t-semiconorm. Whereas the distributivity condition for uninorm $U \in U_{e}^{min}$ over a t-semiconorm has been considered by Theorem 10 in [28], here we only discuss case (ii).

Theorem 4.15. Let $F \in GM$ with F (0, 1) =0 and $U \in U_{e}^{min}$ with e ∈]0, 1 [. U is distributive over F if and only if F = min.

Proof. Suppose that U is distributive over F. Firstly, we can see that F (e, e) ≤ F (1, e) = e. If F (e, e) < e, we have $\begin{matrix} e & > & F (e, e) = U (1, F (e, e)) \\ = & F (U (1, e), U (1, e)) \\ = & F (1, 1) = 1, \end{matrix}$ which leads to a contradiction. Therefore, we get F (e, e) = e. According to Lemma 2.10 and Theorem 2.8 (ii), we have F = min.

Conversely, if F = min, then from Lemma 2.11, U is distributive over F. □

Remark 4.16. Theorem 4.15 is also suitable for a semi-uninorm $N_{e}^{min}$ . However, it is different from Theorem 22 in [13] in which the left-continuity at the point e of F is needed.

Similar to Lemma 4.14 and Theorem 4.15, we have the following results:

Lemma 4.17. [13] Let $F \in GM$ with F (0, 1) = k and $U \in U_{e}^{max}$ with e ∈]0, 1 [. If U is distributive over F, then k ∈ {0, 1}.

Theorem 4.18 Let $F \in GM$ with F (0, 1) =1 and $U \in U_{e}^{max}$ with e ∈]0, 1 [. U is distributive over F if and only if F = max.

The case $F \in GM$ with F (0, 1) =0 has been discussed by Theorem 7 in [28], here we omit it.

4.4 Distributivity of $U \in U_{e}^{ide}$ over $F \in GM$

Lemma 4.19. Let $F \in GM$ that satisfies F (0, 1) = k > 0, and $U \in U_{e}^{ide}$ with e ∈]0, 1 [. If U is distributive over F, then k = 1.

Proof. Suppose that U is distributive over F. For any 0 < y ≤ e, there exists x ∈]0, e [ such that x < ky ≤ y. Thus, $\begin{matrix} U (x, ky) & = & U (x, F (0, y)) = F (U (x, 0), U (x, y)) \\ = & F (0, x \land y) = F (0, x) \\ = & kx . \end{matrix}$ On the other hand, since U is idempotent, we have $U (x, ky) = x \land ky = x .$ Therefore, we get x = kx which implies k = 1. □

According to Lemma 4.19, we only need to consider two cases: (a) F (0, 1) =0, (b) F (0, 1) =1. However, the GM aggregation operator F is a t-seminorm (t-semiconorm, respectively) when F (0, 1) =0 (F (0, 1) =1, respectively). The distributivity of uninorms over t-seminorms (t-semiconorms, respectively) have been discussed by Theorem 11 (Theorem 12, respectively) in [28]. Therefore, we omit it here.

4.5 Distributivity of $U \in U_{e}^{rep}$ over $F \in GM$

Lemma 4.20. Let $F \in GM$ that satisfies F (0, 1) = k, and $U \in U_{e}^{rep}$ with e ∈]0, 1 [. If U is distributive over F, then k ∈ {0, 1}.

Proof. Assume that θ : [0, 1] → [0, + ∞] is a multiplicative generator of representative uninorm U. Since U is distributive over F, we have $\begin{matrix} U (x, k) & = & U (x, F (0, 1)) \\ = & F (U (x, 0), U (x, 1)) \\ = & F (0, 1) = k \end{matrix}$ for any x ∈]0, 1 [. Therefore, by the representation of uninorm U, we have $θ^{- 1} (θ (x) \cdot θ (k)) = k, x \in] 0, 1 [,$ which leads to k = 0 or k = 1. □

Theorem 4.21. Let $F \in GM$ that satisfies F (0, 1) =0, and $U \in U_{e}^{rep}$ with e ∈]0, 1 [. Then U is distributive over F if and only if one of following cases holds:

(i) F = min.

(ii) F is given by $F (x, y) = {\begin{matrix} θ^{- 1} (θ (x) \cdot θ (F (e, θ^{- 1} (\frac{θ (y)}{θ (x)})))), \\ if 0≤ y≤ x≤ 1 and (x,y)≠ (0,0), \\ θ^{- 1} (θ (y) \cdot θ (F (e, θ^{- 1} (\frac{θ (x)}{θ (y)})))), \\ if 0≤ x< y≤ 1, \\ 0, \\ if (x,y)=(0,0), \end{matrix}$ (7) where θ : [0, 1] → [0, + ∞] is a multiplicative generator of representative uninorm U.

Proof. If F (e, e) = e, then from Lemmas 2.10 and 2.11 it follows that F = min.

If F (e, e) ≠ e, then it follows from F (e, e) ≤ F (1, e) = e that F (e, e) < e. Let θ : [0, 1] → [0, + ∞] be a multiplicative generator of U. Due to the distributivity of U over F, we have $\begin{matrix} θ^{- 1} (θ (x) \cdot θ (F (y, z))) \\ = & F (θ^{- 1} (θ (x) \cdot θ (y)), θ^{- 1} (θ (x) \cdot θ (z))) \end{matrix}$ Therefore, $\begin{matrix} θ (x) \cdot θ (F (y, z)) \\ = & θ (F (θ^{- 1} (θ (x) \cdot θ (y)), θ^{- 1} (θ (x) \cdot θ (z)))) . \end{matrix}$ Let θ (x) = u, θ (y) = v, θ (z) = w, and define $G (u, v) = θ (F (θ^{- 1} (u), θ^{- 1} (v))),$ then G : [0, + ∞] ² → [0, + ∞] is a commutative increasing function that satisfies $G (0, v) = 0, G (+ \infty, v) = v, v \in [0, + \infty] .$ Thus, we have $u \cdot G (v, w) = G (uv, uw) .$ Therefore, if v ≤ u and (u, v) ≠ (0, 0), then we have $G (u, v) = G (u \cdot 1, u \cdot \frac{v}{u}) = u \cdot G (1, \frac{v}{u}) .$ By the definition of G, if y ≤ x and (x, y) ≠ (0, 0), we have $\begin{matrix} θ (F (x, y)) & = & θ (F (θ^{- 1} (u), θ^{- 1} (v))) \\ = & G (u, v) = u \cdot G (1, \frac{v}{u}) \\ = & θ (x) \cdot θ (F (e, θ^{- 1} (\frac{θ (y)}{θ (x)}))), \end{matrix}$ which leads to $F (x, y) = θ^{- 1} (θ (x) \cdot θ (F (e, θ^{- 1} (\frac{θ (y)}{θ (x)})))) .$ By the commutativity of F, we get that F has the form of Eq. (7).

Conversely, assume that U is a representative uninorm and F is given in Eq. (7). Then it is easy to prove that U is distributive over F. □

Remark 4.22. Eq. (7) shows that GM aggregation operator F is determined by the function x ↦ F (e, x). For example, let $F (e, x) = θ^{- 1} (\frac{θ (x)}{1 + θ (x)}), x \in [0, 1],$ where θ : [0, 1] → [0, + ∞] is a multiplicative generator of representative uninorm U. Then we have $F (x, y) = θ^{- 1} (\frac{θ (x) θ (y)}{θ (x) + θ (y)}) .$ That is, F is a t-norm with the additive generator $\frac{1}{θ (x)}$ , which coincides with Theorem 13 in [25]. However, F is not unique. For example, if we put $F (e, x) = θ^{- 1} (\frac{θ (x) + θ^{2} (x) + \dots + θ^{n} (x)}{1 + θ (x) + θ^{2} (x) + \dots + θ^{n} (x)}),$ where n is an arbitrary integer, then the representative uninorm U and GM aggregation operator F defined by Eq. (7) satisfy the distributive equation (3).

Example 4.23. Let $θ (x) = \frac{x}{1 - x}, x \in [0, 1 [$ , $U (x, y) = {\begin{matrix} 0, \\ if x=0,y=1 or x=1,y=0, \\ \frac{xy}{(1 - x) (1 - y) + xy}, \\ otherwise, \end{matrix}$ and $F (x, y) = {\begin{matrix} \frac{xy}{x - xy + y} & if (x,y)≠ (0,0), \\ 0 & if (x,y)=(0,0) . \end{matrix}$ Then θ is the multiplicative generator of U, and $F (x, y) = θ^{- 1} (\frac{θ (x) θ (y)}{θ (x) + θ (y)}) .$ Evidently, F satisfies Eq.(7). From Theorem 4.21, it follows that U is distributive over F.

Similar to Theorem 4.21, we have the following result:

Theorem 4.24. Let $F \in GM$ that satisfies F (0, 1) =1, and $U \in U_{e}^{rep}$ with e ∈]0, 1 [. U is distributive over F if and only if F has one of the following two forms:

(i) F = max.

(ii) F is given by $F (x, y) = {\begin{matrix} θ^{- 1} (θ (x) \cdot θ (F (e, θ^{- 1} (\frac{θ (y)}{θ (x)})))), \\ if 0≤ y≤ x≤ 1 and (x,y)≠ (1,1), \\ θ^{- 1} (θ (y) \cdot θ (F (e, θ^{- 1} (\frac{θ (x)}{θ (y)})))), \\ if 0≤ x< y≤ 1, \\ 1, \\ if (x,y)=(1,1), \end{matrix}$ (8) where θ : [0, 1] → [0, + ∞] is a multiplicative generator of representative uninorm U.

Remark 4.25. Analogous to Theorem 4.21, F is determined by the function x ↦ F (e, x). For example, if we put the function $F (e, x) = θ^{- 1} (1 + θ (x)),$ then we have $F (x, y) = θ^{- 1} (θ (x) + θ (y)) .$ That is, F is a t-conorm with additive generator θ (x), which coincides with Theorem 9 in [25]. However, F is not unique. For example, if the functions F (e, x) is given as follows:

$F (e, x) = θ^{- 1} (\frac{1 + θ (x) + θ^{2} (x) + \dots + θ^{n} (x)}{1 + θ (x) + θ^{2} (x) + \dots + θ^{n - 1} (x)}),$

where n is an arbitrary integer, then the representative uninorm U and GM aggregation operator F defined by (7) satisfy the distributive equation (3).

Example 4.26. Let $θ (x) = \frac{x}{1 - x}, x \in [0, 1 [$ , $U (x, y) = {\begin{matrix} 0, \\ if x=0,y=1 or x=1,y=0, \\ \frac{xy}{(1 - x) (1 - y) + xy}, \\ otherwise, \end{matrix}$

and $F (x, y) = {\begin{matrix} \frac{x + y - xy}{1 - xy} & if (x,y)≠ (1,1), \\ 1 & if (x,y)=(1,1) . \end{matrix}$ Then θ is the multiplicative generator of U, and $F (x, y) = θ^{- 1} (θ (x) + θ (y)) .$ Evidently, F satisfies Eq.(7). From Theorem 4.24, it follows that U is distributive over F.

4.6 Distributivity of

U \in U_{e}^{\cos}

over

F \in GM

In this section, we first explore the distributivity of $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ over $F \in GM$ . Analogously we get the corresponding results of distributivity of $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ over $F \in GM$ . Firstly, we begin with the following lemma.

Lemma 4.27. Let $F \in GM$ be a GM aggregation operator that satisfies F (0, 1) = k, and $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ a uninorm with e ∈]0, 1 [. If U is distributive over F, then k ∈ {0, 1}.

Proof. We consider the following two cases:

If U is a conjunctive uninorm, then it follows from the distributivity that $\begin{matrix} U (1, ke) & = & U (1, F (0, e)) \\ = & F (U (1, 0), U (1, e)) \\ = & F (0, 1) = k . \end{matrix}$ Based on the structure of uninorm U, we consider the following three subcases:

If ke < λ, then k = U (1, ke) = ke which implies k = 0 since 0 < e < 1.

If ke > λ, then k = U (1, ke) =1.

If ke = λ, then U (1, ke) =1 or U (1, ke) = ke which also implies k = 1 or k = 0.

If U is a disjunctive uninorm, then we have λ = 0. It follows from U is distributive over F that $\begin{matrix} U (0, k + (1 - k) e) & = & U (0, F (1, e)) \\ = & F (U (0, 1), U (0, e)) \\ = & F (1, 0) = k . \end{matrix}$ If k ∈]0, 1 [, then we have $k = U (0, k + (1 - k) e) = 0,$ which leads to a contradiction.

From all the above, we have k ∈ {0, 1}. □

From Lemma 4.27, to investigate the distributivity of U over F, we only need to consider two cases: (i) F (0, 1) =0; (ii) F (0, 1) =1.

Theorem 4.28. Let $F \in GM$ be a continuous GM aggregation operator that satisfies F (0, 1) =0, and $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ a uninorm with e ∈]0, 1 [. Then U is distributive over F if and only if F has one of the following two forms:

(i) F = min.

(ii) F is given by

$F (x, y) = {\begin{matrix} μ + (1 - μ) T (\frac{x - μ}{1 - μ}, \frac{y - μ}{1 - μ}), \\ if (x,y)∈ [μ,1]2, \\ min, \\ otherwise, \end{matrix}$ (9) where T is given by Eq. (7) in which θ : [0, 1] → [0, + ∞] is a multiplicative generator of the representative uninorm R in Eq. (1) that satisfies $θ (0) = 0, θ (\frac{e - μ}{1 - μ}) = 1, θ (1) = + \infty$ . In other words, R is distributive over the commutative t-seminorm T.

Proof. If F (e, e) = e, then it is direct a consequence from Lemma 2.10. If F (e, e) ≠ e, then it follows from F (e, e) ≤ F (1, e) = e that F (e, e) < e. Suppose that U is distributive over F. Firstly, we prove F is idempotent on [0, μ]. The proof is split into the following two steps:

The first step is to show F is idempotent on [0, λ]. For any x ∈]0, λ [, we have

$\begin{matrix} x & = & U (x, F (1, e)) \\ = & F (U (x, 1), U (x, e)) \\ = & F (x, x) . \end{matrix}$

Thus, $λ = lim_{x \to λ^{-}} F (x, x) \leq F (λ, λ) \leq F (1, λ) = λ$ , which implies F (λ, λ) = λ.

The second step is to show that F is idempotent on [λ, μ]. From F (λ, λ) = λ it follows that

$\begin{matrix} λ & = & F (λ, λ) \\ = & F (U (λ, e), U (λ, e)) \\ = & U (λ, F (e, e)) \end{matrix}$

If F (e, e) < λ, then we have λ = U (λ, F (e, e)) = F (e, e) < λ which leads to a contradiction. So we get λ ≤ F (e, e) < e. If λ ≤ F (e, e) < μ, then for any μ ≤ x < 1, we have

$\begin{matrix} F (x, x) & = & F (U (x, e), U (x, e)) \\ = & U (x, F (e, e)) = x \land F (e, e) \\ = & F (e, e) . \end{matrix}$

Thus, $lim_{x \to 1^{-}} F (x, x) = F (e, e) \neq F (1, 1) = 1$ which contradict with the continuity of F. Therefore, we get F (e, e) ≥ μ. Hence,

$\begin{matrix} μ & = & U (μ, F (e, e)) \\ = & F (U (μ, e), U (μ, e)) \\ = & F (μ, μ) . \end{matrix}$

For any x ∈ [λ, μ], we have

$\begin{matrix} x & = & U (x, μ) = U (x, F (μ, μ)) \\ = & F (U (x, μ), U (x, μ)) \\ = & F (x, x) . \end{matrix}$

Furthermore, we have x = F (x, x) ≤ F (x, y) ≤ F (x, 1) = x for any (x, y) ∈ [0, μ] × [μ, 1]. By the commutativity of F, F (x, y) = min(x, y) for any (x, y) ∈ [0, μ] × [μ, 1] ∪ [μ, 1] × [0, μ]. In addition, from F (x, μ) = F (μ, x) = μ and F (x, 1) = F (1, x) = x for any x ∈ [μ, 1] and Theorem 4.20 we know F has the form of Eq. (9).

Conversely, it can be easily verified that U is distributive over F. □

Distributive operators from Theorem 4.28 for $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ and F given by Eq. (9) can be viewed in Fig. 1.

Fig.1

Distributive operators from Theorem 4.28 for $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ (upper) and F (lower) given by Eq. (9), where R is distributive over the commutativet-seminorm T.

Theorem 4.29. Let $F \in GM$ be a GM aggregation operator that satisfies F (0, 1) =1, and $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ a uninorm with e ∈]0, 1 [. Then U is distributive over F if and only if one of the following cases holds:

(i) F = max.

(ii) There exists a function g : [μ, 1] → [0, + ∞] that satisfies g (μ) =0, g (e) =1 and g (1) =+ ∞ such that U is given by Eq. (1) in which U can be represented as $U (x, y) = g^{- 1} (g (x) \cdot g (y))$ on the square [μ, 1] ², and F is given by

$F (x, y) = {\begin{matrix} max (x, y), \\ if x,y∈ [0,μ], \\ g^{- 1} (g (F (e, y)) \cdot g (x)), \\ if x∈]μ,1] and y∈ [0,μ], \\ g^{- 1} (g (F (x, e)) \cdot g (y)), \\ if x∈ [0,μ] and y∈]μ,1], \\ g^{- 1} (g (y) \cdot g (F (g^{- 1} (\frac{g (x)}{g (y)}), e))), \\ if x,y∈]μ,1] and x≤ y, \\ g^{- 1} (g (x) \cdot g (F (e, g^{- 1} (\frac{g (y)}{g (x)})))), \\ if x,y∈]μ,1] and y< x . \end{matrix}$ (10)

Proof. If F (e, e) = e, then it is directly a consequence from Lemma 2.10. If F (e, e) ≠ e, then from F (e, e) ≥ F (0, e) = e it follows that F (e, e) > e. Now we consider the following cases:

If x, y ∈ [0, μ], then we have $\begin{matrix} x & = & x \land F (e, e) \\ = & U (x, F (e, e)) = F (U (x, e), U (x, e)) \\ = & F (x, x) . \end{matrix}$ Combining that F (0, x) = x, we have F (x, y) = max(x, y) for any x, y ∈ [0, μ].

If x ∈] μ, 1 [ and y ∈ [0, μ], then we have U (x, y) = x ∧ y = y. Assume that θ is multiplicative generator of representative uninorm R in Eq. (1) and let the function g : [μ, 1] → [0, + ∞] is given by $g (x) = θ (\frac{x - μ}{1 - μ}), x \in [μ, 1] .$ then we have g (μ) =0, g (e) =1, g (1) =+ ∞ and $U (x, y) = g^{- 1} (g (x) \cdot g (y)), x, y \in [μ, 1] .$ Furthermore, it follows from F (e, y) ≥ F (e, 0) = e > μ that for any x ∈] μ, 1 [ and y ∈ [0, μ], $\begin{matrix} F (x, y) & = & F (U (x, e), U (x, y)) \\ = & U (x, F (e, y)) \\ = & g^{- 1} (g (x) \cdot g (F (e, y))) . \end{matrix}$ (11) Note that Eq. (11) is also fit for x = 1 and y ∈ [0, μ], since F (1, y) =1.

If x ∈ [0, μ] and y ∈] μ, 1], then analogous to the above item we get $F (x, y) = g^{- 1} (g (y) \cdot g (F (x, e))) .$

If x, y ∈] μ, 1], we consider the following two subcases:

If μ < x ≤ y ≤ 1, then for any z ∈] μ, 1], we have $U (z, F (x, y)) = g^{- 1} (g (z) \cdot g (F (x, y))) .$ On the other hand, we get $\begin{matrix} F (U (z, x), U (z, y)) \\ = & F (g^{- 1} (g (z) \cdot g (x)), g^{- 1} (g (z) \cdot g (y))) . \end{matrix}$ Therefore, by the distributivity of U over F, we get $\begin{matrix} g (z) \cdot g (F (x, y)) \\ = & g (F (g^{- 1} (g (z) \cdot g (x)), g^{- 1} (g (z) \cdot g (y)))) . \end{matrix}$

Let g (x) = u, g (y) = v, g (z) = w and g (F (g^-1 (u), g^-1 (v))) = G (u, v), then we get $w \cdot G (u, v) = G (wu, wv), u, v, w \in] 0, + \infty] .$ Therefore, $G (u, v) = G (v \cdot \frac{u}{v}, v \cdot 1) = v \cdot G (\frac{u}{v}, 1) .$ By the definition of function G, we get $F (x, y) = g^{- 1} (g (y) \cdot g (F (g^{- 1} (\frac{g (x)}{g (y)}), e))) .$

If μ < y < x ≤ 1, then by the community of F, we have $F (x, y) = g^{- 1} (g (x) \cdot g (F (e, g^{- 1} (\frac{g (y)}{g (x)}))) .$

Conversely, if U and F are given by Eq. (1) and (10), respectively, then a routine calculation indicates that U is distributive over F. □

Distributive operators from Theorem 4.29 for $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ and F given by Eq. (10) can be viewed in Fig. 2.

Fig.2

Distributive operators from Theorem 4.29 for $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ (upper) and F (lower) given by Eq. (10), where *1 = g^-1 (g (F (e, y)) · g (x)), *2 = g^-1 (g (F (x, e)) · g (y)), $* 3 = g^{- 1} (g (x) \cdot g (F (e, g^{- 1} (\frac{g (y)}{g (x)}))))$ , $* 4 = g^{- 1} (g (y) \cdot g (F (g^{- 1} (\frac{g (x)}{g (y)}), e))) .$

The distributivity of $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ over $F \in GM$ has been studied. However, as for $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ , the results would be only listed since they can be provedsimilarly.

Lemma 4.30. Let $F \in GM$ be a GM aggregation operator that satisfies F (0, 1) = k, and $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ a uninorm with e ∈]0, 1 [. If U is distributive over F, then k ∈ {0, 1}.

Theorem 4.31. Let $F \in GM$ be a continuous GM aggregation operator that satisfies F (0, 1) =1, and $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ a uninorm with e ∈]0, 1 [. Then U is distributive over F if and only if F has one of the following forms:

(i) F = max.

(ii) F is given by $F (x, y) = {\begin{matrix} ν S (\frac{x}{ν}, \frac{y}{ν}) & if (x,y)∈ [0,ν]2, \\ max & otherwise, \end{matrix}$ (12) where S is given by (7) in which θ : [0, 1] → [0, + ∞] that satisfies $θ (0) = 0, θ (\frac{e}{ν}) = 1, θ (1) = + \infty$ is a multiplicative generator of the representative uninorm R in Eq. (2). In other words, R is distributive over commutative t-semiconorm S.

Distributive operators from Theorem 4.31 for $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ and F can be viewed in Fig. 3.

Fig.3

Distributive operators from Theorem 4.31 for $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ (upper) and F (lower) given by Eq. (13), where R is distributive over commutative t-semiconorm S.

Theorem 4.32. Let $F \in GM$ be a GM aggregation operator that satisfies F (0, 1) =0, and $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ a uninorm with e ∈]0, 1 [. Then U is distributive over F if and only if one of the two following cases holds:

(i) F = min.

(ii) There exists a function g : [0, ν] → [0, + ∞] that satisfies g (0) =0, g (e) =1 and g (ν) =+ ∞ such that U is given by Eq. (2) in which U can be represented as $U (x, y) = g^{- 1} (g (x) \cdot g (y))$ on the square [0, ν] ², and F is given by $F (x, y) = {\begin{matrix} min (x, y), \\ if x,y∈ [ν,1], \\ g^{- 1} (g (F (e, y)) \cdot g (x)), \\ if x∈ [and y∈ ν r1">0,ν and y∈ ν,1], \\ g^{- 1} (g (F (x, e)) \cdot g (y)), \\ if x∈ [ν,1] and y∈ [, \\ g^{- 1} (g (x) \cdot g (F (e, g^{- 1} (\frac{g (y)}{g (x)})))), \\ if x ry ∈ 0 rν and y< x, \\ g^{- 1} (g (y) \cdot g (F (g^{- 1} (\frac{g (x)}{g (y)}), e))), \\ if x ry ∈ 0 rν and x≤ y . \end{matrix}$ (13)

Distributive operators from Theorem 4.32 for $U \equiv 〈 T_{1}, λ, T_{2}, μ, (R, e) 〉_{min}^{\cos}$ and F given by Eq. (13) can be viewed in Fig. 4.

Fig.4

Distributive operators from Theorem 4.32 for $U \equiv 〈 S_{1}, ν, S_{2}, ω, (R, e) 〉_{max}^{\cos}$ (upper) and F (lower) given by Eq. (13), where *1 = g^-1 (g (F (e, y)) · g (x)), *2 = g^-1 (g (F (x, e)) · g (y)) and $* 3 = g^{- 1} (g (x) \cdot g (F (e, g^{- 1} (\frac{g (y)}{g (x)}))))$ , $* 4 = g^{- 1} (g (y) \cdot g (F (g^{- 1} (\frac{g (x)}{g (y)}), e))) .$

5 Conclusion

Distributivity of Mayor’s aggregation operators and other operations has been only studied in [13, 21], although the topic on distributivity has been widely investigated. In this work, we characterized solutions of the distributivity equation for uninorms and Mayor’s aggregation operators. This work enriched the results on distributivity of aggregation operators.

Footnotes

Acknowledgment

We are grateful to the anonymous reviewers and editors for their valuable comments that have improved our paper.

References

Aczél

Lectures on Functional Equations and Their Applications, Acad Press, New York, 1966.

Alsina

, Mayor

, Tomás

M.S.

and Torrens

, Subdistributive De Morgan triplets, Serdica 19 (1993), 258–266.

Alsina

and Trillas

, On almost distributive Lukasiewicz triplets, Fuzzy Sets Syst 50 (1992), 175–178.

Baczyński

and Jayaram

, On the distributivity of fuzzy implications over nilpotent or strict triangular conorms, IEEE Trans Fuzzy Syst 17 (2009), 590–603.

Baczyński

, On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms, Fuzzy Sets Syst 161 (2010), 1406–1419.

Balasubramaniam

and Jagan

, Mohan Rao, On the distributivity of implication operators over T- and S-norms, IEEE Trans Fuzzy Syst 12 (2004), 194–198.

Carbonell

, Mas

, Suner

and Torrens

, On the distributivity and modularity in De Morgan triplets, Int J Uncertain Fuzziness Knowl-Based Syst 4(4) (1996), 351–368.

Drygaś

, Distributivity between semi-t-operators and inullnorms, Fuzzy Sets Syst 264 (2015), 100–109.

Fodor

, Yager

R.R.

and Rybalov

, Structure of uninorms, Int J Uncertain Fuzziness Knowl-Based Syst 5(4) (1997), 411–427.

10.

Fodor

and Rudas

I.J.

, On continuous triangular norms that are migrative, Fuzzy Sets Syst 158 (2007), 1692–1697.

11.

Suárez Garcia

and Gil Álvarez

, Two families of fuzzy integrals, Fuzzy Sets Syst 18 (1986), 67–81.

12.

S.H.

and Li

Z.F.

, The structure of continuous uninorms, Fuzzy Sets Syst 124(1) (2001), 43–52.

13.

Jočić

and Štajner-Papuga

, Distributivity equations and Mayor’s aggregation operators, Knowledge Based Syst 52 (2013), 194–200.

14.

Klement

E.P.

, Mesiar

, Pap

Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.

15.

Liu

, Semi-uninorms and implications on a complete lattice, Fuzzy Sets Syst 191 (2012), 72–82.

16.

Liu

, Distributivity and conditional distributivity of semiuninorms over continuous t-conorms and t-norms, Fuzzy Sets Syst 268 (2015), 27–43.

17.

Mayor

, Contribuciòa l’estudi dels models matemàtics per a la lògica de la vaguetat, Universitat de les Illes Balears, (Ph. D. thesis), (1984).

18.

Mas

, Mayor

and Torrens

, The distributivity condition for uninorms and t-operators, Fuzzy Sets and Systems 128 (2002), 209–225.

19.

Martin

, Mayor

and Torrens

, On locally internal monotonic operations, Fuzzy Sets Syst 137(1) (2003) 27–42.

20.

Qin

and Yang

, Distributive equations of implications based on nilpotent triangular norms, Int J Approx Reasoning 51 (2010), 984–992.

21.

Qin

and Wang

Y.-M.

, Distributivity between semi-toperators and Mayor’s aggregation operators, Inf Sci 346-347 (2016), 6–16.

22.

Ruiz-Aguilera

and Torrens

, Distributivity of strong implications over conjunctive and disjunctive uninorms, Kybernetika 42 (2006), 319–336.

23.

Ruiz-Aguilera

and Torrens

, Distributivity of residual implications over conjunctive and disjunctive uninorms, Fuzzy Sets Syst 158 (2007), 23–37.

24.

Ruiz

and Torrens

, Distributive idempotent uninorms, Int J Uncertain Fuzziness Knowl-Based Syst 11 (2003), 413–428.

25.

Ruiz

and Torrens

, Distributivity and conditional distributivity of a uninorm and a continuous t-conorm, IEEE Trans Fuzzy Syst 14(2) (2006), 180–190.

26.

Ruiz-Aguilera

, Torrens

, De Baets

and Fodor

, Some remarks on the characterization of idempotent uninorms, in: Hullermeir

, Kruse

, Hoffmann

(Eds.), IPMU 2010. IN:LNAI, vol 6178, Springer-Verlag, Berlin. Hendelberg. 2010. pp. 425–434.

27.

, Zong

, Liu

and Xue

, The distributivity equations for semi-t-operators over uninorms, Fuzzy Sets Syst 287 (2016), 172–183.

28.

, Zong

and Liu

, On distributivity equations for uninorms over semi-t-operators, Fuzzy Sets Syst 299 (2016) 41–65.

29.

, Liu

, Ruiz-Aguilera

, Riera

J.V.

and Torrens

, On the distributivity property for uninorms, Fuzzy Sets Syst 287 (2016), 184–202.

30.

, Liu

, Riera

J.V.

, Ruiz-Aguilera

and Torrens

, The distributivity equation for uninorms revisited, Fuzzy Sets Syst 334 (2018), 1–23.

31.

Yager

R.R.

and Rybalov

, Uninorm aggregation operators, Fuzzy Sets Syst 80(1) (1996), 111–120.

Distributivity equation between uninorms and Mayor’s aggregation operators 1

Abstract

Keywords

1 Introduction

2 Preliminaries

4 Distributivity equation for uninorms over Mayor’s aggregation operators

4.1 Distributivity for continuous t-norm T over F ∈ GM

4.3 Distributivity of U ∈ U e min ( U ∈ U e max ) over F ∈ GM

4.4 Distributivity of U ∈ U e ide over F ∈ GM

4.5 Distributivity of U ∈ U e rep over F ∈ GM

Footnotes

Acknowledgment

References

4.1 Distributivity for continuous t-norm T over $F \in GM$

4.3 Distributivity of $U \in U_{e}^{min}$ ( $U \in U_{e}^{max}$ ) over $F \in GM$

4.4 Distributivity of $U \in U_{e}^{ide}$ over $F \in GM$

4.5 Distributivity of $U \in U_{e}^{rep}$ over $F \in GM$