Constructing conjunctive left (right) semi-uninorms and implications satisfying the neutrality principle

Abstract

In this paper, we further investigate the constructions of implications and left (right) semi-uninorms on a complete lattice. We firstly give out the formulas for calculating the upper and lower approximation conjunctive left (right) semi-uninorms of a binary operation. Then, we derive the formulas for calculating the upper and lower approximation NP implications of a binary operation. Finally, we show that the right (left) residuum of the upper approximation conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm of a right (left) infinitely ∨-distributive binary operation is the lower approximation right infinitely ∧-distributive NP implication of the right (left) residuum of the binary operation.

Keywords

Fuzzy connective implication left (right) semi-uninorm upper (lower) approximation neutrality principle (NP)

1 Introduction

In fuzzy logic systems (see [12]), connectives “and”, “or” and “not” are usually modeled by t-norms, t-conorms, and strong negations on [0, 1] (see [1, 17]), respectively. Based on these logical operators on [0, 1], the three fundamental classes of fuzzy implications on [0, 1], i.e., R-, S-, and QL-implications on [0, 1], were defined and extensively studied (see [2 , 27]). But, as was pointed out by Fodor and Keresztfalvi [11], sometimes there is no need of the commutativity or associativity for the connectives “and” and “or”. Thus, many authors investigated implications based on some other operators like weak t-norms [10], pseudo t-norms [38], pseudo-uninorms [28], left and right uninorms [36], semi-uninorms [19], aggregation operators [24, 26] and so on.

Uninorms, introduced by Yager and Rybalov [40], and studied by Fodor et al. [13], are special aggregation operators that have been used in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling (see [14 , 39]). This kind of operation is an important generalization of both t-norms and t-conorms and a special combination of t-norms and t-conorms (see [13, 20]). There are real-life situations when truth functions may not be associative or commutative. By throwing away the commutativity from the axioms of uninorms, Mas et al. introduced the concepts of left and right uninorms on [0, 1] in [21] and later on a finite chain in [22] and Wang and Fang [36, 37] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [19] introduced the concept of semi-uninorms and Su et al. [30] discussed the notions of left and right semi-uninorms on a complete lattice. On the other hand, it is well known that a uninorm (semi-uninorm) U can be conjunctive or disjunctive whenever U (0, 1) =0 or 1, respectively. This fact allows us to use uninorms in defining fuzzy implications (see [7 , 36]).

Constructing fuzzy connectives is an interesting topic. Recently, Jenei and Montagna [16] introduced several new types of constructions of left-continuous t-norms, Wang [34] derived the formulas for calculating the smallest pseudo-t-norm that is stronger than a binary operation and the largest implication that is weaker than a binary operation, Su et al. [30] studied the constructions of left and right semi-uninorms on a complete lattice, Li and Liu [18] derived (U, N)-implications from commutative semiuninorms and pseudo-uninorms, and Su and Wang [29] investigated the constructions of implications and coimplications on a complete lattice. Moreover, Wang [35] discussed the relations between left (right) semi-uninorms and coimplications, Hao et al. [15] investigated the relations between implications and left (right) semi-uninorms, and Niu et al. [25] studied the relations among implications, coimplications and left (right) semi-uninorms, on a completelattice.

This paper continues to explore the research direction started in [15 , 30]. Motivated by these works in [16 , 34], we will further focus on this issue and investigate constructions of the upper and lower approximation conjunctive left (right) semi-uninorms and the upper and lower approximation NP implications on a complete lattice.

This paper is organized as follows. After recalling some necessary definitions and examples about the implications and the left (right) semi-uninorms on a complete lattice in Section 2, we give out the formulas for calculating the upper and lower approximation conjunctive left (right) semi-uninorms in Section 3. In Section 4, we derive the formulas for calculating the upper and lower approximation NP implications. In Section 5, we reveal the relationships between the upper approximation conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms and lower approximation right infinitely ∧-distributive NP implications.

It should be point out that some methods and skills in this paper root in [29, 30].

The knowledge about lattices required in this paper can be found in [4].

2 Implications and left (right) semi-uninorms

Throughout this paper, unless otherwise stated, L always represents any given complete lattice with maximal element 1 and minimal element 0; J stands for any index set.

In this section, we recall some necessary concepts about the implications and left (right) semi-uninorms and illustrate these notions by means of some examples.

Definition 2.1. (Baczyński and Jayaram [2], Bustince et al. [6], De Baets and Fodor [7], Fodor and Roubens [12]) An implication I on L is a hybrid monotonic binary operation (i.e., a function that is decreasing in its first variable and increasing in its second variable) that satisfies the corner conditions I (0, 0) = I (1, 1) =1 and I (1, 0) =0.

An implication I is said to satisfy the neutrality principle with respect to e (w.r.t. e, for short) if I (e, y) = y for any y ∈ L.

An implication that satisfies the neutrality principle is also called an NP implication.

Implications are extensions of the Boolean implication ⇒ (P ⇒ Q meaning that P is sufficient for Q).

Note that for any implication I on L, due to the monotonicity, the absorption principle holds, i.e., I (0, x) = I (x, 1) =1 for any x ∈ L.

Definition 2.2. (Hao et al. [15], Su et al. [30]) A binary operation U on L is called a left semi-uninorm if it satisfies the following two conditions:

(U1)_l there exists a left neutral element, i. e., an element e_L ∈ L satisfying U (e_L, x) = x for all x ∈ L,

(U2) U is increasing in each variable.

Definition 2.3. (Hao et al. [15], Su et al. [30]) A binary operation U on L is called a right semi-uninorm if it satisfies the following two conditions:

(U1)_r there exists a right neutral element, i.e., an element e_R ∈ L satisfying U (x, e_R) = x for all x ∈ L,

(U2) U is increasing in each variable.

Clearly, U (0, 0) =0 and U (1, 1) =1 hold for any left (right) semi-uninorm U on L.

For any left (right) semi-uninorm U on L, U is said to be left-conjunctive and right-conjunctive if U (0, 1) =0 and U (1, 0) =0, respectively. U is called conjunctive if both U (0, 1) =0 and U (1, 0) =0 since it satisfies the classical boundary conditions of AND.

If a left (right) semi-uninorm U on L is associative, then U is also called the left (right) uninorm in[36, 37].

If a left (right) semi-uninorm U with the left (right) neutral element e_L (e_R) has a right (left) neutral element e_R (e_L), then e_L = U (e_L, e_R) = e_R. In this situation, we will denote e_L = e_R by e. Here, U is called the semi-uninorm in [19]. In particular, if the neutral element e = 1, then the semi-uninorm U becomes a t-seminorm (see [31]) or a semi-copula (see [9]); if the neutral element e = 0, then the semi-uninorm U becomes a t-semiconorm (see [8]).

Definition 2.4. (Wang and Fang [36, 37]) A binary operation U on L is called left infinitely ∨-distributive or left arbitrary ∨-distributive if it preserves arbitrary joins in its first argument, i.e., $U (⋁_{j \in J} x_{j}, y) = ⋁_{j \in J} U (x_{j}, y) \forall x_{j}, y \in L;$ left infinitely ∧-distributive or left arbitrary ∧-distributive if it preserves arbitrary meets in its first argument, i.e., $U (⋀_{j \in J} x_{j}, y) = ⋀_{j \in J} U (x_{j}, y) \forall x_{j}, y \in L .$

Definition 2.5. (Wang and Fang [36, 37]) A binary operation U on L is called right infinitely ∨-distributive or right arbitrary ∨-distributive if it preserves arbitrary joins in its second argument, i.e., $U (x, ⋁_{j \in J} y_{j}) = ⋁_{j \in J} U (x, y_{j}) \forall x, y_{j} \in L;$ right infinitely ∧-distributive or right arbitrary ∧-distributive if it preserves arbitrary meets in its second argument, i.e., $U (x, ⋀_{j \in J} y_{j}) = ⋀_{j \in J} U (x, y_{j}) \forall x, y_{j} \in L .$

If a binary operation U is left infinitely ∨-distributive (∧-distributive) and also right infinitely ∨-distributive (∧-distributive), then U is said to be infinitely ∨-distributive (∧-distributive).

When J =∅, noting that the least upper bound of the empty set is 0 and the greatest lower bound of the empty set is 1 (see [4, 5]), we have that $\begin{matrix} U (0, y) = U (⋁_{j \in J} x_{j}, y) = ⋁_{j \in J} U (x_{j}, y) = 0 \\ (U (x, 0) = U (x, ⋁_{j \in J} y_{j}) = ⋁_{j \in J} U (x, y_{j}) = 0) \end{matrix}$ for any x, y ∈ L when U is left (right) infinitely ∨-distributive and $\begin{matrix} U (1, y) = U (⋀_{j \in J} x_{j}, y) = ⋀_{j \in J} U (x_{j}, y) = 1 \\ (U (x, 1) = U (x, ⋀_{j \in J} y_{j}) = ⋀_{j \in J} U (x, y_{j}) = 1) \end{matrix}$ for any x, y ∈ L when U is left (right) infinitely ∧-distributive.

For the sake of convenience, we introduce the following symbols:

$U_{s}^{e_{L}} (L)$ : the set of all left semi-uninorms with the left neutral element e_L on L;

$U_{s}^{e_{R}} (L)$ : the set of all right semi-uninorms with the right neutral element e_R on L;

$U_{cs}^{e_{L}} (L)$ : the set of all conjunctive left semi-uninorms with the left neutral element e_L on L;

$U_{cs}^{e_{R}} (L)$ : the set of all conjunctive right semi-uninorms with the right neutral element e_R on L;

$U_{s \lor}^{e_{L}} (L)$ : the set of all right infinitely ∨-distributive left semi-uninorms with the left neutral element e_L on L;

$U_{\lor s}^{e_{R}} (L)$ : the set of all left infinitely ∨-distributive right semi-uninorms with the right neutral element e_R on L;

$U_{cs \lor}^{e_{L}} (L)$ : the set of all conjunctive right infinitely ∨-distributive left semi-uninorms with the left neutral element e_L on L;

$U_{\lor cs}^{e_{R}} (L)$ : the set of all conjunctive left infinitely ∨-distributive right semi-uninorms with the right neutral element e_R on L;

$I (L)$ : the set of all implications on L;

$I_{\land} (L)$ : the set of all right infinitely ∧-distributive implications on L;

$I^{npe} (L)$ : the set of all implications which satisfy the neutrality principle w.r.t. e on L;

$I_{\land}^{npe} (L)$ : the set of all right infinitely ∧-distributive implications which satisfy the neutrality principle w.r.t. e on L.

Example 2.6. (Su et al. [30]) Let e_L ∈ L, $\begin{matrix} U_{sW}^{e_{L}} (x, y) = {\begin{matrix} y & if x \geq e_{L}, \\ 0 & otherwise, \end{matrix} \\ U_{sM}^{e_{L}} (x, y) = {\begin{matrix} y & if x \leq e_{L}, \\ 1 & otherwise, \end{matrix} \\ U_{csM}^{e_{L}} (x, y) = {\begin{matrix} 0 & if x = 0 or y = 0, \\ y & if 0 < x \leq e_{L}, y \neq 0, \\ 1 & otherwise, \end{matrix} \end{matrix}$ where x and y are elements of L. We know that $U_{sW}^{e_{L}}$ and $U_{sM}^{e_{L}}$ are, respectively, the smallest and greatest elements of $U_{s}^{e_{L}} (L)$ ; $U_{sW}^{e_{L}}$ is the smallest element of $U_{s \lor}^{e_{L}} (L)$ . Moreover, we can see that $U_{sW}^{e_{L}}$ is the smallest element of both $U_{cs}^{e_{L}} (L)$ and $U_{cs \lor}^{e_{L}} (L)$ when e_L ≠ 0; $U_{csM}^{e_{L}}$ is the greatest element of both $U_{cs}^{e_{L}} (L)$ and $U_{cs \lor}^{e_{L}} (L)$ .

Example 2.7. (Su et al. [30]) Let e_R ∈ L, $\begin{matrix} U_{sW}^{e_{R}} (x, y) = {\begin{matrix} x & if y \geq e_{R}, \\ 0 & otherwise, \end{matrix} \\ U_{sM}^{e_{R}} (x, y) = {\begin{matrix} x & if y \leq e_{R}, \\ 1 & otherwise, \end{matrix} \\ U_{csM}^{e_{R}} (x, y) = {\begin{matrix} 0 & if x = 0 or y = 0, \\ x & if 0 < y \leq e_{R}, x \neq 0, \\ 1 & otherwise, \end{matrix} \end{matrix}$ where x and y are elements of L. We know that $U_{sW}^{e_{R}}$ and $U_{sM}^{e_{R}}$ are, respectively, the smallest and greatest elements of $U_{s}^{e_{R}} (L)$ ; $U_{sW}^{e_{R}}$ is the smallest element of $U_{\lor s}^{e_{R}} (L)$ . Moreover, we can see that $U_{sW}^{e_{R}}$ is the smallest element of both $U_{cs}^{e_{R}} (L)$ and $U_{\lor cs}^{e_{R}} (L)$ when e_R ≠ 0; $U_{csM}^{e_{R}}$ is the greatest element of both $U_{cs}^{e_{R}} (L)$ and $U_{\lor cs}^{e_{R}} (L)$ .

3 The upper and lower approximation conjunctive left (right) semi-uninorms

Constructing logic operators is an interesting work. Recently, Jenei and Montagna [16] introduced several new types of constructions of left-continuous t-norms and Su et al. [30] studied the constructions of left and right semi-uninorms on a complete lattice. In this section, we continue the works in [16, 30] and give out the formulas for calculating the upper and lower approximation conjunctive left (right) semi-uninorms of a binary operation.

For any nonempty subfamily {T_j ∣ j ∈ J} of L^L×L, the least upper bound ∨_j∈JT_j and the greatest lower bound ∧_j∈JT_j of $T_{j}^{'} s$ are, respectively, define by $\begin{matrix} (⋁_{j \in J} T_{j}) (x, y) = ⋁_{j \in J} T_{j} (x, y) \forall x, y \in L, \\ (⋀_{j \in J} T_{j}) (x, y) = ⋀_{j \in J} T_{j} (x, y) \forall x, y \in L . \end{matrix}$ If e_L ≠ 0, then it is easy to verify that $\lor_{j \in J} U_{j} \in U_{cs}^{e_{L}} (L)$ for any nonempty subset {U_j ∣ j ∈ J} of $U_{cs}^{e_{L}} (L)$ . Due to Theorem 4.2 in [5] and Example 2.6, we see that $U_{cs}^{e_{L}} (L)$ is a complete lattice with the smallest element $U_{sW}^{e_{L}}$ and greatest element $U_{csM}^{e_{L}}$ . Thus, for a binary operation A on L, if there exists $U \in U_{cs}^{e_{L}} (L)$ such that A ≤ U, then $⋀ {U ∣ A \leq U, U \in U_{cs}^{e_{L}} (L)}$ is the smallest conjunctive left semi-uninorm that is stronger than A on L, we call it the upper approximation conjunctive left semi-uninorm of A and denote it by $[A)_{cs}^{e_{L}}$ ; if there exists $U \in U_{cs}^{e_{L}} (L)$ such that U ≤ A, then $⋁ {U ∣ U \leq A, U \in U_{cs}^{e_{L}} (L)}$ is the largest conjunctive left semi-uninorm that is weaker than A on L, we call it the lower approximation conjunctive left semi-uninorm of A and denote it by $(A]_{cs}^{e_{L}}$ .

Similarly, we introduce the following symbols:

$[A)_{cs}^{e_{R}}$ : the upper approximation conjunctive right semi-uninorm of A;

$(A]_{cs}^{e_{R}}$ : the lower approximation conjunctive right semi-uninorm of A;

$[A)_{cs \lor}^{e_{L}}$ : the upper approximation conjunctive right infinitely ∨-distributive left semi-uninorm of A;

$(A]_{cs \lor}^{e_{L}}$ : the lower approximation conjunctive right infinitely ∨-distributive left semi-uninorm of A;

$[A)_{\lor cs}^{e_{R}}$ : the upper approximation conjunctive left infinitely ∨-distributive right semi-uninorm of A;

$(A]_{\lor cs}^{e_{R}}$ : the lower approximation conjunctive left infinitely ∨-distributive right semi-uninorm of A.

Definition 3.1. (see Su et al. [30]) Let A ∈ L^L×L. Define the upper approximation aggregator A_ua and the lower approximation aggregator A_la of A as follows: $\begin{matrix} A_{ua} (x, y) = ⋁ {A (u, v) ∣ u \leq x, v \leq y} \forall x, y \in L, \\ A_{la} (x, y) = ⋀ {A (u, v) ∣ u \geq x, v \geq y} \forall x, y \in L . \end{matrix}$

Theorem 3.2. (see Su et al. [30]) Let A, B ∈ L^L×L. Then the following statements hold:

A_la ≤ A ≤ A_ua.

(A ∨ B) _ua = A_ua ∨ B_ua and (A ∧ B) _la = A_la ∧ B_la.

A_ua and A_la are increasing in its each variable.

If A is increasing in its each variable, then A_ua = A_la = A.

Theorem 3.3.Let A ∈ L^L×L.

If A is left (right) infinitely ∨-distributive, then A_ua is left (right) infinitely ∨-distributive.

If A is left (right) infinitely ∧-distributive, then A_la is left (right) infinitely ∧-distributive.

Proof. We only prove that statement (1) holds.

If A is left infinitely ∨-distributive, then A is increasing in its first variable, $\begin{matrix} A_{ua} (x, y) & = & ⋁ {A (u, v) ∣ u \leq x, v \leq y} \\ = & ⋁ {A (x, v) ∣ v \leq y} \forall x, y \in L, \\ A_{ua} (⋁_{j \in J} x_{j}, y) & = & ⋁ {A (⋁_{j \in J} x_{j}, v) ∣ v \leq y} \\ = & ⋁ {⋁_{j \in J} A (x_{j}, v) ∣ v \leq y} \\ = & ⋁_{j \in J} (⋁ {A (x_{j}, v) ∣ v \leq y}) \\ = & ⋁_{j \in J} A_{ua} (x_{j}, y) \forall x_{j}, y \in L (j \in J), \end{matrix}$ i.e., A_ua is left infinitely ∨-distributive.

Similarly, we can show that A_ua is right infinitely ∨-distributive when A is right infinitely ∨-distributive. □

Su et al. [30] have constructed the upper and lower approximation left (right) semi-uninorms on a complete lattice. Below, we give out the formulas for calculating the upper and lower approximation conjunctive left (right) semi-uninorms of a binary operation.

Theorem 3.4.Let A ∈ L^L×L and 0 ≠ e_L ∈ L.

If $A \leq U_{csM}^{e_{L}}$ , then $[A)_{cs}^{e_{L}} = U_{sW}^{e_{L}} \lor A_{ua}$ .

If $U_{sW}^{e_{L}} \leq A$ , then $(A]_{cs}^{e_{L}} = U_{csM}^{e_{L}} \land A_{la}$ .

If $A \leq U_{csM}^{e_{L}}$ and A is right infinitely ∨-distributive, then $[A)_{cs \lor}^{e_{L}} = U_{sW}^{e_{L}} \lor A_{ua}$ . Moreover, if A is increasing in its first variable, then $[A)_{cs \lor}^{e_{L}} = U_{sW}^{e_{L}} \lor A$ .

Proof. When e_L ≠ 0, $U_{sW}^{e_{L}}$ is the smallest element of both $U_{cs}^{e_{L}} (L)$ and $U_{cs \lor}^{e_{L}} (L)$ by Example 2.6.

(1) Let $U_{1} = U_{sW}^{e_{L}} \lor A_{ua}$ . If $A \leq U_{csM}^{e_{L}}$ , then U₁ ≥ A, $A_{ua} \leq (U_{csM}^{e_{L}})_{ua} = U_{csM}^{e_{L}}$ . Thus, $U_{sW}^{e_{L}} \leq U_{1} \leq U_{csM}^{e_{L}}$ . It implies that U₁ (1, 0) = U₁ (0, 1) =0 and U₁ (e_L, y) = y for any y ∈ L. By Theorem 3.2(3) and the monotonicity of $U_{sW}^{e_{L}}$ , we can see that U₁ is increasing in its each variable. So, $U_{1} \in U_{cs}^{e_{L}} (L)$ . If A ≤ U and $U \in U_{cs}^{e_{L}} (L)$ , then U = U_ua ≥ A_ua and $U \geq U_{sW}^{e_{L}} \lor A_{ua} = U_{1}$ . Therefore, $[A)_{cs}^{e_{L}} = U_{sW}^{e_{L}} \lor A_{ua}$ .

(2) Let $U_{2} = U_{csM}^{e_{L}} \land A_{la}$ . If $U_{sW}^{e_{L}} \leq A$ , then $A_{la} \geq (U_{sW}^{e_{L}})_{la} = U_{sW}^{e_{L}}$ and $U_{sW}^{e_{L}} \leq U_{2} \leq U_{csM}^{e_{L}}$ . Thus, U₂ (1, 0) = U₂ (0, 1) =0 and U₂ (e_L, y) = y for any y ∈ L. By Theorem 3.2(3) and the monotonicity of $U_{csM}^{e_{L}}$ , we know that U₂ is increasing in its each variable. So, $U_{2} \in U_{cs}^{e_{L}} (L)$ . If U ≤ A and $U \in U_{cs}^{e_{L}} (L)$ , then U = U_la ≤ A_la and $U \leq U_{csM}^{e_{L}} \land A_{la} = U_{2}$ . Therefore, $(A]_{cs}^{e_{L}} = U_{csM}^{e_{L}} \land A_{la}$ .

(3) Let $U_{3} = U_{sW}^{e_{L}} \lor A_{ua}$ . If $A \leq U_{csM}^{e_{L}}$ , then $U_{3} \in U_{cs}^{e_{L}} (L)$ by statement (1). Noting that A is right infinitely ∨-distributive, we can see that A_ua is also right infinitely ∨-distributive by Theorem 3.3(1). Thus, U₃ is right infinitely ∨-distributive and $U_{3} \in U_{cs \lor}^{e_{L}} (L)$ . By the proof of (1), we have that $[A)_{cs \lor}^{e_{L}} = U_{sW}^{e_{L}} \lor A_{ua}$ .

Moreover, if A is increasing in its first variable, then A_ua = A by Theorem 3.2(4) and so $[A)_{cs \lor}^{e_{L}} = U_{sW}^{e_{L}} \lor A$ . □

By the proofs of Theorem 3.6(3) in [30] and Theorem 3.4(3), we can see that Theorem 3.4(3) improves Theorem 3.6(3) in [30]. In fact, by removing the condition A is increasing in its first variable in Theorem 3.6(3) in [30], we have that $[A)_{s \lor}^{e_{L}} = U_{sW}^{e_{L}} \lor A_{u}$ by the proof of Theorem 3.4(3).

Example 3.5. Let L = {0, a, b, 1} be a lattice, where 0 < a < 1, 0 < b < 1, a ∨ b = 1 and a ∧ b = 0. Define A ∈ L^L×L as follows:

A	a	b	1
0	0	a	a
a	0	a	a
b	1	a	1
1	1	1	1

We can see that

U_{sW}^{1_{L}} \leq A

, A is increasing in its first variable and right infinitely ∨-distributive. Let

U = U_{csM}^{1_{L}} \land A

. Then

U	a	b	1
0	0	0	0
a	0	0	a
b	a	0	1
1	a	b	1

Noting that U (b, a ∨ b) = U (b, 1) =1 ≠ a = U (b, a) ∨ U (b, b), we see that U is not right infinitely ∨-distributive.

This example illustrates that an analog of Theorem 3.4(3) may not hold for calculating the lower approximation conjunctive right infinitely ∨-distributive left semi-uninorm of a binary operation.

Similarly, for calculating the upper and lower approximation conjunctive right semi-uninorms of a binary operation, we have the following theorem.

Theorem 3.6.Let A ∈ L^L×L and 0 ≠ e_R ∈ L.

If $A \leq U_{csM}^{e_{R}}$ , then $[A)_{cs}^{e_{R}} = U_{sW}^{e_{R}} \lor A_{ua}$ .

If $U_{sW}^{e_{R}} \leq A$ , then $(A]_{cs}^{e_{R}} = U_{csM}^{e_{R}} \land A_{la}$ .

If $A \leq U_{csM}^{e_{R}}$ and left infinitely ∨-distributive, then $[A)_{\lor cs}^{e_{R}} = U_{sW}^{e_{R}} \lor A_{ua}$ . Moreover, if A is increasing in its second variable, then $[A)_{\lor cs}^{e_{R}} = U_{sW}^{e_{R}} \lor A$ .

4 The upper and lower approximation implications which satisfy the neutrality principle

Recently, Su and Wang [29] have investigated the constructions of implications and coimplications on a complete lattice. This section continues to explore the research direction started in [29]. In this section, we will study the constructions of the upper and lower approximation implications which satisfy the neutrality principle.

Definition 4.1. (Wang and Fang [36]) Let U be a binary operation on L. Define $I_{U}^{L}, I_{U}^{R} \in L^{L \times L}$ as follows: $\begin{matrix} I_{U}^{L} (x, y) = ⋁ {z \in L ∣ U (z, x) \leq y} \forall x, y \in L, \\ I_{U}^{R} (x, y) = ⋁ {z \in L ∣ U (x, z) \leq y} \forall x, y \in L . \end{matrix}$ Here, $I_{U}^{L}$ and $I_{U}^{R}$ are, respectively, called the left and right residuum of the binary operation U.

For any binary operation U on L and x, y ∈ L, it is straightforward to verify that

$I_{U}^{L} (x, 1) = I_{U}^{R} (x, 1) = 1$ .

$x \leq I_{U}^{L} (y, U (x, y))$ and $y \leq I_{U}^{R} (x, U (x, y))$ .

If U (1, 0) =0, then $I_{U}^{L} (0, y) = 1$ and if U (0, 1) =0, then $I_{U}^{R} (0, y) = 1$ .

Due to Theorems 4.4 and 4.5 in [36], we know that U and $I_{U}^{R}$ satisfy the following right residual principle: $U (x, z) \leq y \Leftrightarrow z \leq I_{U}^{R} (x, y) \forall x, y, z \in L$ when a binary operation U is right infinitely ∨-distributive; U and $I_{U}^{L}$ satisfy the following left residual principle: $U (z, x) \leq y \Leftrightarrow z \leq I_{U}^{L} (x, y) \forall x, y, z \in L$ when U is left infinitely ∨-distributive.

When U is a left (right) semi-uninorm on L, it is easy to see that $I_{U}^{L}$ and $I_{U}^{R}$ are all decreasing in the first variable and increasing in the second one by Definition 4.1.

Example 4.2. For some left or right semi-uninorms in Examples 2.6 and 2.7, a simple computation shows that $\begin{matrix} I_{U_{sW}^{e_{L}}}^{R} (x, y) = {\begin{matrix} y & if x \geq e_{L}, \\ 1 & otherwise, \end{matrix} \\ I_{U_{sW}^{e_{R}}}^{L} (x, y) = {\begin{matrix} y & if x \geq e_{R}, \\ 1 & otherwise, \end{matrix} \\ I_{U_{csM}^{e_{L}}}^{R} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ y & if 0 < x \leq e_{L}, y \neq 1, \\ 0 & otherwise, \end{matrix} \\ I_{U_{csM}^{e_{R}}}^{L} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ y & if 0 < x \leq e_{R}, y \neq 1, \\ 0 & otherwise, \end{matrix} \end{matrix}$ where x and y are elements of L. We can see that $I_{U_{sW}^{e_{L}}}^{R}$ is the greatest element of both $I^{{npe}_{L}} (L)$ and $I_{\land}^{{npe}_{L}} (L)$ when e_L ≠ 0; $I_{U_{sW}^{e_{R}}}^{L}$ is the greatest element of both $I^{{npe}_{R}} (L)$ and $I_{\land}^{{npe}_{R}} (L)$ when e_R ≠ 0; $I_{U_{csM}^{e_{L}}}^{R}$ is the smallest element of both $I^{{npe}_{L}} (L)$ and $I_{\land}^{{npe}_{L}} (L)$ ; and $I_{U_{csM}^{e_{R}}}^{L}$ is the smallest element of both $I^{{npe}_{R}} (L)$ and $I_{\land}^{{npe}_{R}} (L)$ .

It is easy to verify that if J≠ ∅, then $I_{j} \in I^{npe} (L) \forall j \in J \Rightarrow ⋀_{j \in J} I_{j} \in I^{npe} (L) .$ Due to Theorem 4.2 in [5], we see that $I^{npe} (L)$ is also a complete lattice with the smallest element $I_{U_{csM}^{e}}^{R}$ and greatest element $I_{U_{sW}^{e}}^{R}$ when e ≠ 0. Thus, for a binary operation A on L, if there exists $I \in I^{npe} (L)$ such that A ≤ I, then $⋀ {I ∣ A \leq I, I \in I^{npe} (L)}$ is the smallest implication that is stronger than A and satisfies the neutrality principle w.r.t. e on L. Here, we call it the upper approximation NP implication of A and denote it by $[A)_{I}^{npe}$ . Similarly, if there exists $I \in I^{npe} (L)$ such that I ≤ A, then $⋁ {I ∣ I \leq A, I \in I^{npe} (L)}$ is the largest implication that is weaker than A and satisfies the neutrality principle w.r.t. e on L. Here, we call it the lower approximation NP implication of A and denote it by $(A]_{I}^{npe}$ .

Likewise, for a binary operation A on L, we may introduce the following symbols:

$[A)_{I}^{npe \land}$ : the upper approximation right infinitely ∧-distributive NP implication of A.

$(A]_{I}^{npe \land}$ : the lower approximation right infinitely ∧-distributive NP implication of A.

Definition 4.3. (see Su and Wang [29]) Let A ∈ L^L×L. Define the upper approximation implicator A_ui and the lower approximation implicator A_li of A as follows: $\begin{matrix} A_{ui} (x, y) = ⋁ {A (u, v) ∣ u \geq x, v \leq y} \forall x, y \in L, \\ A_{li} (x, y) = ⋀ {A (u, v) ∣ u \leq x, v \geq y} \forall x, y \in L . \end{matrix}$

Theorem 4.4. (see Su and Wang [29]) Let A, B ∈ L^L×L. Then the following statements hold:

A_li ≤ A ≤ A_ui.

(A ∨ B) _ui = A_ui ∨ B_ui and (A ∧ B) _li = A_li ∧ B_li.

A_ui and A_li are hybrid monotonic.

If A is hybrid monotonic, then A_li = A = A_ui.

Theorem 4.5. Let A ∈ L^L×L.

(1) If A is right infinitely ∨-distributive, then A_ui is also right infinitely ∨-distributive, $\begin{matrix} (I_{A}^{R})_{li} = I_{A_{ua}}^{R}, (I_{A}^{R})_{ui} \leq I_{A_{la}}^{R}, \\ A_{ua} (x, (I_{A}^{R})_{li} (x, y)) \leq y \forall x, y \in L . \end{matrix}$ Moreover, if A is increasing in its first variable, then $A_{la} (x, (I_{A}^{R})_{ui} (x, y)) \leq y \forall x, y \in L .$

(2) If A is right infinitely ∧-distributive, then A_li is also right infinitely ∧-distributive.

(3) If A is left infinitely ∨-distributive, then $\begin{matrix} (I_{A}^{L})_{li} = I_{A_{ua}}^{L}, (I_{A}^{L})_{ui} \leq I_{A_{la}}^{L}, \\ A_{ua} ((I_{A}^{L})_{li} (x, y), x) \leq y \forall x, y \in L . \end{matrix}$ Moreover, if A is increasing in its second variable, then $A_{la} ((I_{A}^{L})_{ui} (x, y), x) \leq y \forall x, y \in L .$

Proof. We only prove that statement (1) holds.

Assume that A is a right infinitely ∨-distributive binary operation on L. By the proof of Theorem 3.3(1), we can see that A_ui is also right infinitely ∨-distributive. By Definitions 3.1 and 4.1, the monotonicity of A and $I_{A}^{R}$ , and the right residual principle, we have that $\begin{matrix} I_{A_{ua}}^{R} (x, y) = ⋁ {z \in L ∣ A_{ua} (x, z) \leq y} \\ = ⋁ {z \in L ∣ \lor {A (u, v) ∣ u \leq x, v \leq z} \leq y} \\ = ⋁ {z \in L ∣ \lor {A (u, z) ∣ u \leq x} \leq y} \\ = ⋁ {z \in L ∣ A (u, z) \leq y \forall u \leq x} \\ = ⋁ {z \in L ∣ z \leq I_{A}^{R} (u, y) \forall u \leq x} \\ = ⋁ {z \in L ∣ z \leq ⋀_{u \leq x} I_{A}^{R} (u, y)} \\ = ⋀_{u \leq x} I_{A}^{R} (u, y) \forall x, y \in L, \end{matrix}$ $\begin{matrix} (I_{A}^{R})_{li} (x, y) = ⋀ {I_{A}^{R} (u, v) ∣ u \leq x, v \geq y} \\ = ⋀ {I_{A}^{R} (u, y) ∣ u \leq x} = ⋀_{u \leq x} I_{A}^{R} (u, y) \forall x, y \in L . \end{matrix}$ Thus, $(I_{A}^{R})_{li} = I_{A_{ua}}^{R}$ . Similarly, for any x, y ∈ L we have that $\begin{matrix} (I_{A}^{R})_{ui} (x, y) = ⋁ {I_{A}^{R} (u, y) ∣ u \geq x}, \\ A_{la} (x, z) = ⋀ {A (u_{1}, v) ∣ u_{1} \geq x, v \geq z} \\ = ⋀ {A (u_{1}, z) ∣ u_{1} \geq x}, \end{matrix}$ $(I_{A_{la}}^{R}) (x, y) = ⋁ {z \in L ∣ \land {A (u_{1}, z) ∣ u_{1} \geq x} \leq y} .$ If u ≥ x, let $z = I_{A}^{R} (u, y)$ , then $\begin{matrix} A (u, z) = A (u, ⋁ {c \in L ∣ A (u, c) \leq y}) \\ = ⋁ {A (u, c) ∣ A (u, c) \leq y} \leq y, \\ ⋀ {A (u_{1}, z) ∣ u_{1} \geq x} \leq A (u, z) \leq y, \\ (I_{A}^{R})_{ui} (x, y) \leq (I_{A_{la}}^{R}) (x, y) . \end{matrix}$ So, $(I_{A}^{R})_{ui} \leq I_{A_{la}}^{R}$ . Due to Theorem 3.3(1), we know that A_ua is right infinitely ∨-distributive. Thus, for any x, y ∈ L, $\begin{matrix} A_{ua} (x, (I_{A}^{R})_{li} (x, y)) = A_{ua} (x, I_{A_{ua}}^{R} (x, y)) \\ = A_{ua} (x, ⋁ {z \in L ∣ A_{ua} (x, z) \leq y}) \\ = ⋁ {A_{ua} (x, z) ∣ z \in L, A_{ua} (x, z) \leq y} \leq y . \end{matrix}$ Moreover, if A is increasing in its first variable, then it follows from the monotonicity of A, the hybrid monotonicity of $I_{A}^{R}$ , and Theorems 3.2 and 4.4 that A_la = A and $(I_{A}^{R})_{ui} = I_{A}^{R}$ . Thus, $A_{la} (x, (I_{A}^{R})_{ui} (x, y)) = A (x, I_{A}^{R} (x, y)) \leq y \forall x, y \in L .$ □

Now, we give out the formulas for calculating the upper and lower approximation implications which satisfy the neutrality principle.

Theorem 4.6.Let A ∈ L^L×L and e_L ≠ 0.

If $A \leq I_{U_{sW}^{e_{L}}}^{R}$ , then $[A)_{I}^{{npe}_{L}} = I_{U_{csM}^{e_{L}}}^{R} \lor A_{ui}$ .

If $A \geq I_{U_{csM}^{e_{L}}}^{R}$ , then $(A]_{I}^{{npe}_{L}} = I_{U_{sW}^{e_{L}}}^{R} \land A_{li}$ .

If $A \geq I_{U_{csM}^{e_{L}}}^{R}$ and A is right infinitely ∧-distributive, then $(A]_{I}^{{npe}_{L} \land} = I_{U_{sW}^{e_{L}}}^{R} \land A_{li}$ . Moreover, if A is decreasing in its first variable, then $(A]_{I}^{{npe}_{L} \land} = I_{U_{sW}^{e_{L}}}^{R} \land A$ .

Proof. We only prove that statements (1) and (3) hold.

(1) Let $I_{1} = I_{U_{csM}^{e_{L}}}^{R} \lor A_{ui}$ . If $A \leq I_{U_{sW}^{e_{L}}}^{R}$ , then I₁ ≥ A and $I_{U_{csM}^{e_{L}}}^{R} \leq I_{1} \leq I_{U_{sW}^{e_{L}}}^{R}$ by Theorem 4.4. Thus, I₁ (0, 0) = I₁ (1, 1) =1, I₁ (1, 0) =0 and I₁ (e_L, y) = y for any y ∈ L. By Theorem 4.4(3) and the hybrid monotonicity of $I_{U_{csM}^{e_{L}}}^{R}$ , we know that I₁ is hybrid monotonic. So, $I_{1} \in I^{{npe}_{L}} (L)$ . If A ≤ I and $I \in I^{{npe}_{L}} (L)$ , then I = I_ui ≥ A_ui and $I \geq I_{U_{csM}^{e_{L}}}^{R} \lor A_{ui} = I_{1}$ . Therefore, $[A)_{I}^{{npe}_{L}} = I_{U_{csM}^{e_{L}}}^{R} \lor A_{ui}$ .

(3) Let $I_{3} = I_{U_{sW}^{e_{L}}}^{R} \land A_{li}$ . If $A \geq I_{U_{csM}^{e_{L}}}^{R}$ , then I₃ ≤ A, $A_{li} \geq (I_{U_{csM}^{e_{L}}}^{R})_{li} = I_{U_{csM}^{e_{L}}}^{R}, I_{U_{csM}^{e_{L}}}^{R} \leq I_{3} \leq I_{U_{sW}^{e_{L}}}^{R} .$ Thus, $I_{3} \in I^{{npe}_{L}} (L)$ . It follows from Theorem 4.5(2) that A_li is right infinitely ∧-distributive. So, I₃ is right infinitely ∧-distributive, i.e., $I_{3} \in I_{\land}^{{npe}_{L}} (L)$ . By the proof of statement (1), we know that $(A]_{I}^{{npe}_{L} \land} = I_{U_{sW}^{e_{L}}}^{R} \land A_{li}$ .

Moreover, if A is decreasing in its first variable, then A_li = A by Theorem 4.4(4) and $(A]_{I}^{{npe}_{L} \land} = I_{U_{sW}^{e_{L}}}^{R} \land A$ . □

Example 4.7. Let L = {0, a, b, c, 1} and e_L = b.

Define A ∈ L^L×L as follows:

A	0	a	b	c	1
0	1	1	1	1	1
a	c	1	1	1	1
b	0	0	0	c	1
c	0	a	a	c	1
1	0	0	0	0	1

It is easy to see that

A \leq I_{U_{sW}^{e_{L}}}^{R}

, A_ui = A, and A is decreasing in its first variable and right infinitely ∧-distributive. Let

I = I_{U_{csM}^{e_{L}}}^{R} \lor A

. Then

I	0	a	b	c	1
0	1	1	1	1	1
a	c	1	1	1	1
b	0	a	b	c	1
c	0	a	a	c	1
1	0	0	0	0	1

By Theorem 4.6, we see that I is an NP implication w.r.t. b.

Analogous to Theorem 4.6, we have the following theorem.

Theorem 4.8.Let A ∈ L^L×L and e_R ≠ 0.

If $A \leq I_{U_{sW}^{e_{R}}}^{L}$ , then $[A)_{I}^{{npe}_{R}} = I_{U_{csM}^{e_{R}}}^{L} \lor A_{ui}$ .

If $A \geq I_{U_{csM}^{e_{R}}}^{L}$ , then $(A]_{I}^{{npe}_{R}} = I_{U_{sW}^{e_{R}}}^{L} \land A_{li}$ .

If $A \geq I_{U_{csM}^{e_{R}}}^{L}$ and A is right infinitely ∧-distributive, then $(A]_{I}^{{npe}_{R} \land} = I_{U_{sW}^{e_{R}}}^{L} \land A_{li}$ . Moreover, if A is decreasing in its first variable, then $(A]_{I}^{{npe}_{R} \land} = I_{U_{sW}^{e_{R}}}^{L} \land A$ .

5 The relations between the upper approximation conjunctive left (right) semi-uninorms and lower approximation implications which satisfy the neutrality principle

In this section, we reveal the relationships between the upper approximation conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms and lower approximation right infinitely ∧-distributive implications which satisfy the neutrality principle.

Theorem 5.1.Let e_L, e_R ≠ 0 and A ∈ L^L×L.

If $A \leq U_{csM}^{e_{L}}$ and A is right infinitely ∨-distributive, then $(I_{A}^{R}]_{I}^{{npe}_{L} \land} = I_{[A)_{cs \lor}^{e_{L}}}^{R}$ .

If $A \leq U_{csM}^{e_{R}}$ and A is left infinitely ∨-distributive, then $(I_{A}^{L}]_{I}^{{npe}_{R} \land} = I_{[A)_{\lor cs}^{e_{R}}}^{L}$ .

Proof. We only prove that statement (1) holds.

Assume that $A \leq U_{csM}^{e_{L}}$ and A is right infinitely ∨-distributive. Then it follows from Theorem 4.6 in [36] and Definition 4.1 that $I_{A}^{R}$ is right infinitely ∧-distributive and $I_{A}^{R} \geq I_{U_{csM}^{e_{L}}}^{R}$ . Thus, $(I_{A}^{R}]_{I}^{{npe}_{L} \land} = I_{U_{sW}^{e_{L}}}^{R} \land (I_{A}^{R})_{li}$ by Theorem 4.6(3). Moreover, it follows from Theorems 3.3(1) and 3.4(3) and the right residual principle that $\begin{matrix} I_{[A)_{cs \lor}^{e_{L}}}^{R} (x, y) = ⋁ {z \in L ∣ [A)_{cs \lor}^{e_{L}} (x, z) \leq y} \\ = ⋁ {z \in L ∣ (U_{sW}^{e_{L}} \lor A_{ua}) (x, z) \leq y} \\ = ⋁ {z \in L ∣ U_{sW}^{e_{L}} (x, z) \lor A_{ua} (x, z) \leq y} \\ = ⋁ {z \in L ∣ U_{sW}^{e_{L}} (x, z) \leq y, A_{ua} (x, z) \leq y} \\ = ⋁ {z \in L ∣ z \leq I_{U_{sW}^{e_{L}}}^{R} (x, y), z \leq I_{A_{ua}}^{R} (x, y)} \\ = ⋁ {z \in L ∣ z \leq I_{U_{sW}^{e_{L}}}^{R} (x, y) \land I_{A_{ua}}^{R} (x, y)} \\ = (I_{U_{sW}^{e_{L}}}^{R} \land I_{A_{ua}}^{R}) (x, y) \forall x, y \in L, \end{matrix}$ i.e., $I_{[A)_{cs \lor}^{e_{L}}}^{R} = I_{U_{sW}^{e_{L}}}^{R} \land I_{A_{ua}}^{R}$ . By Theorem 4.5(1), we know that $(I_{A}^{R})_{li} = I_{A_{ua}}^{R}$ . Therefore, $(I_{A}^{R}]_{I}^{{npe}_{L} \land} = I_{U_{sW}^{e_{L}}}^{R} \land (I_{A}^{R})_{li} = I_{U_{sW}^{e_{L}}}^{R} \land I_{A_{ua}}^{R} = I_{[A)_{cs \lor}^{e_{L}}}^{R} .$ □

If P and Q are two propositions, then the generalized modus ponens (GMP) rule (see [7]) gives a lower bound for the truth value of Q when the truth values of propositions P and P ⇒ Q are known. Let $U = [A)_{cs \lor}^{e_{L}}$ . Then U is right infinitely ∨-distributive and $(I_{A}^{R}]_{I}^{{npe}_{L} \land} = I_{U}^{R}$ by Theorem 5.1. Thus, $[A)_{cs \lor}^{e_{L}}$ and $(I_{A}^{R}]_{I}^{{npe}_{L} \land}$ satisfy the GMP rule by Theorem 4.3(5) in [36].

Generally speaking, $(A]_{cs}^{e_{L}}$ and $[I_{A}^{R})_{I}^{{npe}_{L}}$ don’t satisfy the GMP rule even if $U_{sW}^{e_{L}} \leq A$ and A is right infinitely ∨-distributive.

Example 5.2. For the binary operation A on L in Example 3.5, we have that A_la = A,

(I_{A}^{R})_{ui} = I_{A}^{R}

$I_{A}^{R}$	0	a	b	1
0	a	1	a	1
a	a	1	a	1
b	0	b	0	1
1	0	0	0	1

I	0	a	b	1
0	1	1	1	1
a	a	1	1	1
b	0	1	b	1
1	0	a	b	1

where

I = I_{U_{csM}^{1_{L}}}^{R} \lor (I_{A}^{R})_{ui}

. By Theorems 3.4 and 4.6, we see that

\begin{matrix} (A]_{cs}^{1_{L}} (b, [I_{A}^{R})_{I}^{np 1_{L}} (b, a)) \\ = U_{csM}^{1_{L}} (b, [I_{A}^{R})_{I}^{np 1_{L}} (b, a)) \\ ⋀ A (b, [I_{A}^{R})_{I}^{np 1_{L}} (b, a)) \\ = U_{csM}^{1_{L}} (b, I (b, a)) ⋀ A (b, I (b, a)) \\ = U_{csM}^{1_{L}} (b, 1)) ⋀ A (b, 1) = 1 ≰ a, \end{matrix}

i.e.,

(A]_{cs}^{e_{L}}

and

[I_{A}^{R})_{I}^{{npe}_{L}}

don’t satisfy the GMP rule.

Moreover, we know that $I_{A}^{R} \leq I_{U_{sW}^{1_{L}}}^{R}$ , $I_{A}^{R}$ is decreasing in its first variable and right infinitely ∧-distributive, $I (b, a \land b) = I (b, 0) = 0 \neq b = I (b, a) \land I (b, b),$ i.e., I isn’t right infinitely ∧-distributive.

This example illustrates that an analog of Theorem 4.6(3) may not hold for calculating the upper approximation right infinitely ∧-distributive implication which satisfies the neutrality principle.

Finally, we give out some conditions such that the lower approximation conjunctive left (right) semi-uninorms of a binary operation and upper approximation NP implication of right (left) residuum of the binary operation satisfy the GMP rule.

Theorem 5.3.Let e_L, e_R ≠ 0 and A ∈ L^L×L.

If $U_{sW}^{e_{L}} \leq A$ , A (x, y) ≤ y for any x ∈ (0, e_L] and y ∈ L, and A is increasing in its first variable and right infinitely ∨-distributive, then $(A]_{cs}^{e_{L}}$ and $[I_{A}^{R})_{I}^{{npe}_{L}}$ satisfy the GMP rule.

If $U_{sW}^{e_{R}} \leq A$ , A (x, y) ≤ x for any x ∈ L and y ∈ (0, e_R], and A is increasing in its second variable and left infinitely ∨-distributive, then $(A]_{cs}^{e_{R}}$ and $[I_{A}^{L})_{I}^{{npe}_{R}}$ satisfy the GMP rule in the form $(A]_{cs}^{e_{R}} ([I_{A}^{L})_{I}^{{npe}_{R}} (x, y), x) \leq y \forall x, y \in L .$

Proof. We only prove that statement (1) holds.

Assume that $U_{sW}^{e_{L}} \leq A$ , A is increasing in its first variable and right infinitely ∨-distributive. Then, $I_{A}^{R} \leq I_{U_{sW}^{e_{L}}}^{R}$ , A_la = A and $(I_{A}^{R})_{ui} = I_{A}^{R}$ by the proof of Theorem 4.5. By Theorem 3.4(2), we see that $\begin{matrix} (A]_{cs}^{e_{L}} (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) \\ = U_{csM}^{e_{L}} (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) ⋀ A (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) . \end{matrix}$ By Example 4.2 and Theorem 4.6(1), we know that $\begin{matrix} [I_{A}^{R})_{I}^{{npe}_{L}} (x, y) = I_{U_{csM}^{e_{L}}}^{R} (x, y) \lor I_{A}^{R} (x, y) \\ = {\begin{matrix} 1 & if x = 0 or y = 1, \\ y \lor I_{A}^{R} (x, y) & if 0 < x \leq e_{L}, y \neq 1, \\ I_{A}^{R} (x, y) & otherwise . \end{matrix} \end{matrix}$ Thus, when $0 < x \leq e_{L}, y \neq 1 and [I_{A}^{R})_{I}^{{npe}_{L}} (x, y) \neq 0$ , $\begin{matrix} (A]_{cs}^{e_{L}} (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) \\ = (y \lor I_{A}^{R} (x, y)) \land A (x, y \lor I_{A}^{R} (x, y)), \\ (A]_{cs}^{e_{L}} (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) \\ = {\begin{matrix} 0 & if x = 0 or [I_{A}^{R})_{I}^{{npe}_{L}} (x, y) = 0, \\ A (x, 1) & if 0 < x \leq e_{L}, y = 1, \\ A (x, I_{A}^{R} (x, y)) & otherwise . \end{matrix} \end{matrix}$ If A (x, y) ≤ y for any x ∈ (0, e_L] and y ∈ L, then $A (x, y \lor I_{A}^{R} (x, y)) = A (x, y) ⋁ A (x, I_{A}^{R} (x, y)) \leq y .$ Therefore, $(A]_{cs}^{e_{L}} (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) \leq y$ for all x, y ∈ L, i.e., $(A]_{cs}^{e_{L}}$ and $[I_{A}^{R})_{I}^{{npe}_{L}}$ satisfy the GMP rule. □

By Example 5.2, we can see that the condition A (x, y) ≤ y for any x ∈ (0, e_L] and y ∈ L in Theorem 5.3(1) is also essential for a complete lattice L.

Let L = [0, 1] and 0 < e_L ≤ 1. Assume that $U_{sW}^{e_{L}} \leq A$ , A is increasing in its first variable and right infinitely ∨-distributive on [0, 1]. If A (x, y) > y for some 0 < x ≤ e_L and y ≠ 1, then A (x, z) ≥ A (x, y) > y when z ≥ y and hence $I_{A}^{R} (x, y) \leq y$ . Thus, $(y \lor I_{A}^{R} (x, y)) ⋀ A (x, y \lor I_{A}^{R} (x, y)) \leq y .$ By the proof of Theorem 5.3, we see that $(A]_{cs}^{e_{L}} (x, [I_{A}^{R})_{I}^{{npe}_{L}} (x, y)) \leq y \forall x, y \in [0, 1] .$ This illustrates that the condition A (x, y) ≤ y for any x ∈ (0, e_L] and y ∈ L in Theorem 5.3(1) is unnecessary when L = [0, 1].

When L = [0, 1], is the condition A is increasing in its first variable in Theorem 5.3(1) redundant? This is an unsolved problem.

6 Conclusions and future works

Constructing fuzzy connectives is an interesting topic. In this paper, motivated by these works in [16 , 34], we give out the formulas for calculating the upper and lower approximation conjunctive left (right) semi-uninorms of a binary operation, derive the formulas for calculating the upper and lower approximation implications which satisfy the neutrality principle, and show that the right (left) residuum of the upper approximation conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms of a binary operation are the lower approximation right infinitely ∧-distributive NP implications of the right (left) residuum of the binary operation.

In a forthcoming paper, we will further investigate the constructions of left (right) semi-uninorms and coimplications on a complete lattice.

Footnotes

Acknowledgments

The authors wish to thank Associate Editor and the anonymous referees for their valuable comments and advice.

This work is supported by the Jiangsu Provincial Natural Science Foundation of China (BK20161313) and National Natural Science Foundation of China (11571006).

References

Alsina

, Frank

M.J.

, and Schweizer

, Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, Singapore, 2006.

Baczyński

, and Jayaram

, Fuzzy Implication, Studies in Fuzziness and Soft Computing, Vol. 231, Springer, Berlin, 2008.

Baczyński

, and Jayaram

, QL-implications: Some properties and intersections, Fuzzy Sets and Systems161 (2010), 158–188.

Birkhoff

, Lattice Theory, American Mathematical Society Colloquium Publishers, Providence, 1967.

Burris

, and Sankappanavar

H.P.

, A Course in Universal Algebra, World Publishing Corporation, Beijing, 1981.

Bustince

, Burillo

, and Soria

, Automorphisms, negations and implication operators, Fuzzy Sets and Systems134 (2003), 209–229.

De Baets

, and Fodor

, Residual operators of uninorms, Soft Computing3 (1999), 89–100.

De Cooman

, and Kerre

E.E.

, Order norms on bounded partially ordered sets, J Fuzzy Math2 (1994), 281–310.

Durante

, Klement

E.P.

, Mesiar

, and Sempi

, Conjunctors and their residual implicators: Characterizations and construction methods, Mediterranean Journal of Mathematics4 (2007), 343–356.

10.

Fodor

J.C.

, Strict preference relations based on weak t-norms, Fuzzy Sets and Systems43 (1991), 327–336.

11.

Fodor

J.C.

, and Keresztfalvi

, Nonstandard conjunctions and implications in fuzzy logic, International Journal of Approximate Reasoning12 (1995), 69–84.

12.

Fodor

J.C.

, and Roubens

, Fuzzy Preference Modelling and Multicriteria Decision Support, Theory and Decision Library, Series D: System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, 1994.

13.

Fodor

J.C.

, Yager

R.R.

, and Rybalov

, Structure of uninorms, Internat J Uncertainty, Fuzziness and Knowledge-Based Systems5 (1997), 411–427.

14.

Gabbay

, and Metcalfe

, Fuzzy logics based on [0; 1)-continuous uninorms, Arch Math Logic46 (2007), 425–449.

15.

Hao

X.Y.

, Niu

M.X.

, and Wang

Z.D.

, The relations between implications and left (right) semi-uninorms on a complete lattice, Internat J Uncertainty, Fuzziness and Knowledge-Based Systems23 (2015), 245–261.

16.

Jenei

, and Montagna

, A general method for constructing leftcontinuous t-norms, Fuzzy Sets and Systems136 (2003), 263–282.

17.

Klement

E.P.

, Mesiar

, and Pap

, Triangular Norms, Trends in Logic, Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dodrecht, 2000.

18.

Z.B.

, and Liu

H.W.

, Generalization of (U, N)-implications derived from commutative semiuninorms and pseudo-uninorms, Journal of Intelligent and Fuzzy Systems29 (2015), 2177–2184.

19.

Liu

H.W.

, Semi-uninorm and implications on a complete lattice, Fuzzy Sets and Systems191 (2012), 72–82 .

20.

Mas

, Massanet

, Ruiz-Aguilera

, and Torrens

, A survey on the existing classes of uninorms, Journal of Intelligent and Fuzzy Systems29 (2015), 1021–1037.

21.

Mas

, Monserrat

, and Torrens

, On left and right uninorms, Internat J Uncertainty, Fuzziness and Knowledge-Based Systems9 (2001), 491–507.

22.

Mas

, Monserrat

, and Torrens

, On left and right uninorms on a finite chain, Fuzzy Sets and Systems146 (2004), 3–17.

23.

Mas

, Monserrat

, and Torrens

, Two types of implications derived from uninorms, Fuzzy Sets and Systems158 (2007), 2612–2626.

24.

Musilek

, Guanlao

, and Barreiro

, Genetic programming of fuzzy aggregation operations, Journal of Intelligent and Fuzzy Systems16 (2005), 107–118.

25.

Niu

M.X.

, Hao

X.Y.

, and Wang

Z.D.

, Relations among implications, coimplications and left (right) semi-uninorms, Journal of Intelligent and Fuzzy Systems29 (2015), 927–938 .

26.

Ouyang

, On fuzzy implications determined by aggregation operators, Information Sciences193 (2012), 153–162.

27.

Shi

, Van Gasse

, Ruan

, and Kerre

E.E.

, On dependencies and independencies of fuzzy implication axioms, Fuzzy Sets and Systems161 (2010), 1388–1405.

28.

, and Wang

Z.D.

, Pseudo-uninorms and coimplications on a complete lattice, Fuzzy Sets and Systems224 (2013), 53–62.

29.

, and Wang

Z.D.

, Constructing implications and coimplications on a complete lattice, Fuzzy Sets and Systems247 (2014), 68–80.

30.

, Wang

Z.D.

, and Tang

K.M.

, Left and right semi-uninorms on a complete lattice, Kybernetika49 (2013), 948–961.

31.

Suárez García

, and Gil Álvarez

, Two families of fuzzy intergrals, Fuzzy Sets and Systems18 (1986), 67–81.

32.

Takács

, Uninorm-based models for FLC systems, Journal of Intelligent and Fuzzy Systems19 (2008), 65–73.

33.

Tsadiras

A.K.

, and Margaritis

K.G.

, The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons, Fuzzy Sets and Systems93 (1998), 263–274.

34.

Wang

Z.D.

, Generating pseudo-t-norms and implication operators, Fuzzy Sets and Systems157 (2006), 398–410.

35.

Wang

Z.D.

, Left (right) semi-uninorms and coimplications on a complete lattice, Fuzzy Sets and Systems287 (2016), 227–239.

36.

Wang

Z.D.

, and Fang

J.X.

, Residual operators of left and right uninorms on a complete lattice, Fuzzy Sets and Systems160 (2009), 22–31.

37.

Wang

Z.D.

, and Fang

J.X.

, Residual coimplicators of left and right uninorms on a complete lattice, Fuzzy Sets and Systems160 (2009), 2086–2096.

38.

Wang

Z.D.

, and Yu

Y.D.

, Pseudo-t-norms and implication operators on a complete Brouwerian lattice, Fuzzy Sets and Systems132 (2002), 113–124.

39.

Yager

R.R.

, Uninorms in fuzzy system modeling, Fuzzy Sets and Systems122 (2001), 167–175.

40.

Yager

R.R.

, and Rybalov

, Uninorm aggregation operators, Fuzzy Sets and Systems80 (1996), 111–120.