Relations among implications,coimplications and left (right) semi-uninorms

Abstract

In this paper, we further study implications, coimplications and left (right) semi-uninorms on a complete lattice. We firstly show that the N-dual operation of the right (left) residual implication, which is generated by a left (right)-conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm, is the right (left) residual coimplication generated by its N-dual operation. As a dual result, the N-dual operation of the right (left) residual coimplication, which is generated by a left (right)-disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorm, is the right (left) residual implication generated by its N-dual operation. Then, we demonstrate that the N-dual operations of the left (right) semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left (right) semi-uninorms. Finally, we reveal the relationships between conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms induced by implications and disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms induced by coimplications, where both implications and coimplications satisfy the neutrality principle.

Keywords

Fuzzy connective implication coimplication left (right) semi-uninorm neutrality principle

1 Introduction

Uninorms, introduced by Yager and Rybalov [27], and studied by Fodor et al. [10], are special aggregation operators that have been proven useful in many fields like fuzzy logic, expert systems, neural networks, aggregation, and fuzzy system modeling (see [11 , 26]). 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 [10]). But, there are real-life situations when truth functions cannot be associative or commutative (see [7, 8]). By throwing away the commutativity from the axioms of uninorms, Mas et al. introduced the concepts of left and right uninorms in [14, 15], and Wang and Fang [22, 23] studied the left and right uninorms on a complete lattice. By removing the associativity and commutativity from the axioms of uninorms, Liu [12] introduced the concept of semi-uninorms, and Su et al. [21] discussed the notion of left and right semi-uninorms, on a complete lattice. On the other hand, it is well known that a uninorm (semi-uninorm, left and right uninorms) U is conjunctive or disjunctive whenever U (0, 1) =0 or 1, respectively. This fact allows us to use uninorms (semi-uninorm, left and right uninorms and so on) in defining fuzzy implications and coimplications (see [5 , 23]).

In this paper, based on [5 , 23] we further study implications, coimplications and left (right) semi-uninorms on a complete lattice. The organization of this study is as follows. Section 2 recalls some necessary concepts, results and examples about implications, coimplications, left (right) semi-uninorms and N-dual operations. In Section 3, we show that the N-dual operation of the right (left) residual implication, which is generated by a left (right)-conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm, is the right (left) residual coimplication, which is generated by its N-dual operation. As a dual result, the N-dual operation of the right (left) residual coimplication, which is generated by a left (right)-disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorm, is the right (left) residual implication, which is generated by its N-dual operation. Then, we demonstrate that the N-dual operations of the left (right) semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left (right) semi-uninorms. In Section 4, we reveal the relationships between conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms induced by implications and disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms induced by coimplications, and prove that the join-semilattice of all conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms is order-reversing isomorphic to the meet-semilattice of all right infinitely ∧-distributive implications, the meet-semilattice of all right infinitely ∧-distributive implications is order-reversing isomorphic to the join-semilattice of all right infinitely ∨-distributive coimplications, and the join-semilattice of all right infinitely ∨-distributive coimplications is order-reversing isomorphic to the meet-semilattice of all disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms; where all implications and coimplications satisfy the neutrality principle.

It should be point out that some results and methods in this paper originate from some already known results on the interval [0, 1] (for example, see [1 , 17]).

The knowledge about lattices required in this paper can be found in [2].

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.

2 Implications, coimplications, left (right) semi-uniorms and N-dual operations

In this section, we briefly recall some concepts, results and examples which will be used in the paper.

Definition 2.1. [1 , 9] An implication I on Lis a hybrid monotonous (with non-increasing first and non-decreasing second partial mappings) binary operation that satisfies the boundary conditions I (0, 0) = I (1, 1) =1 and I (1, 0) =0. A coimplication C on L is a hybrid monotonous binary operation that satisfies the corner conditions C (0, 0) = C (1, 1) =0 and C (0, 1) =1.

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

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

Definition 2.2. [22, 23] A binary operation U on L is called left (right) infinitely ∨-distributive or left (right) arbitrary ∨-distributive if it preserves arbitrary joins in its first (second) argument, i.e., $\begin{matrix} U (⋁_{j \in J} x_{j}, y) = ⋁_{j \in J} U (x_{j}, y) \forall x_{j}, y \in L \\ (U (x, ⋁_{j \in J} y_{j}) = ⋁_{j \in J} U (x, y_{j}) \forall x, y_{j} \in L); \end{matrix}$ left (right) infinitely ∧-distributive or left (right) arbitrary ∧-distributive if it preserves arbitrary meets in its first (second) argument, i.e., $\begin{matrix} U (⋀_{j \in J} x_{j}, y) = ⋀_{j \in J} U (x_{j}, y) \forall x_{j}, y \in L \\ (U (x, ⋀_{j \in J} y_{j}) = ⋀_{j \in J} U (x, y_{j}) \forall x, y_{j} \in L) . \end{matrix}$ 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).

Let 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 [2, 3]), 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:

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

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

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

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

Example 2.3. [20] Let $\begin{matrix} I_{W} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ 0 & otherwise, \end{matrix} \\ I_{M} (x, y) = {\begin{matrix} 0 & if (x, y) = (1, 0), \\ 1 & otherwise, \end{matrix} \\ C_{W} (x, y) = {\begin{matrix} 1 & if (x, y) = (0, 1), \\ 0 & otherwise, \end{matrix} \\ C_{M} (x, y) = {\begin{matrix} 0 & if x = 1 or y = 0, \\ 1 & otherwise, \end{matrix} \end{matrix}$ where x and y are elements of L. Then I _W and I _M are, respectively, the smallest and greatest elements of $I (L)$ and I _W is also the smallest element of $I_{\land} (L)$ . C _W and C _M are, respectively, the smallest and greatest elements of $C (L)$ and C _M is also the largest element of $C_{\lor} (L)$ .

Moreover, by Examples 2.4 and 2.5 in [20], we know that $I_{\land} (L)$ is not a join-semilattice and $C_{\lor} (L)$ is not a meet-semilattice.

Definition 2.4. [21] A binary operation U on L is called a left (right) semi-uninorm if it satisfies the following two conditions:

there exists a left (right) neutral element, i. e., an element e _L ∈ L (e _R ∈ L) satisfying U (e _L, x) = x (U (x, e _R) = x) for all x ∈ L,

U is non-decreasing in each variable.

If a left (right) semi-uninorm U is associative, then U is the left (right) uninorm in [22, 23]. 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. Let e = e _L = e _R. Here, U is the semi-uninorm in [12].

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. U is said to be left-disjunctive and right-disjunctive if U (1, 0) =1 and U (0, 1) =1, respectively. We call U disjunctive if both U (1, 0) =1 and U (0, 1) =1 by a similarreason.

Now, for the sake of convenience, we list the following symbols:

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

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

$U_{s \land}^{e_{L}} (L)$ ( $U_{\land s}^{e_{R}} (L)$ ): the set of all right (left) infinitely ∧-distributive left (right) semi-uninorms with left (right) neutral element e _L (e _R) on L;

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

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

$U_{ds}^{e_{L}} (L)$ ( $U_{ds}^{e_{R}} (L)$ ): the set of all disjunctive left (right) semi-uninorms with left (right) neutral element e _L (e _R) on L;

$U_{ds \land}^{e_{L}} (L)$ ( $U_{\land ds}^{e_{R}} (L)$ ): the set of all disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms with left (right) neutral element e _L (e _R) on L.

Example 2.5. Let e _L ∈ L, $\begin{matrix} U_{sW}^{e_{L}} (x, y) = {\begin{matrix} y & if x \geq e_{L}, \\ 0 & otherwise, \end{matrix} \\ U_{sM}^{e_{L}} (x, y) = {\begin{matrix} y & if x \leq e_{L}, \\ 1 & otherwise, \end{matrix} \\ U_{csM}^{e_{L}} (x, y) = {\begin{matrix} 0 & if x = 0 or y = 0, \\ y & if 0 < x \leq e_{L}, y \neq 0, \\ 1 & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} U_{dsW}^{e_{L}} (x, y) = {\begin{matrix} 1 & if x = 1 or y = 1, \\ y & if e_{L} \leq x < 1, \\ 0 & otherwise, \end{matrix} \end{matrix}$ where x and y are elements of L. By virtue of Example 2.5 in [21], 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 $U_{s \lor}^{e_{L}} (L)$ ; and $U_{sM}^{e_{L}}$ is the greatest element of $U_{s \land}^{e_{L}} (L)$ . Moreover, it is easy to see that $U_{csM}^{e_{L}}$ is the greatest element of $U_{cs \lor}^{e_{L}} (L)$ ; $U_{dsW}^{e_{L}}$ is the smallest element of $U_{ds \land}^{e_{L}} (L)$ ; $U_{sW}^{e_{L}}$ is the smallest element of $U_{cs \lor}^{e_{L}} (L)$ when e _L ≠ 0; and $U_{sM}^{e_{L}}$ is the greatest element of $U_{ds \land}^{e_{L}} (L)$ when e _L ≠ 1.

Example 2.6. Let e _R ∈ L, $\begin{matrix} U_{sW}^{e_{R}} (x, y) = {\begin{matrix} x & if y \geq e_{R}, \\ 0 & otherwise, \end{matrix} \\ U_{sM}^{e_{R}} (x, y) = {\begin{matrix} x & if y \leq e_{R}, \\ 1 & otherwise, \end{matrix} \\ U_{csM}^{e_{R}} (x, y) = {\begin{matrix} 0 & if x = 0 or y = 0, \\ x & if 0 < y \leq e_{R}, x \neq 0, \\ 1 & otherwise, \end{matrix} \\ U_{dsW}^{e_{R}} (x, y) = {\begin{matrix} 1 & if x = 1 or y = 1, \\ x & if e_{R} \leq y < 1, \\ 0 & otherwise, \end{matrix} \end{matrix}$ where x and y are elements of L. By virtue of Example 2.6 in [21], 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 $U_{\lor s}^{e_{R}} (L)$ ; and $U_{sM}^{e_{R}}$ is the greatest element of $U_{\land s}^{e_{R}} (L)$ . Moreover, it is easy to see that $U_{csM}^{e_{R}}$ is the greatest element of $U_{\lor cs}^{e_{R}} (L)$ ; $U_{dsW}^{e_{R}}$ is the smallest element of $U_{\land ds}^{e_{R}} (L)$ ; $U_{sW}^{e_{R}}$ is the smallest element of $U_{\lor cs}^{e_{R}} (L)$ when e _R ≠ 0; and $U_{sM}^{e_{R}}$ is the greatest element of $U_{\land ds}^{e_{R}} (L)$ when e _R ≠ 1.

Definition 2.7. [13] A mapping N : L → L is called a negation if

N (0) =1 and N (1) =0,

x ≤ y, x, y ∈ L ⇒ N (y) ≤ N (x).

A negation N is called strong if it is an involution, i. e., N (N (x)) = x for any x ∈ L.

Theorem 2.8. [24] Let x_j ∈ L (j ∈ J). If N is a strong negation on L, then $N (⋁_{j \in J} x_{j}) = ⋀_{j \in J} N (x_{j}), N (⋀_{j \in J} x_{j}) = ⋁_{j \in J} N (x_{j}) .$

Definition 2.9. [4] Consider a strong negation N on L. The N-dual operation of a binary operation A on L is the binary operation A _N on L defined by $A_{N} (x, y) = N^{- 1} (A (N (x), N (y))) \forall x, y \in L .$

Note that (A _N) _{N
^-1} = (A _N) _N = A for any binary operation A on L.

For any nonempty subfamily {A _j ∣ j ∈ J} of L ^L×L, the least upper bound ∨_j∈J A _j and the greatest lower bound ∧_j∈J A _j of are, respectively, defined by $\begin{matrix} (⋁_{j \in J} A_{j}) (x, y) = ⋁_{j \in J} A_{j} (x, y) \forall x, y \in L, \\ (⋀_{j \in J} A_{j}) (x, y) = ⋀_{j \in J} A_{j} (x, y) \forall x, y \in L . \end{matrix}$

The following theorem lists some properties of N-dual operations.

Theorem 2.10. [4 , 21] Let A, B be two binary operations and N a strong negation on L. Then the following statements hold:

(A ∧ B) _N = A _N ∨ B _N and (A ∨ B) _N = A _N ∧ B _N .

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

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

If A is non-increasing (non-decreasing) in its ith variable, then A _N is non-increasing (non-decreasing) in its ith variable (i = 1, 2).

The N-dual operation of an implication is a coimplication and the N-dual operation of a coimplication is an implication.

The N-dual operation of a left (right) semi-uninorm with a left (right) neutral element e _L (e _R) is a left (right) semi-uninorm with a left (right) neutral element N (e _L) (N (e _R)).

3 The residual implications and coimplicatons generated by left (right) semi-uninorms and the left (right) semi-uninorms induced by implications and coimplications

Recently, De Baets and Fodor [5] investigated the residual operators of uninorms on [0, 1], Torrens et al. [16, 17] studied the implications and coimplications derived from uninorms on [0, 1]. Now, we consider the residual implications and coimplications generated by left (right) semi-uninorms on a complete lattice.

Definition 3.1. [22] Let U be a binary operation on L. Define $I_{U}^{L}, I_{U}^{R} \in L^{L \times L}$ as follows: $\begin{matrix} I_{U}^{L} (x, y) = ⋁ {z \in L | U (z, x) \leq y}, \\ I_{U}^{R} (x, y) = ⋁ {z \in L | U (x, z) \leq y} \forall x, y \in L . \end{matrix}$ Here, $I_{U}^{L}$ and $I_{U}^{R}$ are, respectively, called the left and right residuums of U.

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 non-increasing in the first variable and non-decreasing in the second one by Definition 3.1.

For any operation U on L and x, y ∈ L, it follows from Theorems 4.1 and 4.2 in [22] 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 $I_{U}^{R} (0, y) = 1$ .

If U is a left (right) semi-uninorm with the left (right) neutral element e _L (e _R), then $I_{U}^{R} (e_{L}, y) = y (I_{U}^{L} (e_{R}, y) = y)$ for any y ∈ L.

By virtue of Theorems 3.1, 3.3 and 3.4 in [12], we see that if U is a left (right)-conjunctive left (right) semi-uninorm with the left (right) neutral element e _L (e _R), then $I_{U}^{R} (I_{U}^{L})$ is an implication which satisfies the neutrality principle w.r.t. e _L (e _R); if U is a left (right)-conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm with the left (right) neutral element e _L (e _R), then $I_{U}^{R} (I_{U}^{L})$ is a right infinitely ∧-distributive implication and $\begin{matrix} I_{U}^{R} (x, y) = \max {z \in L | U (x, z) \leq y} \\ (I_{U}^{L} (x, y) = \max {z \in L | U (z, x) \leq y}) . \end{matrix}$ Here, $I_{U}^{L}$ and $I_{U}^{R}$ are, respectively, called the left and right residual implication generated by the left (right) semi-uninorm U.

By Theorems 4.4 and 4.5 in [22] or Theorems 3.3 and 3.4 in [12], we know that if a binary operation U is right infinitely ∨-distributive, then U and $I_{U}^{R}$ satisfy the generalized modus ponens (GMP) rule (see [5]) $U (x, I_{U}^{R} (x, y)) \leq y$ and the following right residual (implication) principle: $U (x, z) \leq y \Leftrightarrow z \leq I_{U}^{R} (x, y) \forall x, y, z \in L;$ if U is left infinitely ∨-distributive, then U and $I_{U}^{L}$ satisfy GMP rule in the form $U (I_{U}^{L} (x, y), x) \leq y$ and the following left residual (implication) principle: $U (z, x) \leq y \Leftrightarrow z \leq I_{U}^{L} (x, y) \forall x, y, z \in L .$

Example 3.2. For some left or right semi-uninorms in Examples 2.5 and 2.6, a simple computationshows that

$\begin{matrix} I_{U_{sW}^{e_{R}}}^{L} (x, y) = {\begin{matrix} y & if x \geq e_{R}, \\ 1 & otherwise, \end{matrix} \\ I_{U_{sW}^{e_{L}}}^{R} (x, y) = {\begin{matrix} y & if x \geq e_{L}, \\ 1 & otherwise, \end{matrix} \\ I_{U_{csM}^{e_{L}}}^{L} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ e_{L} & if 0 < x \leq y < 1, \\ 0 & otherwise, \end{matrix} \\ I_{U_{csM}^{e_{L}}}^{R} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ y & if 0 < x \leq e_{L}, y \neq 1, \\ 0 & otherwise, \end{matrix} \\ I_{U_{csM}^{e_{R}}}^{L} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ y & if 0 < x \leq e_{R}, y \neq 1, \\ 0 & otherwise, \end{matrix} \\ I_{U_{csM}^{e_{R}}}^{R} (x, y) = {\begin{matrix} 1 & if x = 0 or y = 1, \\ e_{R} & if 0 < x \leq y < 1, \\ 0 & otherwise . \end{matrix} \end{matrix}$ When e _L, e _R ∈ L \ {0}, we see that $I_{U_{csM}^{e_{L}}}^{L}$ and $I_{U_{csM}^{e_{R}}}^{R}$ are two implications; $I_{U_{csM}^{e_{L}}}^{R}$ and $I_{U_{csM}^{e_{R}}}^{L}$ are, respectively, the smallest elements of $I_{\land}^{{npe}_{L}} (L)$ and $I_{\land}^{{npe}_{R}} (L)$ ; and $I_{U_{sW}^{e_{L}}}^{R}$ and $I_{U_{sW}^{e_{R}}}^{L}$ are the greatest elements of $I_{\land}^{{npe}_{L}} (L)$ and $I_{\land}^{{npe}_{R}} (L)$ , respectively.

Definition 3.3. [23] Let U be a binary operation on L. Define $C_{U}^{L}, C_{U}^{R} \in L^{L \times L}$ as follows: $\begin{matrix} C_{U}^{L} (x, y) = ⋀ {z \in L | y \leq U (z, x)} \forall x, y \in L, \\ C_{U}^{R} (x, y) = ⋀ {z \in L | y \leq U (x, z)} \forall x, y \in L . \end{matrix}$ Here, $C_{U}^{L}$ and $C_{U}^{R}$ are, respectively, called the left and right deresiduums of U.

For any operation U on L, it follows from Theorems 3.1 and 3.2 in [23] that

$C_{U}^{L} (x, 0) = C_{U}^{R} (x, 0) = 0$ for any x ∈ L.

For any x, y ∈ L, $C_{U}^{L} (y, U (x, y)) \leq x$ and $C_{U}^{R} (x, U (x, y)) \leq y$ .

If U is right-disjunctive, then $C_{U}^{L} (1, y) = 0$ and if U is left-disjunctive, then $C_{U}^{R} (1, y) = 0$ .

If U is a left (right) semi-uninorm with the left (right) neutral element e _L (e _R), then $C_{U}^{R} (e_{L}, y) = y (C_{U}^{L} (e_{R}, y) = y)$ for any y ∈ L.

By virtue of Definition 3.3, it is easy to see that $C_{U}^{L}$ and $C_{U}^{R}$ are all non-increasing in the first variable and non-decreasing in the second one when U is a left (right) semi-uninorm; $C_{U}^{L} (e, x) = C_{U}^{R} (e, x) = x$ for any x ∈ L when U is a semi-uninorm with the neutral element e.

Example 3.4. For some left or right semi-uninorms in Examples 2.5 and 2.6, a simple computation shows that $\begin{matrix} C_{U_{sM}^{e_{R}}}^{L} (x, y) = {\begin{matrix} y & if x \leq e_{R}, \\ 0 & otherwise, \end{matrix} \\ C_{U_{sM}^{e_{L}}}^{R} (x, y) = {\begin{matrix} y & if x \leq e_{L}, \\ 0 & otherwise, \end{matrix} \\ C_{U_{dsW}^{e_{L}}}^{L} (x, y) = {\begin{matrix} 0 & if x = 1 or y = 0, \\ e_{L} & if 0 < y \leq x < 1, \\ 1 & otherwise, \end{matrix} \\ C_{U_{dsW}^{e_{L}}}^{R} (x, y) = {\begin{matrix} 0 & if x = 1 or y = 0, \\ y & if e_{L} \leq x < 1, \\ 1 & otherwise, \end{matrix} \\ C_{U_{dsW}^{e_{R}}}^{L} (x, y) = {\begin{matrix} 0 & if x = 1 or y = 0, \\ y & if e_{R} \leq x < 1, y \neq 0, \\ 1 & otherwise, \end{matrix} \\ C_{U_{dsW}^{e_{R}}}^{R} (x, y) = {\begin{matrix} 0 & if x = 1 or y = 0, \\ e_{R} & if 0 < y \leq x < 1, \\ 1 & otherwise . \end{matrix} \end{matrix}$ When e _L, e _R ∈ L \ {1}, we see that $C_{U_{dsW}^{e_{L}}}^{L}$ and $C_{U_{dsW}^{e_{R}}}^{R}$ are two coimplications, $C_{U_{sM}^{e_{L}}}^{R}$ and $C_{U_{sM}^{e_{R}}}^{L}$ are, respectively, the smallest elements of $C_{\lor}^{np e_{L}} (L)$ and $C_{\lor}^{{npe}_{R}} (L)$ ; and $C_{U_{dsW}^{e_{L}}}^{R}$ and $C_{U_{dsW}^{e_{R}}}^{L}$ are the greatest elements of $C_{\lor}^{np e_{L}} (L)$ and $C_{\lor}^{{npe}_{R}} (L)$ , respectively.

Theorem 3.5. (1) If $U \in U_{s}^{e_{L}} (L)$ is left-disjunctive, then $C_{U}^{R} \in C (L)$ .

(2) If $U \in U_{s}^{e_{R}} (L)$ is right-disjunctive, then $C_{U}^{L} \in C (L)$ .

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

If U is a left-disjunctive left semi-uninorm with the left neutral element e _L, then $C_{U}^{R}$ is non-increasing in its first and non-decreasing in its second variable and $C_{U}^{R} (1, 1) = 0$ . Moreover, $C_{U}^{R} (0, 0) = ⋀ {z \in L | 0 \leq U (0, z)} = 0 .$ By the non-decreasingness of U, we see that $\begin{matrix} C_{U}^{R} (0, 1) = ⋀ {z \in L | U (0, z) = 1} \\ \geq & ⋀ {z \in L | z = U (e_{L}, z) \geq U (0, z) = 1} = 1 . \end{matrix}$ Thus, $C_{U}^{R}$ is a coimplication on L.□

Moreover, if $U \in U_{s \land}^{e_{L}} (L)$ ( $U \in U_{\land s}^{e_{R}} (L)$ ) is left (right)-disjunctive, then it follows from Theorems 3.1 and 3.2 in [12], Theorem 3.5 in [23] and Theorem 3.5 that $C_{U}^{R} \in C_{\lor} (L)$ ( $C_{U}^{L} \in C_{\lor} (L)$ ) and $\begin{matrix} C_{U}^{R} (x, y) = \min {z \in L | y \leq U (x, z)} \\ (C_{U}^{L} (x, y) = \min {z \in L | y \leq U (z, x)}) . \end{matrix}$ Here, $C_{U}^{L}$ and $C_{U}^{R}$ are, respectively, called the left and right residual coimplication generated by the left (right) semi-uninorm U.

If P and Q are two propositions, then the property $U (x, C_{U}^{R} (x, y)) \geq y$ is a generalization of the following tautology Q ⇒ (P ∨ (PnotLeftarrowQ)) in classical logic and is in some sense dual to the modus ponens [4]. By Theorems 3.3 and 3.4 in [23], we know that U and $C_{U}^{R}$ satisfy the generalized dual modus ponens rule and the following right residual (coimplication) principle: $y \leq U (x, z) \Leftrightarrow C_{U}^{R} (x, y) \leq z \forall x, y, z \in L$ when U is a right infinitely ∧-distributive left semi-uninorm on L; U and $C_{U}^{L}$ satisfy the generalized dual modus ponens rule in the form: $U (C_{U}^{L} (x, y), x) \geq y$ and the following left residual (coimplication) principle: $y \leq U (z, x) \Leftrightarrow C_{U}^{L} (x, y) \leq z \forall x, y, z \in L$ when U is left infinitely ∧-distributive right semi-uninorm on L.

The following theorem reveals the relationships between the residual implications and the residual coimplications.

Theorem 3.6. Let U be a binary operation and N strong negation on L. Then

$(I_{U}^{L})_{N} = C_{U_{N}}^{L}$ and $(C_{U}^{L})_{N} = I_{U_{N}}^{L}$ .

$(I_{U}^{R})_{N} = C_{U_{N}}^{R}$ and $(C_{U}^{R})_{N} = I_{U_{N}}^{R}$ .

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

Noting that the strong negation N is a bijection, by Definition 2.9, we have that $\begin{matrix} (I_{U}^{L})_{N} (x, y) = N (I_{U}^{L} (N (x), N (y))) \\ = & N (⋁ {z \in L | U (z, N (x)) \leq N (y)}) \\ = & ⋀ {N (z) \in L | N (U (N (N (z)), N (x))) \geq y} \\ = & ⋀ {N (z) \in L | y \leq U_{N} (N (z), x)} \\ = & ⋀ {u \in L | y \leq U_{N} (u, x)} = C_{U_{N}}^{L} (x, y) \forall x, y \in L . \end{matrix}$ Thus, $(I_{U}^{L})_{N} = C_{U_{N}}^{L}$ . Moreover, $(I_{U_{N}}^{L})_{N} = C_{(U_{N})_{N}}^{L} = C_{U}^{L}$ and so $(C_{U}^{L})_{N} = I_{U_{N}}^{L}$ .□

By virtue of Theorem 3.6, we see that the N-dual operation of the right (left) residual implication, which is generated by a left (right)-conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm, is the right (left) residual coimplication generated by its N-dual operation and the N-dual operation of the right (left) residual coimplication, which is generated by a left (right)-disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorm, is the right (left) residual implication generated by its N-dual operation.

Liu [12] discussed the semi-uninorms induced by implications, and Su and Wang [19] studied the pseudo-uninorms induced by coimplications. Below, we investigate the left (right) semi-uninorms induced by implications and coimplications on a complete lattice.

Definition 3.7. [12] Let I be an implication on L. Define two induced operators $U_{I}^{L}$ and $U_{I}^{R}$ of I as follows: $\begin{matrix} U_{I}^{L} (x, y) = ⋀ {z \in L | x \leq I (y, z)} \forall x, y \in L, \\ U_{I}^{R} (x, y) = ⋀ {z \in L | y \leq I (x, z)} \forall x, y \in L . \end{matrix}$

Clearly, $U_{I}^{R}$ = $C_{I}^{R}$ , $U_{I}^{L} (0, x) = U_{I}^{R} (x, 0) = 0$ , $U_{I}^{L} (1, x) = U_{I}^{R} (x, 1)$ for any x ∈ L. It is easy to see that $U_{I}^{L}$ and $U_{I}^{R}$ are all non-decreasing in its each variable.

Moreover, for any implication I, it follows from Definition 3.7 that $U_{I}^{L} (I (x, y), x) \leq y, U_{I}^{R} (x, I (x, y)) \leq y \forall x, y \in L,$ i.e., $U_{I}^{L}$ and I, $U_{I}^{R}$ and I satisfy the GMP rule.

Definition 3.8. [19] Let C be a coimplication on L. Define two induced operators $U_{C}^{L}$ and $U_{C}^{R}$ of C as follows: $\begin{matrix} U_{C}^{L} (x, y) = ⋁ {z \in L | C (y, z) \leq x} \forall x, y \in L, \\ U_{C}^{R} (x, y) = ⋁ {z \in L | C (x, z) \leq y} \forall x, y \in L . \end{matrix}$

Clearly, $U_{C}^{R}$ = $I_{C}^{R}$ , $U_{C}^{L} (1, x) = U_{C}^{R} (x, 1) = 1$ ; $U_{C}^{L} (0, x) = U_{C}^{R} (x, 0) = \lor {z \in L | C (x, z) = 0}$ for any x ∈ L. It is also easy to see that $U_{C}^{L}$ and $U_{C}^{R}$ are all non-decreasing in its each variable.

Moreover, for any coimplication C, it follows from Definition 3.8 that $y \leq U_{C}^{L} (C (x, y), x), y \leq U_{C}^{R} (x, C (x, y)) \forall x, y \in L .$ These explain that $U_{C}^{L}$ and C, $U_{C}^{R}$ and C satisfy the generalized dual modus ponens rule.

Example 3.9. For I _W, I _M, C _W and C _M in Example 2.3, we have that $\begin{matrix} U_{I_{W}}^{L} (x, y) = U_{I_{W}}^{R} (x, y) = {\begin{matrix} 0 & if x = 0 or y = 0, \\ 1 & otherwise, \end{matrix} \\ U_{I_{M}}^{L} (x, y) = {\begin{matrix} \land_{a \in L \ {0}} a & if x > 0, y = 1, \\ 0 & otherwise, \end{matrix} \\ U_{I_{M}}^{R} (x, y) = {\begin{matrix} \land_{a \in L \ {0}} a & if x = 1, y > 0, \\ 0 & otherwise . \end{matrix} \\ U_{C_{M}}^{L} (x, y) = U_{C_{M}}^{R} (x, y) = {\begin{matrix} 1 & if x = 1 or y = 1, \\ 0 & otherwise, \end{matrix} \\ U_{C_{W}}^{L} (x, y) = {\begin{matrix} \lor_{a \in L \ {1}} a & if x < 1, y = 0, \\ 1 & otherwise, \end{matrix} \\ U_{C_{W}}^{R} (x, y) = {\begin{matrix} \lor_{a \in L \ {1}} a & if x = 0, y < 1, \\ 1 & otherwise . \end{matrix} \end{matrix}$ Thus, these operations induced by implications I _W and I _M and coimplications C _W and C _M are neither left semi-uninorms nor right semi-uninorms on L.

Now, we find some conditions such that these operations induced by implications and coimplications are left or right semi-uninorms.

Theorem 3.10. Let $I \in I (L)$ and $C \in C (L)$ .

If I and C satisfies the neutrality principle w.r.t. e _L, then $U_{I}^{R}, U_{C}^{R} \in U_{s}^{e_{L}} (L)$ . Moreover, if $I \in I_{\land} (L)$ and $C \in C_{\lor} (L)$ , then $U_{I}^{R} \in U_{s \lor}^{e_{L}} (L)$ and $U_{C}^{R} \in U_{s \land}^{e_{L}} (L)$ .

If I and C satisfies the neutrality principle w.r.t. e _R, then $U_{I}^{L}, U_{C}^{L} \in U_{s}^{e_{R}} (L)$ . Moreover, if $I \in I_{\land} (L)$ and $C \in C_{\lor} (L)$ , then $U_{I}^{L} \in U_{\lor s}^{e_{R}} (L)$ and $U_{C}^{L} \in U_{\land s}^{e_{R}} (L)$ .

Here, $U_{I}^{R}$ ( $U_{I}^{L}$ ) and $U_{C}^{R}$ ( $U_{C}^{L}$ ) are called the left (right) semi-uninorms induced by the implication I and the coimplication C, respectively.

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

Assume that $C \in C (L)$ . Then $U_{C}^{R}$ is non-decreasing in each variable. If C satisfies the neutrality principle w.r.t. e _L, then $\begin{matrix} U_{C}^{R} (e_{L}, y) = ⋁ {z \in L | C (e_{L}, z) \leq y} \\ = & ⋁ {z \in L | z \leq y} = y \forall y \in L . \end{matrix}$ So, $U_{C}^{R} \in U_{s}^{e_{L}} (L)$ . Moreover, if C is a right infinitely ∨-distributive, then it follows from Theorem 5.3 in [19] that $U_{C}^{R}$ is right infinitely ∧-distributive. Thus, $U_{C}^{R} \in U_{s \land}^{e_{L}} (L)$ .

Similarly, we can show that $U_{I}^{R} \in U_{s}^{e_{L}} (L)$ when I satisfies the neutrality principle w.r.t. e _L and $U_{I}^{R} \in U_{s \lor}^{e_{L}} (L)$ when $I \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _L.□

When $I \in I (L)$ , I (0, x) =1 for any x ∈ L and hence it follows from Definition 3.7 that $U_{I}^{L} (1, 0) = U_{I}^{R} (0, 1) = 0$ . Thus, $U_{I}^{L}$ and $U_{I}^{R}$ in Theorem 3.10 are, respectively, the conjunctive left and right semi-uninorms induced by the implication I.

When $C \in C (L)$ , C (1, x) =0 for any x ∈ L and hence it follows from Definition 3.8 that $U_{C}^{L} (0, 1) = U_{C}^{R} (1, 0) = 1$ . Thus, $U_{C}^{L}$ and $U_{C}^{R}$ in Theorem 3.10 are, respectively, the disjunctive left and right semi-uninorms induced by the coimplication C.

By virtue of Theorems 4.2 and 4.3 in [12] and Theorems 5.1 and 5.2 in [19], we know that I, $U_{I}^{L}$ and $U_{I}^{R}$ satisfy the following adjunction conditions: $\begin{matrix} x \leq I (y, z) \Leftrightarrow U_{I}^{L} (x, y) \leq z \forall x, y, z \in L, \\ y \leq I (x, z) \Leftrightarrow U_{I}^{R} (x, y) \leq z \forall x, y, z \in L \end{matrix}$ when I is a right infinitely ∧-distributive implication on L; C, $U_{C}^{L}$ and $U_{C}^{R}$ satisfy the following adjunction conditions: $\begin{matrix} C (y, z) \leq x \Leftrightarrow z \leq U_{C}^{L} (x, y) \forall x, y, z \in L, \\ C (x, z) \leq y \Leftrightarrow z \leq U_{C}^{R} (x, y) \forall x, y, z \in L \end{matrix}$ when C is a right infinitely ∨-distributive coimplication on L.

The following theorem reveals the relationships between the left (right) semi-uninorms induced by implications and coimplications.

Theorem 3.11. Let I be an implication, C a coimplication and N a strong negation on L. Then

$(U_{C}^{L})_{N} = U_{C_{N}}^{L}$ and $(U_{I}^{L})_{N} = U_{I_{N}}^{L}$ .

$(U_{C}^{R})_{N} = U_{C_{N}}^{R}$ and $(U_{I}^{R})_{N} = U_{I_{N}}^{R}$ .

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

If I is an implication and C a coimplication, then it follows from Theorem 2.10 that I _N is a coimplication and C _N an implication. By Definition 2.9, we see that $\begin{matrix} (U_{C}^{L})_{N} (x, y) = N (U_{C}^{L} (N (x), N (y))) \\ = & N (⋁ {z \in L | C (N (y), z) \leq N (x)}) \\ = & ⋀ {N (z) \in L | C (N (y), z) \leq N (x)} \\ = & ⋀ {N (z) \in L | N (C (N (y), N (N (z)))) \geq x} \end{matrix}$

$\begin{matrix} = & ⋀ {N (z) \in L | C_{N} (y, N (z)) \geq x} \\ = & ⋀ {u \in L | C_{N} (y, u) \geq x} \\ = & (U_{C_{N}}^{L}) (x, y) \forall x, y \in L . \end{matrix}$ Thus, $(U_{C}^{L})_{N} = U_{C_{N}}^{L}$ .

We can prove in an analogous way that $(U_{I}^{L})_{N} = U_{I_{N}}^{L}$ .□

By Theorems 3.10 and 3.11, we know that the N-dual operation of the left (right) semi-uninorm induced by an implication, which satisfies the neutrality principle w.r.t. e _L (e _R), is the left (right) semi-uninorm induced by its N-dual operation. As a dual result, the N-dual operation of the left (right) semi-uninorm induced by a coimplication, which satisfies the neutrality principle w.r.t. e _L (e _R), is the left (right) semi-uninorm induced by its N-dual operation.

4 The relations between conjunctive left (right) semi-uninorms induced by implications and disjunctive left (right) semi-uninorms induced by coimplications

By Theorem 2.10, we know that the N-dual operations of an implication and a coimplication are, respectively, a coimplication and an implication and the N-dual operation of a left (right) semi-uninorm is a left (right) semi-uninorm. By virtue of Theorem 3.6, we see that the N-dual operations of the right (left) residual implication and coimplication, which are generated by a left (right) semi-uninorm, are, respectively, the right (left) residual coimplication and implication, which are generated by its N-dual operation. By Theorem 3.11, we know that the N-dual operations of the left (right) semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left (right) semi-uninorms.

In the final section, we reveal the relationships between conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms induced by implications and disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms induced by coimplications on a complete lattice.

Theorem 4.1. (1) If $U \in U_{s \lor}^{e_{L}} (L)$ ( $U \in U_{\lor s}^{e_{R}} (L)$ )is left (right)-conjunctive, then $I_{U}^{R} (I_{U}^{L}) \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _L (e _R) and $U_{I_{U}^{R}}^{R} = U$ ( $U_{I_{U}^{L}}^{L} = U$ ).

If $U \in U_{s \land}^{e_{L}} (L)$ ( $U \in U_{\land s}^{e_{R}} (L)$ ) is left (right)-disjunctive, then $C_{U}^{R} (C_{U}^{L}) \in C_{\lor} (L)$ satisfies the neutrality principle w.r.t. e _L (e _R) and $U_{C_{U}^{R}}^{R} = U$ ( $U_{C_{U}^{L}}^{L} = U$ ).

If $I \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _L (e _R), then

$U_{I}^{R} \in U_{s \lor}^{e_{L}} (L)$ ( $U_{I}^{L} \in U_{\lor s}^{e_{R}} (L)$ ) is conjunctive and $I_{U_{I}^{R}}^{R} = I$ ( $I_{U_{I}^{L}}^{L} = I$ ).

If $C \in C_{\lor} (L)$ satisfies the neutrality principle w.r.t. e _L (e _R), then

$U_{C}^{R} \in U_{s \land}^{e_{L}} (L)$ ( $U_{C}^{L} \in U_{\land s}^{e_{R}} (L)$ ) is disjunctive and $C_{U_{C}^{R}}^{R} = C$ ( $C_{U_{C}^{L}}^{L} = C$ ).

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

(1) If U is a left-conjunctive right infinitely ∨-distributive left semi-uninorm, then $I_{U}^{R} \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _L by Theorem 3.1 in [12] and Theorem 4.6 in [22]. Moreover, it follows from the right residual (implication) principlethat $\begin{matrix} U_{I_{U}^{R}}^{R} (x, y) = ⋀ {z \in L | y \leq I_{U}^{R} (x, z)} \\ = & ⋀ {z \in L | U (x, y) \leq z} = U (x, y) \forall x, y \in L . \end{matrix}$ Thus, $U_{I_{U}^{R}}^{R} = U$ .

Similarly, we can show that $I_{U}^{L} \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _R and $U_{I_{U}^{L}}^{L} = U$ when U is a right-conjunctive left infinitely ∨-distributive right semi-uninorm.

(3) If $I \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _L, then $U_{I}^{R}$ is a conjunctive right infinitely ∨-distributive left semi-uninorm by Theorem 3.10. Moreover, it follows from the adjunction condition that $\begin{matrix} I_{U_{I}^{R}}^{R} (x, y) = ⋁ {z \in L | U_{I}^{R} (x, z) \leq y} \\ = & ⋁ {z \in L | z \leq I (x, y)} = I (x, y) \forall x, y \in L . \end{matrix}$ Therefore, $I_{U_{I}^{R}}^{R} = I$ .

Similarly, we can see that $U_{I}^{L} \in U_{\lor s}^{e_{R}} (L)$ is conjunctive and $I_{U_{I}^{L}}^{L} = I$ when $I \in I_{\land} (L)$ satisfies the neutrality principle w.r.t. e _R.□

Example 4.2. Let L = [0, 1], $U (x, y) = {\begin{matrix} \frac{1}{4} xy & if y = 0 or x < \frac{1}{2}, \\ y & if x = \frac{1}{2}, \\ 1 & otherwise . \end{matrix}$ Then, $U \in U_{s \lor}^{\frac{1}{2}} ([0, 1])$ is left-conjunctive. By Definition 3.1, we have that $\begin{matrix} I_{U}^{R} (x, y) = \sup {z \in [0, 1] ∣ U (x, z) \leq y} \end{matrix}$ $\begin{matrix} = & {\begin{matrix} 1 & if x = 0 or y = 1, \\ \min {1, \frac{4 y}{x}} & if 0 < x < \frac{1}{2}, \\ y & if x = \frac{1}{2}, \\ 0 & otherwise . \end{matrix} \end{matrix}$ Thus, $I_{U}^{R} \in I_{\land} ([0, 1])$ satisfies the neutrality principle w.r.t. $\frac{1}{2}$ . By Definition 3.7, we see that $\begin{matrix} U_{I_{U}^{R}}^{R} (x, y) = \inf {z \in [0, 1] ∣ y \leq I_{U}^{R} (x, z)} \\ = & {\begin{matrix} \frac{1}{4} xy & if y = 0 or x < \frac{1}{2}, \\ y & if x = \frac{1}{2}, \\ 1 & otherwise, \end{matrix} \end{matrix}$ i.e., $U_{I_{U}^{R}}^{R} = U$ .

Theorem 4.3. (1) If e _L ≠ 0 (e _R ≠ 0), then $U_{cs \lor}^{e_{L}} (L)$ ( $U_{\lor cs}^{e_{R}} (L)$ ) is a complete lattice with the smallest element $U_{sW}^{e_{L}}$ ( $U_{sW}^{e_{R}}$ ) and greatest element $U_{csM}^{e_{L}}$ ( $U_{csM}^{e_{R}}$ ).

If e _L ≠ 1 (e _R ≠ 1), then $U_{ds \land}^{e_{L}} (L)$ ( $U_{\land ds}^{e_{R}} (L)$ ) is a complete lattice with the smallest element $U_{dsW}^{e_{L}}$ ( $U_{dsW}^{e_{R}}$ ) and greatest element

$U_{sM}^{e_{L}}$ ( $U_{sM}^{e_{R}}$ ).

If e _L ≠ 0 (e _R ≠ 0), then $I_{\land}^{{npe}_{L}} (L)$ ( $I_{\land}^{{npe}_{R}} (L)$ ) is a complete lattice with the smallest element $I_{U_{csM}^{e_{L}}}^{R}$ ( $I_{U_{csM}^{e_{R}}}^{L}$ ) and greatest element $I_{U_{sW}^{e_{L}}}^{R}$ ( $I_{U_{sW}^{e_{R}}}^{L}$ ).

If e _L ≠ 1 (e _R ≠ 1), then $C_{\lor}^{{npe}_{L}} (L)$ ( $C_{\lor}^{{npe}_{R}} (L)$ ) is a complete lattice with the smallest element $C_{U_{sM}^{e_{L}}}^{R}$ ( $C_{U_{sM}^{e_{R}}}^{L}$ ) and greatest element $C_{U_{dsW}^{e_{L}}}^{R}$ ( $C_{U_{dsW}^{e_{R}}}^{L}$ ).

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

(1) Suppose that $U_{j} \in U_{cs \lor}^{e_{L}} (L) (j \in J)$ and J≠ ∅. Clearly, $⋁_{j \in J} U_{j} \in U_{cs}^{e_{L}} (L)$ . Moreover, for any index set K and any x, y _k ∈ L (k ∈ K), we have that $\begin{matrix} (⋁_{j \in J} U_{j}) (x, ⋁_{k \in K} y_{k}) = ⋁_{j \in J} U_{j} (x, ⋁_{k \in K} y_{k}) \\ = & ⋁_{j \in J} ⋁_{k \in K} U_{j} (x, y_{k}) = ⋁_{k \in K} ⋁_{j \in J} U_{j} (x, y_{k}) \\ = & ⋁_{k \in K} (⋁_{j \in J} U_{j} (x, y_{k})) = ⋁_{k \in K} ((⋁_{j \in J} U_{j}) (x, y_{k})) . \end{matrix}$ Hence, $⋁_{j \in J} U_{j} \in U_{cs \lor}^{e_{L}} (L)$ . By virtue of Theorem 4.2 in [3] and Example 2.5, we see that $U_{cs \lor}^{e_{L}} (L)$ is a complete lattice with the smallest element $U_{sW}^{e_{L}}$ and greatest element $U_{csM}^{e_{L}}$ when e _L ≠ 0.

(3) Assume that e _L ≠ 0, $I_{j} \in I_{\land}^{{npe}_{L}} (L) (j \in J)$ , and J≠ ∅. Clearly, $⋀_{j \in J} I_{j} \in I^{npe L} (L)$ . Moreover, for any index set K and any x, y _k ∈ L (k ∈ K), we see that $\begin{matrix} (⋀_{j \in J} I_{j}) (x, ⋀_{k \in K} y_{k}) = ⋀_{j \in J} I_{j} (x, ⋀_{k \in K} y_{k}) \end{matrix}$

$\begin{matrix} = & ⋀_{j \in J} ⋀_{k \in K} I_{j} (x, y_{k}) = ⋀_{k \in K} ⋀_{j \in J} I_{j} (x, y_{k}) \\ = & ⋀_{k \in K} (⋀_{j \in J} I_{j} (x, y_{k})) = ⋀_{k \in K} ((⋀_{j \in J} I_{j}) (x, y_{k})) . \end{matrix}$ Hence, $⋀_{j \in J} I_{j} \in I_{\land}^{npe L} (L)$ . By virtue of Theorem 4.2 in [3] and Example 3.2, we know that $I_{\land}^{{npe}_{L}} (L)$ is a complete lattice with the smallest element $I_{U_{csM}^{e_{L}}}^{R}$ and greatest element $I_{U_{sW}^{e_{L}}}^{R}$ .□

Define four mappings φ ₁, φ ₂, ψ ₁ and ψ ₂ as follows: $\begin{matrix} φ_{1} : U_{cs \lor}^{e_{L}} (L) \to I_{\land}^{np e_{L}} (L), φ_{1} (U) = I_{U}^{R}, \\ φ_{2} : U_{ds \land}^{e_{L}} (L) \to C_{\lor}^{{npe}_{L}} (L), φ_{2} (U) = C_{U}^{R}, \\ ψ_{1} : U_{\lor cs}^{e_{R}} (L) \to I_{\land}^{{npe}_{R}} (L), ψ_{1} (U) = I_{U}^{L}, \\ ψ_{2} : U_{\land ds}^{e_{R}} (L) \to C_{\lor}^{{npe}_{R}} (L), ψ_{2} (U) = C_{U}^{L} . \end{matrix}$ Then it follows from Theorem 4.1 that φ ₁, φ ₂, ψ ₁ and ψ ₂ are all invertible, $\begin{matrix} φ_{1}^{- 1} (I) = U_{I}^{R} \forall I \in I_{\land}^{{npe}_{L}} (L), \\ φ_{2}^{- 1} (C) = U_{C}^{R} \forall C \in C_{\lor}^{{npe}_{L}} (L), \\ ψ_{1}^{- 1} (I) = U_{I}^{L} \forall I \in I_{\land}^{{npe}_{R}} (L), \\ ψ_{2}^{- 1} (C) = U_{C}^{L} \forall C \in C_{\lor}^{{npe}_{R}} (L) . \end{matrix}$ Moreover, we have the following theorem.

Theorem 4.4. (1) ( $U_{cs \lor}^{e_{L}} (L), \lor$ ) and $(U_{\lor cs}^{e_{R}} (L), \lor)$ are order-reversing isomorphic to $(I_{\land}^{{npe}_{L}} (L), \land)$ and $(I_{\land}^{np e_{R}} (L)), \land)$ , respectively.

$(U_{ds \land}^{e_{L}} (L), \land)$ and $(U_{\land ds}^{e_{R}} (L), \land)$ are order-reversing isomorphic to $(C_{\lor}^{np e_{L}} (L), \lor)$ and $(C_{\lor}^{{npe}_{R}} (L), \lor)$ , respectively.

$(U_{cs \lor}^{e_{L}} (L), \lor)$ and $(U_{\lor cs}^{e_{R}} (L), \lor)$ are order-reversing isomorphic to $(U_{ds \land}^{N (e_{L})} (L), \land)$ and $(U_{\land ds}^{N (e_{R})} (L), \land)$ ,

respectively.

$(I_{\land}^{{npe}_{L}} (L), \land)$ and $(I_{\land}^{{npe}_{R}} (L), \land)$ are, respectively, order-reversing isomorphic to $(C_{\lor}^{npN (e_{L})} (L), \lor)$ and $(C_{\lor}^{npN (e_{R})} (L), \lor)$ .

Proof. (1) If $U_{1}, U_{2} \in U_{cs \lor}^{e_{L}} (L)$ , then it is easy to see that $U_{1} \lor U_{2} \in U_{cs \lor}^{e_{L}} (L)$ . Moreover, it follows from the right residual (implication) principle that $\begin{matrix} I_{(U_{1} \lor U_{2})}^{R} (x, y) = ⋁ {z \in L | (U_{1} \lor U_{2}) (x, z) \leq y} \\ = & ⋁ {z \in L | U_{1} (x, z) \lor U_{2} (x, z) \leq y} \\ = & ⋁ {z \in L | U_{1} (x, z) \leq y, U_{2} (x, z) \leq y} \end{matrix}$

$\begin{matrix} = & ⋁ {z \in L | z \leq I_{U_{1}}^{R} (x, y), z \leq I_{U_{2}}^{R} (x, y)} \\ = & ⋁ {z \in L | z \leq I_{U_{1}}^{R} (x, y) \land I_{U_{2}}^{R} (x, y)} \\ = & (I_{U_{1}}^{R} \land I_{U_{2}}^{R}) (x, y) \forall x, y \in L, \end{matrix}$ i.e., φ ₁ (U ₁ ∨ U ₂) = φ ₁ (U ₁) ∧ φ ₁ (U ₂). Thus, φ ₁ is an order-reversing isomorphism of ( $U_{cs \lor}^{e_{L}} (L), \lor$ ) onto $(I_{\land}^{{npe}_{L}} (L), \land)$ .

(2) If $U_{1}, U_{2} \in U_{ds \land}^{e_{L}} (L)$ , then $U_{1} \land U_{2} \in U_{ds \land}^{e_{L}} (L)$ . Moreover, it follows from the right residual (coimplication) principle that $\begin{matrix} C_{(U_{1} \land U_{2})}^{R} (x, y) = ⋀ {z \in L | y \leq (U_{1} \land U_{2}) (x, z)} \\ = & ⋀ {z \in L | y \leq U_{1} (x, z) \land U_{2} (x, z)} \\ = & ⋀ {z \in L | y \leq U_{1} (x, z), y \leq U_{2} (x, z)} \\ = & ⋀ {z \in L | C_{U_{1}}^{R} (x, y) \leq z, C_{U_{2}}^{R} (x, y) \leq z} \end{matrix}$

$\begin{matrix} = & ⋀ {z \in L | C_{U_{1}}^{R} (x, y) \lor C_{U_{2}}^{R} (x, y) \leq z} \\ = & (C_{U_{1}}^{R} \lor C_{U_{2}}^{R}) (x, y) \forall x, y \in L, \end{matrix}$ i.e., φ ₂ (U ₁ ∧ U ₂) = φ ₂ (U ₁) ∨ φ ₂ (U ₂). So, φ ₂ is an order-reversing isomorphism of $(U_{ds \land}^{e_{L}} (L), \land)$ onto $(C_{\lor}^{np e_{L}} (L), \lor)$ .

(3) Define $f : U_{cs \lor}^{e_{L}} (L) \to U_{ds \land}^{N (e_{L})} (L)$ as follows: $f (U) = U_{N} \forall U \in U_{cs \lor}^{e_{L}} (L)$ .

(i) If $U \in U_{cs \lor}^{e_{L}} (L)$ , then it follows from Theorem 2.10 (2) and (6) that U _N is a right infinitely ∧-distributive left semi-uninorm with the left neutral element N (e _L). Noting that U is a conjunctive left semi-uninorm, we have that $\begin{matrix} U_{N} (1, 0) = N^{- 1} (U (N (1), (N (0))) \\ = & N^{- 1} (U (0, 1)) = N^{- 1} (0) = 1, \\ U_{N} (0, 1) = N^{- 1} (U (N (0), (N (1))) \\ = & N^{- 1} (U (1, 0)) = N^{- 1} (0) = 1 . \end{matrix}$ Thus, $U_{N} \in U_{ds \land}^{N (e_{L})} (L)$ and so f is a morphism of $U_{cs \lor}^{e_{L}} (L)$ into $U_{ds \land}^{N (e_{L})} (L)$ .

(ii) If $U_{1}, U_{2} \in U_{cs \lor}^{e_{L}} (L)$ and f (U ₁) = f (U ₂), then $\begin{matrix} (U_{1})_{N} = (U_{2})_{N}, \\ U_{1} = ((U_{1})_{N})_{N} = ((U_{2})_{N})_{N} = U_{2} . \end{matrix}$ Moreover, for any $U \in U_{ds \land}^{N (e_{L})} (L)$ , we have that $U_{N} \in U_{cs \lor}^{e_{L}} (L)$ and f (U _N) = (U _N) _N = U. Thus, f is a bijection.

(iii) If $U_{1}, U_{2} \in U_{cs \lor}^{e_{L}} (L)$ , then it follows from Theorem 2.10 (1) that $\begin{matrix} f (U_{1} \lor U_{2}) = (U_{1} \lor U_{2})_{N} \\ = & (U_{1})_{N} \land (U_{2})_{N} = f (U_{1}) \land f (U_{2}) . \end{matrix}$ Therefore, f is an order-inversing isomorphism of $(U_{cs \lor}^{e_{L}} (L), \lor)$ onto $(U_{ds \land}^{N (e_{L})} (L), \land)$ .

(4) Define $g : I_{\land}^{{npe}_{L}} (L) \to C_{\lor}^{npN (e_{L})} (L)$ as follows: $g (I) = I_{N} \forall I \in I_{\land}^{{npe}_{L}} (L)$ . If $I \in I_{\land}^{{npe}_{L}} (L)$ , then $I_{N} \in C_{\lor} (L)$ by Theorem 2.10 (3) and (5) and $\begin{matrix} I_{N} (N (e_{L}), x)) = N^{- 1} (I (N (N (e_{L})), N (x))) \\ = & N^{- 1} (I (e_{L}, N (x)) = N^{- 1} (N (x)) = x \forall x \in L . \end{matrix}$ Thus, $I_{N} \in C_{\lor}^{npN (e_{L})} (L)$ and g is a morphism of $I_{\land}^{{npe}_{L}} (L)$ into $C_{\lor}^{npN (e_{L})} (L)$ . Moreover, by the proof of statement (3), we see that g is an order-inversing isomorphism of $(I_{\land}^{{npe}_{L}} (L), \land)$ onto $(C_{\lor}^{npN (e_{L})} (L), \lor)$ .□

By Theorems 4.1, 4.3 and 4.4, we can get two relational graphs as follows:

5 Conclusions and future works

In this paper, we have discussed the residual implications and coimplications generated by left (right) semi-uninorms and the left (right) semi-uninorms induced by implications and coimplications. We have shown that the N-dual operation of the right (left) residual implication, which is generated by a left (right)-conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm, is the right (left) residual coimplication generated by its N-dual operation and the N-dual operation of the right (left) residual coimplication, which is generated by a left (right)-disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorm, is the right (left) residual implication generated by its N-dual operation. We have demonstrated that that the N-dual operations of the left (right) semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left (right) semi-uninorms. We have revealed the relationships between conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms induced by implication and disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms induced by coimplication; and proved that the join-semilattice of all conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms is order-reversing isomorphic to the meet-semilattice of all right infinitely ∧-distributive implications, the meet-semilattice of all right infinitely ∧-distributive implications is order-reversing isomorphic to the join-semilattice of all right infinitely ∨-distributive coimplications, and the join-semilattice of all right infinitely ∨-distributive coimplications is order-reversing isomorphic to the meet-semilattice of all disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms; where all implications and coimplications satisfy the neutrality principle.

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

Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments and suggestions on an earlier version of this paper.

This work is supported by Science Foundation of Yancheng Teachers University (13YSYJB0108).

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