Generalizations of ( U,N )-implications derived from commutative semi-uninorms and pseudo-uninorms

Abstract

In this paper, two classes of generalized (U, N)-implications derived from commutative semi-uninorms and pseudo-uninorms are investigated. Then some properties of U_I,N operations derived from implications are explored. Last the necessary and sufficient conditions such that a (U_CS, N)-operation (or (U_P, N)-operation) is an implication are given out.

Keywords

Fuzzy connective fuzzy implication commutative semi-uninorm pseudo-uninorm

1 Introduction

Fuzzy logic has been widely applied in many fields, such as approximate reasoning, decision making, expert systems and control, etc.. In fuzzy logic, fuzzy implications (or simply, implications) are probably the most important tools for managing the conditional statement “If p, then q”, with p and q fuzzy statements. Hence, fuzzy implications play a vital role in almost all fields where fuzzy logic applies [1 –9].

As the diversity of the real applications, using different models to perform fuzzy implications is necessary. The importance of having different models for fuzzy implications is also pointed out [10]. Various implications are put forward in literature. As known (S, N)-implications are one of important classes of implications. Baczyński and Jayaram [11] have characterized (S, N)-implications generated from triangular conorms (t-conorms, for short) and continuous (strict, strong) negation. Many researchers generalized (S, N)-implications by taking a different class of aggregation functions instead of the usual t-conorms. Bustince, et al. [12] introduced the (TS, N)-operation derived from a function TS _λ,f and a fuzzy negation N. They also showed the necessary and sufficient conditions to guarantee that a (TS, N)-operation is in fact a (TS, N)-implication. Aguiló, et al. [13] introduced the (G _f,N, N)-implication derived from a dual representable aggregation function G _f,N and a fuzzy negation N. Ouyang [14] introduced a S-implication like operator I _S derived from a binary function A and a strong negation N and discussed its some basic properties.

(U, N)-implications, as a generalization of (S, N)-implications, are also an important class of implications, where a t-conorm S is replaced by a uninorm U. A uninorm is a binary operation with monotonicity, associativity, commutativity and a neutral element e ∈ (0, 1). Baczyński and Jayaram [15] have also characterized (U, N)-implications generated from disjunctive uninorms and strict (strong) negations.

By removing the commutativity and associativity, Liu [24] introduced semi-uninorms U _S on a completelattice and discussed some of their properties. Baczyński and Jayaram [15] used to indicate that characterization of the family of implications derived from some non-commutative connectives seems worthy of an attempt. However, in some cases, for the convenience, commutativity (or associativity) is important and required to generate implications. Hence, in this paper we will extend the (U, N)-implications by the commutative semi-uninorms and pseudo-uninorms. Some properties and characterisations of these two classes of implications will be investigated.

The remain parts of this paper are organized as follows. In Section 2, we review some concepts and conclusions about semi-uninorms, (U, N)-implications, etc.. In Section 3, we discuss the properties of implications derived from semi-uninorms, commutative semi-uninorms and pseudo-uninorms. Section 4 explores some properties of U _I,N operations derived from implications. (U _CS, N)- and (U _P, N)-implications are characterized in Section 5. The final section concludes the paper.

2 Preliminaries

In this section, we will recall some concepts and conclusions employed in the rest of this paper.

Definition 2.1. (Liu [24]) A binary operation U _S on [0, 1] is called a semi-uninorm if it satisfies thefollowing conditions:

(U1) there exists a neutral element e ∈ [0, 1], i.e., U _S (e, x) = U _S (x, e) = x for all x ∈ [0, 1];

(U2) U _S is non-decreasing in each variable.

A semi-uninorm U _S is called a commutative semi-uninorm, denoted by U _CS, if it is commutative. A semi-uninorm U _S is called a pseudo-uninorm, denoted by U _P, if it is associative. A semi-uninorm U _S on [0, 1] is a uninorm if it is commutative and associative. As an exception, if the neutral element e = 1 (e = 0), then the semi-uninorm U _S on [0, 1] becomes a triangular seminorm T (triangular semiconorm S) [25] or a semi-copula [26, 27]. It is clear that U _S (0, 0) =0 and U _S (1, 1) =1 hold for any semi-uninorm U _S on [0, 1], hence a semi-uninorm is a binary aggregation operator with a neutral element e ∈ [0, 1]. A semi-uninorm U _S is said to be left-conjunctive (right-conjunctive) if U _S (0, 1) =0 (U _S (1, 0) =0). U _S is said to be conjunctive if both U _S (0, 1) =0 and U _S (1, 0) =0. Obviously, U _S (0, x) =0 (U _S (x, 0) =0) (∀x ∈ [0, 1]) holds for any left-conjunctive (right-conjunctive) semi-uninorm U _S. If U _S (1, 0) =1 (U _S (0, 1) =1), then we call the semi-uninorm U _S left-disjunctive (right-disjunctive). We call the semi-uninorm U _S disjunctive if both U _S (1, 0) =1 and U _S (0, 1) =1.

Example 2.2. (i) Let U _S : [0, 1] ² → [0, 1] be a binary function defined as follows:

$\begin{matrix} U_{S} (x, y) = \\ {\begin{matrix} 0 & if (x, y) \in [0, \frac{1}{3})^{2}, \\ max {x, y} & if (x, y) \in [\frac{1}{3}, 1]^{2}, \\ min {x, y} & if (x, y) \in [0, \frac{1}{3}] \times {\frac{1}{3}} \\ or (x, y) \in {\frac{1}{3}} \times [0, \frac{1}{3}], \\ \frac{1}{4} x + \frac{3}{4} y & otherwise, \end{matrix} \end{matrix}$ (1) then U _S is a semi-uninorm with neutral element $e = \frac{1}{3}$ .

(ii) Let U _CS : [0, 1] ² → [0, 1] be a binary function defined as follows:

$\begin{matrix} U_{CS} (x, y) = \\ {\begin{matrix} 0 & if (x, y) \in [0, \frac{1}{3})^{2}, \\ max {x, y} & if (x, y) \in [\frac{1}{3}, 1)^{2} \\ or x = 1 or y = 1, \\ min {x, y} & if (x, y) \in [0, \frac{1}{3}] \times {\frac{1}{3}} \\ or (x, y) \in {\frac{1}{3}} \times [0, \frac{1}{3}], \\ \frac{x + y}{2} & otherwise, \end{matrix} \end{matrix}$ (2) then U _CS is a commutative semi-uninorm with neutralelement $\frac{1}{3}$ .

(iii) (Su and Wang [21]) Let ${\underline{U}}_{P}, {\bar{U}}_{P} : [0, 1]^{2} \to [0, 1]$ be binary functions with neutral elements e ∈ (0, 1) defined as follows: $\begin{matrix} {\underline{U}}_{P} (x, y) = {\begin{matrix} 0, & if (x, y) \in [0, e)^{2}, \\ max (x, y) & if (x, y) \in [e, 1]^{2}, \\ min (x, y) & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} {\bar{U}}_{P} (x, y) = {\begin{matrix} min (x, y), & if (x, y) \in [0, e]^{2}, \\ 1, & if (x, y) \in (e, 1]^{2}, \\ max (x, y), & otherwise . \end{matrix} \end{matrix}$ Then ${\underline{U}}_{P}$ and ${\bar{U}}_{P}$ are the smallest and largest pseudo-uninorms with neutral element e on [0, 1], respectively. Moreover, ${\underline{U}}_{P}$ is conjunctive and ${\bar{U}}_{P}$ is disjunctive.

(iv) (Klement, Mesiar and Pap [2]) Let U _P : [0, 1] ² → [0, 1] be a binary function defined as follows: $\begin{matrix} U_{P} (x, y) = {\begin{matrix} 0 & if (x, y) \in [0, \frac{1}{2}] \times [0, 1), \\ min (x, y) & otherwise . \end{matrix} \end{matrix}$ Then U _P is a conjunctive pseudo-uninorm with neutral element 1 on [0, 1].

Following we will review some basic notions on fuzzy implications, fuzzy negations, etc., and more details can be found in [1, 15].

Definition 2.3. (Baczyński, Jayaram [1, 15]) A binary operator I : [0, 1] ² → [0, 1] is called a fuzzy implication (for short, implication) if, for all x, y ∈ [0, 1], it satisfies the following conditions:

(I1) I is nonincreasing in the first variable,

(I2) I is nondecreasing in the second variable,

(I3) I (0, 0) =1,

(I4) I (1, 1) =1,

(I5) I (1, 0) =0.

The set of all fuzzy implications will be denotedby

FI

Definition 2.4. (Baczyński, Jayaram [1, 15]) Let e ∈ ∖ (0, 1). An implication I is said to have (NP_e) the left neutrality property, if I (e, y) = y holds for any y ∈ [0, 1]; (OP_e) the order property, if x ≤ y ⇔ I (x, y) ≥ e holds for any x, y ∈ [0, 1]; (IP_e) the identity principle, if I (x, x) ≥ e holds for any x ∈ [0, 1].

Definition 2.5. (Baczyński, Jayaram [1, 15]) (i) A function N : [0, 1] → [0, 1] is called a fuzzy negation if N (0) =1, N (1) =0, and N is decreasing; (ii) A fuzzy negation N is called strict if, N is strictly decreasing and continuous; (iii) A fuzzy negation N is called strong if it is an involution, i.e., N (N (x)) = x, x ∈ [0, 1].

Lemma 2.6. (Bazyński, Jayaram [11, 16]). If N is a continuous fuzzy negation, then function N : [0, 1] → [0, 1] defined by: $\begin{matrix} N (x) = {\begin{matrix} N^{(- 1)} (x), & if x \in (0, 1], \\ 1, & if x = 0, \end{matrix} \end{matrix}$ is a strictly decreasing fuzzy negation. Moreover, $\begin{matrix} N^{(- 1)} = N, \\ N \circ N = {id}_{[0, 1]}, \\ N \circ N |_{Ran (N)} = {id}_{Ran (N)} . \end{matrix}$

Definition 2.7. (De Baets [18]) Consider a fuzzy negation N on [0, 1]. The N-dual operation of a binary operation A on [0, 1] is the binary operation A _N on [0, 1] defined by: $\begin{matrix} A_{N} (x, y) = N^{- 1} (A (N (x), N (y))), \forall x, y \in [0, 1] . \end{matrix}$

Definition 2.8. (Baczyński, Jayaram [1, 15]) Let I : [0, 1] ² → [0, 1] be any function and α ∈ [0, 1]. If $N_{I}^{α}$ given by: $\begin{matrix} N_{I}^{α} (x) = I (x, α), x \in [0, 1], \end{matrix}$ is a fuzzy negation, then it is called the natural negation of I with respect to α.

Definition 2.9. UN (Baczyński, Jayaram [1, 15]) A function I : [0, 1] ² → [0, 1] is called a (U, N)-operation, if there exist a uninorm U and a fuzzy negation N such that $\begin{matrix} I_{U, N} (x, y) = U (N (x), y), x, y \in [0, 1] . \end{matrix}$

3 Generalizations of (U, N)-implications

As pointed out by Fodor and Keresztfalvi [23], sometimes there is no need of the commutativity and associativity for connectives “and” and “or”. By throwingaway the associativity and commutativity respectively, we will investigate the (U, N)- implications derived from commutative semi-uninorms and pseudo-uninorms.

Definition 3.1. A function I : [0, 1] ² → [0, 1] is called a (U _S, N)-operation, if there exist a semi-uninorm U _S with neutral element e ∈ [0, 1] and a fuzzy negation N on [0, 1] such that $\begin{matrix} I (x, y) = U_{S} (N (x), y), x, y \in [0, 1] . \end{matrix}$

If I is a (U _S, N)-operation generated from a semi-uninorms U _S and a fuzzy negation N, then I is denoted by I _{U_S,N}. Particularly, if U _S is commutative, then (U _S, N)-operation can write as (U _CS, N)-operation and I is denoted by I _{U_CS,N}. If U _S is associative, i.e., U _S is a pseudo-uninorm, then (U _S, N)-operation can write as (U _P, N)-operation and I is denoted by I _{U_P,N}. It is clear that (U _CS, N)-operations and (U _P, N)-operations are generalizations of (U, N)-operations. If U _CS (U _P) is associative (commutative) then corresponding (U _CS, N)-operations ((U _P, N)-operations) become (U, N)-operations.

Proposition 3.2. Let I_{U_S,N} be a (U_S, N)-operation, U_S be a semi-uninorm with neutral element e ∈ (0, 1) and N be a fuzzy negation on [0, 1], then

I _{U_S,N} satisfies (I1),(I2) and (I5);

If N (e) = e, then I _{U_S,N} satisfies NP_e, i.e. I _{U_S,N} (e, y) = y;

If N is a continuous negation on [0, 1], then there exist some α ∈]0, 1 [ such that I _{U_S,N} satisfies NP_α;

$N_{I_{U_{S}, N}}^{e} = N$ ;

If U _S is commutative, then I _{U_S,N} satisfiesR-CP(N) with respect to N;

If U _S is commutative and N is continuous, then I _{U_S,N} satisfies L-CP $(N)$ with respect to $N$ ;

If U _S is commutative and N is strong, then I _{U_S,N} satisfies CP(N) with respect to N;

If U _S is associative and N is strong, then

\begin{matrix} I_{U_{P}, N} (N (x), I_{U_{P}, N} (y, z)) \\ = I_{U_{P}, N} ((I_{U_{P}, N})_{N} (x, y), z) \end{matrix}

holds for all x, y, z ∈ [0, 1].

Proof. (i) Since a semi-uninorm U _S is non-decreasing in each variable and a fuzzy negation N is decreasing, I _{U_S,N} satisfies (I1) and (I2). Moreover, $\begin{matrix} I_{U_{S}, N} (1, 0) = U_{S} (N (1), 0) = U_{S} (0, 0) = 0, \end{matrix}$ i.e. I _{U_S,N} satisfies (I5). (ii) If N (e) = e, then for any y ∈ [0, 1], $\begin{matrix} I_{U_{S}, N} (e, y) = U_{S} (N (e), y) = U_{S} (e, y) = y, \end{matrix}$ i.e. I _{U_S,N} satisfies NP_e. (iii) If N is continuous and e ∈ (0, 1), then there exists some α such that N (α) = e. For any y ∈ [0, 1], $\begin{matrix} I_{U_{S}, N} (α, y) = U_{S} (N (α), y) = U_{S} (e, y) = y, \end{matrix}$ i.e. I _{U_S,N} satisfies NP_α. (iv) For any x ∈ [0, 1], $\begin{matrix} N_{I_{U_{S}, N}}^{e} (x) & = I_{U_{S}, N} (x, e) \\ = U_{S} (N (x), e) \\ = N (x), \end{matrix}$ i.e., $N_{I_{U_{S}, N}}^{e} = N$ . (v) For x, y ∈ [0, 1], since U _CS,N is commutative and N is a negation, we get: $\begin{matrix} I_{U_{S}, N} (x, N (y)) & = U_{S} (N (x), N (y)) \\ = U_{S} (N (y), N (x)) \\ = I_{U_{S}, N} (y, N (x)), \end{matrix}$ i.e. I _{U_S,N} satisfies R-CP with respect to N.

(vi) Since U _S is commutative and N is continuous, then for any x, y ∈ [0, 1], $\begin{matrix} I_{U_{S}, N} (N (x), y) & = U_{S} (N \circ N (x), y) \\ = U_{S} (x, y) = U_{S} (y, x) \\ = U_{S} (N \circ N (y), x) \\ = I_{U_{S}, N} (N (y), x), \end{matrix}$ i.e. I _{U_S,N} satisfies L-CP with respect to $N$ . (vii) For x, y ∈ [0, 1], since U _CS,N is commutative and N is a strong negation, we get: $\begin{matrix} I_{U_{S}, N} (x, y) & = U_{S} (N (x), y) \\ = U_{S} (N (N (y)), N (x)) \\ = I_{U_{S}, N} (N (y), N (x)) \end{matrix}$ i.e. I _{U_S,N} satisfies CP with respect to N. (viii) Since U _S is associative and N is strong, then for any x, y, z ∈ [0, 1], $\begin{matrix} I_{U_{P}, N} (N (x), I_{U_{P}, N} (y, z)) \\ = U_{P} (x, U_{P} (N (y), z)) \\ = U_{P} (U_{P} (x, N (y)), z)) \\ = I_{U_{P}, N} (N (I_{U_{P}, N} (N (x), N (y))), z) \\ = I_{U_{P}, N} ((I_{U_{P}, N})_{N} (x, y), z) . \end{matrix}$

Remark 3.3. Baczyński and Jayaram[1] pointed out that for any binary function I and a strong negation N, I satisfies CP(N) if and only if I satisfies L-CP(N) if and only if I satisfies R-CP(N). Hence, for a commutative semi-uniorm and a strong negation N, I _{U_S,N} satisfies CP(N), L-CP(N) and R-CP(N) by Proposition 3 (vii).

Example 3.4. Let U _S be a semi-uninorm given in Equation (1) and N (x) =1 - x, then $\begin{matrix} I_{U_{S}, N} (x, y) = \\ {\begin{matrix} 0 & if \frac{2}{3} < x \leq 1 \\ and 0 \leq y < \frac{1}{3}, \\ max {1 - x, y} & if 0 \leq x \leq \frac{2}{3} \\ and \frac{1}{3} \leq y \leq 1, \\ min {1 - x, y} & if (\frac{2}{3} \leq x \leq 1 and y = \frac{1}{3}) \\ or (x = \frac{2}{3} and 0 \leq y \leq \frac{1}{3}), \\ \frac{1}{4} - \frac{1}{4} x + \frac{3}{4} y & otherwise . \end{matrix} \end{matrix}$

It is clear that I _{U_S,N} satisfies (I1), (I2) and (I5) but not (I3) and (I4).

From Example 3.4, we know that a (U _S, N)-operation generated from a semi-uninorm U _S and a fuzzy negation N is not always a fuzzy implication. Let U _S be a semi-uninorm with neutral element e ∈ (0, 1) and N be a fuzzy negation on [0, 1]. The operation I _{U_S,N} is a fuzzy implication if and only if U _S is a disjunctive semi-uninorm, i.e. U _CS (0, 1) = U _CS (1, 0) =1.

Example 3.5. Let U _S be a disjunctive semi-uninorm defined as follows: $\begin{matrix} U_{S} (x, y) = \\ {\begin{matrix} 0 & if (x, y) \in [0, \frac{1}{3})^{2}, \\ max {x, y} & if (x, y) \in [\frac{1}{3}, 1)^{2} \\ or x = 1 or y = 1, \\ min {x, y} & if (x, y) \in [0, \frac{1}{3}] \times {\frac{1}{3}} \\ or (x, y) \in {\frac{1}{3}} \times [0, \frac{1}{3}], \\ \frac{1}{4} x + \frac{3}{4} y & otherwise, \end{matrix} \end{matrix}$ and N (x) =1 - x, then $\begin{matrix} I_{U_{S}, N} (x, y) = \\ {\begin{matrix} 0 & if \frac{2}{3} < x \leq 1 and 0 \leq y < \frac{1}{3}, \\ max {1 - x, y} & if (0 < x \leq \frac{2}{3} and \frac{1}{3} \leq y < 1) \\ or x = 0 or y = 1, \\ min {1 - x, y} & if (\frac{2}{3} \leq x \leq 1 and y = \frac{1}{3}) \\ or (x = \frac{2}{3} and 0 \leq y \leq \frac{1}{3}), \\ \frac{1}{4} - \frac{1}{4} x + \frac{3}{4} y & otherwise . \end{matrix} \end{matrix}$

It is clear that I _{U_S,N} satisfies (I1)-(I5) and hence is a fuzzy implication.

Proposition 3.6. Let I_{U_S,N} be a (U_S, N)-implication generated from a disjunctive semi-uninorm U_S with neutral element e ∈ (0, 1) and a continuous fuzzy negation N. Let α ∈ (0, 1) be an arbitrary but fixed number. Then the following statements are equivalent:

(i) $N_{I_{U_{S}, N}}^{α} = N$ ,

(ii) α = e.

Proof. (i) ⇒ (ii) Since N is continuous, there exists an e ₀ ∈ (0, 1) such that e = N (e ₀). If $N_{I_{U_{S}, N}}^{α} = N$ , then we get $\begin{matrix} e & = N (e_{0}) \\ = N_{I_{U_{S}, N}}^{α} (e_{0}) \\ = I_{U_{S}, N} (e_{0}, α) \end{matrix}$

$\begin{matrix} = U_{S} (N (e_{0}), α) \\ = U (e, α) \\ = α . \end{matrix}$ (ii) ⇒ (i) If α = e, then for all x ∈ [0, 1], $\begin{matrix} N_{I_{U_{S}, N}}^{α} (x) & = I_{U_{S}, N} (x, α) \\ = I_{U_{S}, N} (x, e) \\ = U_{S} (N (x), e) \\ = N (x) . \end{matrix}$

4 U_I,N operations derived from implications

In this section, we will explore some properties of U _I,N operations derived from implications.

Definition 4.1. (Baczy $\overset{´}{n}$ ski, Jayaram [1, 15]) Let $I \in FI$ and N be a fuzzy negation. A binary operation U _I,N on [0, 1] is defined as follows: $U_{I, N} (x, y) = I (N (x), y), x, y \in [0, 1] .$ (3)

Proposition 4.2. Let $I \in FI$ and N be a fuzzy negation. Then for all x ∈ [0, 1], we have

(i) U_I,N is increasing in each variable;

(ii)U _I,N (x, 1) = U _I,N (1, x) =1. In particular, U _I,N (0, 1) = U _I,N (1, 0) =1, i.e., U _I,N is disjunctive;

(iii) If I satisfies NP_N (e), where e ∈ (0, 1) is a constant, then e is a left neutral element of U _I,N. If $N_{I}^{e} \circ N = {id}_{[0, 1]}$ , then e is a right neutral element of U _I,N. Further, if I satisfies both NP_N (e) and $N_{I}^{e} \circ N = {id}_{[0, 1]}$ , then U_I,N is a semi-uninorm with the neutral element e.

Proof. For the proof of statements (i) and (ii), we refer to the proof of Proposition 6.1 (i) and (ii)[15]. (iii) If I satisfies NP_N (e), then $\begin{matrix} U_{I, N} (e, x) = I (N (e), x) = x, \end{matrix}$ i.e., U _I,N has a left neutral element. If $N_{I}^{e} \circ N = {id}_{[0, 1]}$ , then $\begin{matrix} U_{I, N} (x, e) = I (N (x), e) = N_{I}^{e} \circ N (x) = x, \end{matrix}$ i.e., U _I,N has a right neutral element. Further, if Isatisfies both NP_N (e) and $N_{I}^{e} \circ N = {id}_{[0, 1]}$ , then U _I,N is a semi-uninorm with the neutral element e.

Lemma 4.3. Let I_{U_S,N} be a (U_S, N)-implication generated from a disjunctive semi-uninorm U_S and a continuous fuzzy negation N, then U_S and N are uniquely determined.

Proof. By Proposition 3, we have that $N = N_{I_{U_{S}, N}}^{e}$ is uniquely determined. We assume that there exist two semi-uninorms U _S1 and U _S2 such that, for all x, y ∈ [0, 1], $\begin{matrix} I_{U_{S}, N} (x, y) = U_{S 1} (N (x), y) = U_{S 2} (N (x), y) . \end{matrix}$

Fixed arbitrarily x ₀, y ₀ ∈ [0, 1]. Since N is continuous there is an x ₁ ∈ [0, 1] such that N (x ₁) = x ₀. Thus, $\begin{matrix} U_{S 1} (x_{0}, y_{0}) & = U_{S 1} (N (x_{1}), y_{0}) \\ = I_{U_{S}, N} (x_{1}, y_{0}) \\ = U_{S 2} (N (x_{1}), y_{0}) \\ = U_{S 2} (x_{0}, y_{0}), \end{matrix}$ i.e., U _S1 = U _S2. Thus, U _S is uniquely determined.

5 Characterizations of (U_CS, N)- and (U_P, N)-implications

In this part, we will characterize the (U _CS, N)- and (U _P, N)-implications generated from commutative semi-uninorms and pseudo-uninorms.

Proposition 5.1. Let $I \in FI$ and N be a fuzzy negation. Then for all x ∈ [0, 1], we have (i) U_I,N is commutative if and only if I satisfiesL-CP(N); (ii) If I satisfies NP_N (e) (or $N_{I}^{e} \circ N = {id}_{[0, 1]}$ ) andL-CP(N), then U_I,N is a commutative semi-uninorm with the neutral element e.

Proof. (i) If U _I,N is commutative, then $\begin{matrix} I (N (x), y) & = U_{I, N} (x, y) \\ = U_{I, N} (y, x) \\ = I (N (y), x), \end{matrix}$ i.e., I satisfies L - CP (N). Conversely, we can obtain the conclusion by retracing the above steps. (ii) If I satisfies NP_N (e) and L-CP(N), from (i) and Proposition 4.2 (iii), U _I,N is a commutative semi-uninorm with the neutral element e. For the case that I satisfies L-CP(N) and $N_{I}^{e} \circ N = {id}_{[0, 1]}$ , the proof is similar.

Theorem 5.2. For a binary operator I : [0, 1] ² → [0, 1], the following statements are equivalent: (i) I is a (U_CS, N)-implication generated from some disjunctive commutative semi-uninorm U _CS , which has neutral element e, and some strict negation N. (ii) I satisfies (I1), (I3), NPN ^-1 (e) and L-CP(N^- 1), where N is a strict negation.

Moreover, the representation of (U _CS, N)-implication is unique in this case.

Proof. (i) ⇒ (ii) It is clear that I satisfies (I1) and (I3). For N is strict, we have $\begin{matrix} I (N^{- 1} (e), y) & = U (N \circ N^{- 1} (e), y) \\ = U (e, y) \\ = y, \end{matrix}$ for all y ∈ [0, 1], i.e., I satisfies (NP _N^-1(e)). By Proposition 3.4 (iv) and Proposition 1.5.2 [1], I satisfies R - CP (N), furthermore, I satisfies L - CP (N ^-1).

(ii) ⇒ (i) I satisfies (I1) and L - CP (N ^-1), by Lemma 1.5.13 [1], I satisfies (I2). Since I satisfies (I3) and L - CP (N ^-1), then $\begin{matrix} I (1, 1) & = I (N^{- 1} (0), 1) \\ = I (N^{- 1} (1), 0) \\ = I (0, 0) \\ = 1, \end{matrix}$ i.e., I satisfies (I4). I satisfies (I1) and (NP _N^-1(e)), then $\begin{matrix} 0 \leq I (1, 0) \leq I (N^{- 1} (e), 0) = 0 . \end{matrix}$

That is I satisfies (I5). Thus, I is a fuzzy implication. By virtue of Proposition 5.1 (ii), U _I,N^-1 is a commutative semi-uninorm. Further, for any x, y ∈ [0, 1] $\begin{matrix} I_{U_{I, N^{- 1}}, N} (x, y) & = U_{I, N^{- 1}} (N (x), y) \\ = I (N^{- 1} (N (x)), y) \\ = I (x, y) . \end{matrix}$ By Lemma 4, the representation of (U _CS, N)- implication is unique in this case.

Wang and Fang introduced pseudo-uninorms on a complete lattice in [20]. Su, Wang [21] and Su, Liu [22], respectively, further studied pseudo-uninorms, coimplications and residual coimplications generated from pseudo-uninorms on a complete lattice. Here, we will characterize the (U _P, N)-implications derived from pseudo-uninorms.

Proposition 5.3. Let $I \in FI$ and N be a strong fuzzy negation. Then for all x ∈ [0, 1] (i) If I satisfies the following equality

$I (N (x), I (y, z)) = I (I_{N} (x, y), z),$ (4) for all x, y, z ∈ [0, 1], then U _I,N is associative, where I _N is a N-dual operation of I. (ii) If I satisfies NP_N (e), $N_{I}^{e} \circ N = {id}_{[0, 1]}$ and Equation (4), then U _I,N is a pseudo-uninorm with the neutralelement e.

Proof. (i) For any x, y, z ∈ [0, 1], we have $\begin{matrix} U_{I, N} (x, U_{I, N} (y, z)) \\ = I (N (x), I (N (y), z)) \\ = I (I_{N} (x, N (y)), z) by Equation (Eq 1) \\ = I (N^{- 1} (I (N (x), y)), z) by Definition 2.10 \\ = I (N (I (N (x), y)), z) by N is strong \\ = I (N (U_{I, N} (x, y), z) \\ = U_{I, N} (U_{I, N} (x, y), z), \end{matrix}$ i.e., U _I,N is associative. (ii) If I satisfies NP_N (e) and $N_{I}^{e} \circ N = {id}_{[0, 1]}$ , by Proposition 4.2, U _I,N is a semi-uninorm with the neutral element e. Further, I satisfies Eq. (4), by (i), U _I,N is associative. Hence, U _I,N is a pseudo-uninorm with the neutral element e.

Theorem 5.4. For a binary operator I : [0, 1] ² → [0, 1], the following statements are equivalent:

(i) I is a (U _P, N)-implication generated from some disjunctive pseudo-uninorm U _P, which has neutral element e, and some strong negation N.

(ii) I satisfies (I1)-(I4), NP_N (e), $N_{I}^{e} \circ N = {id}_{[0, 1]}$ and Equation (4), where N is a strong negation.

Moreover, the representation of (U _P, N)-implication is unique in this case.

Proof. We can prove (i) ⇒ (ii) easily by the Proposition 3.2 (ii), (iv), (viii) and Lemma 4.3. (ii) ⇒ (i) I satisfies (I1) and (NP _N^-1(e)), then $\begin{matrix} 0 \leq I (1, 0) \leq I (N^{- 1} (e), 0) = 0 . \end{matrix}$ That is I satisfies (I5). Thus, I is a fuzzy implication. By virtue of Proposition 5.3, I is a (U _P, N)-implication generated from some disjunctive pseudo-uninorm. Further, by Lemma 4.3, the representation of (U _P, N)-implication is unique in this case.

Remark 5.5. Bacyński and Jayaram have characterized (U, N)-implications generated from disjunctive uninorms and strict (strong) negations (Theorem 6.5 [15]). From the associativity and commutativity of uninorms, (U, N)-implications satisfies EP. There need less conditions ((I1), (I3), EP and the function $N_{I}^{e}$ is a strict (strong) negation) to characterize (U, N)-implications. However, commutative semi-unnorms (or pseudo-uninorms) have not the associativity (or commutativity) and EP is independent on NP, OP, IP, CP and continuity (see Proposition 4.1[29]), thus, there need more conditions to characterize (U _CS, N)-implications (or (U _P, N)-implications).

6 Conclusions

In this paper we introduced the (U _CS, N)- and (U _P, N)-operations derived from commutative semi-uninorms and pseudo-uninorms. The necessary and sufficient conditions that a (U _CS, N)-operation or (U _P, N)-operation is an implication were given out. Some properties and characterizations of these two classes of implications were investigated. As the generalizations of (U, N)-implications, these implications can be applied in some fields. For exploring the applications, in the future, we will introduce the distributivity of these two class of implications over t-norms and t-conorms.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61174099) and the Research Found for the Doctoral Program of Higher Education of China (No. 20120131110001). The authors would like to express their thanks to the editors and the anonymous referees for their valuable comments and kind help.

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Generalizations of ( U,N )-implications derived from commutative semi-uninorms and pseudo-uninorms

Abstract

Keywords

1 Introduction

2 Preliminaries

4 UI,N operations derived from implications

Acknowledgments

References

4 U_I,N operations derived from implications