In clasical logic, it is possible to combine the uniary negation operator ¬ with any other binary operator in order to generate the other binary operators. In this paper, we introduce the concept of (N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G and two different fuzzy negations N∗ and N are used for the generalization of the implication p → q ≡ ¬ [p ∧ ¬ (¬ p ∨ q)] . We show that (N∗, O, N, G)-implication are fuzzy implication without any restricted conditions. Further, we also study that some properties of (N∗, O, N, G)-implication that are necessary for the development of this paper. The key contribution of this paper is to introduced the concept of circledcircG,N-compositions on (N∗, O, N, G)-implications. If - or -implications constructed from the tuples or satisfy a certain property P, we now investigate whether circledcircG,N-composition of - and -implications satisfies the same property or not. If not, then we attempt to characterise those implications -, -implications satisfying the property P such that circledcircG,N-composition of - and -implications also satisfies the same property. Further, we introduced sup-circledcircO-composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Subsequently, we show that under which condition sup-circledcircO-composition of (N∗, O, N, G)-implications are fuzzy implication. We also study the intersections between families of fuzzy implications, including RO-implications (residual implication), (G, N)-implications, QL-implications, D-implications and (N∗, O, N, G)-implications.
Fuzzy implications play an important role in both theoretical and applied aspects of fuzzy set theory while generalizing the classical implication which takes values in the set {0, 1} , to fuzzy logic where the truth values lie in the unit interval [0, 1] . Fuzzy implications play a key role in approximate reasoning, fuzzy control, decision theory, control theory, expert systems. Baldwin introduced a new approach to approximate reasoning by utilizing fuzzy logic concept [6]. Further, Baldwin and Pilsworth [7], introduced axiomatic technique to implication for approximate reasoning using fuzzy logic. Yager [35] introduced a new classes of implication operators and their application in approximate reasoning. Based on fuzzy logic, Deng and Heijmans, introduced the concept of grey-scale morphology [14]. D’eer et al. [13] initiated the concept of noise tolerant fuzzy rough sets applying implicator conjunctor theory. Fang and Hu [25], applied fuzzy implicators and complicators to construct granular fuzzy rough set theory. Based on quantitative database, Yan and Chen discovered a cover set of ARsi with hierarchy [36]. Dombi and Baczynski emphasized the application of fuzzy implications in fuzzy control theory [20]. Kerre and Nachtegael developed novel fuzzy implication strategies for image processing models [28].
Brief review of fuzzy implications
The different techniques of obtaining fuzzy implications found in the literature, so far, can be largely categorized based on the underlying operators from where they are generated: (a) from other fuzzy logical connectives; Baczynski and Jayaram [1], introduced (S, N)-implications and their characterization based on t-conorm S and fuzzy negation N . Dubois and Prade defined residuated implication (R - implication) based on t-norm T and discussed the application of R-implication in approximate reasoning [21]. Baczynski and Jayaram introduced QL-opertion and QL-implication based on t-conorm S, t-norm T and fuzzy negation N and discussed some of their basic properties [4]. Ruiz-Aguilera and Torrens introduced the concept of residual implication (R - implication) based on conjunctive and disjunctive uninorm [34]. Pinheiro et al. [33], introduced (T, N)-implications and their characterization based on t-norm T and fuzzy negation N . (b) From monotone functions over the unit interval [0, 1] ; Massanet and Torrens [30], developed a new class of fuzzy implications called h-implications and their generalizations. Yager [35], introduced some new classes of implication (f - and g - implications) operators and their application in approximate reasoning. (c) From given fuzzy implications; Durante et al. [22], developed the concept of residual ordinal sum implications and their characterizations. Massanet and Torrens [31], applied threshold generation technique for the construction of a new implication and discussed its application. Further, Massanet and Torrens introduced vertical threshold generation technique for the construction of new fuzzy implication and developed its properties [32].
Now notice that both t-norms and t-conorms are associative aggregation functions, which enables any QL-implication function to comply with a variety of properties (as discussed, for example, by Baczynski and Jayaram [2] and Baczynski et al. [5]). Interestingly, Fodor and Keresztfalvi [26], Bustince et al. [10, 11], and Dimuro et al. [16] have demonstrated that there are a variety of situations where the associativity of the aggregation operators is not necessary nor always needed, including decision-making [8, 12, 23], classification issues [24, 29], and image processing [27].
Bustince et al. [9], introduced a non associative binary aggregation function called the overlap function O and discussed the migrativity of overlap function. Further, Bustince et al. [11], introduced another non associative binary aggregation function called grouping function G and in addition they studied generalized bientropic functions for fuzzy modeling of pairwise comparisons. Dimuro and Bedregal [15], developed archimedean overlap functions and further they studied some properties like ordinal sum, cancellation, idempotency and limiting properties. Based on grouping function G and fuzzy negation N, Dimuro et al. [16] presented the concept of (G, N)-implication and analyzed its characterization. Based on overlap function, Dimuro and Bedregal introduced the concept of residual implications and analyzed some of their properties [17]. Dimuro et al. [18] presented the idea of QL-operation and QL-implication constructed from grouping G function, overlap O function and fuzzy negation N and discussed a conditions for QL-operations become QL-implications. Further, Dimuro et al. [19] studied law of O-conditionality for fuzzy implications constructed from grouping G functions and overlap O functions.
Motivation of our research
In classical logic, it is possible to combine the unary negation operator ¬ with any other binary operator in order to generate the other binary operators. In this paper, we introduce the concept of (N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G and two different fuzzy negations N∗ and N are used for the generalization of the implication p → q ≡ ¬ [p ∧ ¬ (¬ p ∨ q)] . We show that (N∗, O, N, G)-implication are fuzzy implication without any restricted conditions. Further, we also study that some properties of (N∗, O, N, G)-implication that are necessary for the development of this paper.
The study of distributive laws involving fuzzy implications has attracted much attention not only due to their theoretical aesthetics but also due to their applicational value. In this paper, we present the important distributive laws involving (N∗, O, N, G)-implications denoted by IN∗,O,N,G with respect to greatest fuzzy negation N = Nbarwedge . The distributive laws are
where O, O(1), O(2), O(3) : [0, 1] 2 → [0, 1] are four overlap functions and G, G(1), G(2), G(3) : [0, 1] 2 → [0, 1] are four grouping functions. Another contribution in this paper is the exchange principle IN∗,O,N,G (e, IN∗,O,N,G (u, v)) = IN∗,O,N,G (u, IN∗,O,N,G (e, v)) under certain conditions.
The key contribution of this paper is to introduce the concept of ⊚G,N-compositions on (N∗, O, N, G)-implications. Let or be two fuzzy implications constructed from the tuples or satisfying certain property P . We now investigate whether ⊚G,N-composition denoted by satisfies the same property or not. If not, then we attempt to characterize those implications which satisfy the property P such that also satisfies the same property? Further, we introduce sup-⊚O-composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Subsequently, we show that under which condition sup-⊚O-composition of (N∗, O, N, G)-implications are fuzzy implication. We study and discuss the intersections between families of fuzzy implications, including RO-implications, (G, N)-implications, (O, G, N)-implications, D-implications and (N∗, O, N, G)-implications.
The structure of this paper
The remainder of the paper is organized as follows. Section 2 presents basic concepts that are necessary to develop the paper, including the concepts related to fuzzy negations, grouping functions, overlap functions and fuzzy negations. In Section 3, we introduce the concept of (N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G and two different fuzzy negations N∗ and N which are necessary for the generalization of the implication p → q ≡ ¬ [p ∧ ¬ (¬ p ∨ q)] . We show that (N∗, O, N, G)-implications are fuzzy implication without any restricted conditions. In Section 4 we deal with the most important distributive laws involving (N∗, O, N, G)-implications with respect to greatest fuzzy negation N = Nbarwedge . In Section 5, we develop the idea of ⊚G,N-compositions on (N∗, O, N, G)-implications. If or are fuzzy implications constructed from the tuples or which satisfy the property P then we investigate whether ⊚G,N-composition denoted by satisfies the same property or not? In section 6, we establish sup-⊚O-composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Furthermore, we show that under which condition sup-⊚O-composition of (N∗, O, N, G)-implications are fuzzy implication. Section 7 is devoted to the intersections between families of fuzzy implications, including RO-implications, (G, N)-implications, QL-implications, D-implications and (N∗, O, N, G)-implications. In the final section, our researches is concluded.
Preliminary
In this section, we recall the basic concepts that are necessary to develop the paper, including the concepts related to fuzzy negations, grouping functions, overlap functions, fuzzy negations and fuzzy negations.
Definition 1. [18] A fuzzy negation is a function N : [0, 1] → [0, 1] satisfying the following conditions:
(i) The boundary conditions: N (0) =1 and N (1) =0 ;
(ii) N is decreasing: if e ≤ u then N (u) ≤ N (e) .
There are other properties that a fuzzy negation N may satisfy, as the following:
(iii) N is frontier: ∀e∈ [0, 1] : N (e) ∈ {0, 1} ⇔ e = 0 ∨ e = 1 ;
(iv) N is crisp: ∀e∈ [0, 1] : N (e) ∈ {0, 1} ;
(v) N is non-filling: ∀e∈ [0, 1] : N (e) =1 ⇔ e = 0 ;
(vi) N is strong: ∀e ∈ [0, 1] : N (N (e)) = e (involutive property) .
(vii) The greatest fuzzy negation N⊼ : [0, 1] → [0, 1] is defined by
(viii) The lowest fuzzy negation N⊻ : [0, 1] → [0, 1] is defined by
Definition 2. [18] A fuzzy implication is a bivariate function I : [0, 1] 2 → [0, 1] satisfying the following properties for all e, u, v ∈ [0, 1] :
(i) First place antitonicity: if e ≤ u then I (u, v)≤ I (e, v) ;
(ii) Second place isotonicity: if u ≤ v then I (e, u)≤ I (e, v) ;
(iii) Boundary condition 1, I (0, 0) =1 ;
(iv) Boundary condition 2, I (1, 1) =1 ;
(v) Boundary condition 3, I (1, 0) =0 .
There are other properties that a fuzzy implication I may satisfy, as the following:
(vi) The lowest falsity property:∀ e, u ∈ [0, 1] , I (e, u) =0 ⇔ e = 1 and u = 0 ;
(vii) The lowest truth property: ∀ e, u ∈ [0, 1] , I (e, u) =1 ⇔ e = 0 or u = 1 .
Definition 3. [10] An overlap function is a bivariate function O : [0, 1] 2 → [0, 1] satisfying the following properties for all e, u ∈ [0, 1] :
(i) O is commutative that is, O (e, u) = O (u, e) ;
(ii) O (e, u) =0 if and only if e = 0 or u = 0 ;
(iii) O (e, u) =1 if and only if e = u = 1 ;
(iv) O is increasing;
(v) O is continuous.
If 1 is the neutral element of an overlap function O, then it is denoted by neO = 1 .
Definition 4. [11] A grouping function is a bivariate function G : [0, 1] 2 → [0, 1] satisfying the following properties for all e, u ∈ [0, 1] :
(i) G is commutative: G (e, u) = G (u, e) ;
(ii) G (e, u) =0 if and only if e = u = 0 ;
(iii) G (e, u) =1 if and only if e = 1 or u = 1 ;
(iv) G is increasing;
(v) G is continuous.
If 0 is the neutral element of grouping function G, then it is denoted by neG = 0 .
(N∗, O, N, G)-implication constructed from non
associative structures
In classical logic, it is possible to combine the unary negation operator ¬ with any other binary operator in order to generate the other binary operators. In this section, we introduce the concept of (N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G and two different fuzzy negations N∗ and N are used for the generalization of the implication p → q ≡ ¬ [p ∧ ¬ (¬ p ∨ q)] . We establish that (N∗, O, N, G)-implications are fuzzy implication without any restricted conditions. Further, we disclose some properties of (N∗, O, N, G)-implication that are necessary for the development of this paper.
Definition 5. A function I : [0, 1] 2 → [0, 1] is called (N∗, O, N, G)-implication if there exists an overlap function O : [0, 1] 2 → [0, 1] , a grouping function G : [0, 1] 2 → [0, 1] and two fuzzy negations N∗, N : [0, 1] → [0, 1] , such that
(N∗, O, N, G)-implication is denoted by IN∗,O,N,G . Observe that, whenever N∗ and N are continuous fuzzy negations, then IN∗,O,N,G is also continuous.
Theorem 1.The IN∗,O,N,G is a fuzzy implication.
Proof. We prove that IN∗,O,N,G is a fuzzy implication. For this
Let e, u, v ∈ [0, 1] and e ≤ v . Then N (v) ≤ N (e) . This implies that G ((N (v) , u) ≤ G ((N (e) , u) . Also N (G ((N (v) , u)) ≥ N (G ((N (e) , u)) . This implies that O (N (G (N (v) , u)) , v) ≥ O (N (G (N (e) , u)) , e) . Thus N∗ (O (N (G (N (v) , u)) , v)) ≤ N∗ (O (N (G (N (e) , u)) , e)) . Therefore IN∗,O,N,G (v, u) ≤ IN∗,O,N,G (e, u) . Further let e, u, v ∈ [0, 1] and u ≤ v . Then G ((N (e) , u) ≤ G ((N (e) , v) . Also N (G (N (e) , u)) ≥ N (G (N (e) , v)) . This implies that O (N (G (N (e) , u)) , e) ≥ O (N (G (N (e) , v)) , e) . Thus N∗ (O (N (G (N (e) , u)) , e)) ≤ N∗ (O (N (G (N (e) , v)) , e)) . Therefore IN∗,O,N,G (e, u) ≤ IN∗,O,N,G (e, v) .□
Example 1. (i) Consider the overlap function a grouping function G =any G, a fuzzy negation N∗ = any negation N∗ and a fuzzy negation
Then
Let e = 0 . Then
Let e > 0 . Then
Thus
Hence
Let e, u, v ∈ [0, 1] and e ≤ v . Then This implies that Thus Therefore Further, let e, u, v ∈ [0, 1] and u ≤ v . Then This implies that Thus
Hence Therefore is a fuzzy implication.
(ii) Consider the overlap function O (e, u) = Tmin = min { e, u } , a grouping function G = Smax = max { e, u } , a fuzzy negation N∗ = any negation N∗ and a fuzzy negation
Then
Let e = 0 . Then
Let e > 0 . Then
Thus
Hence IN∗,Tmin,N⊻,Smax is a fuzzy implication.
(iii) Consider the overlap function a grouping function G (e, u) = SP (e, u) = e + u - eu, a fuzzy negation N∗ = any fuzzy negation N∗ and a fuzzy negation N (e) = 1 - e . Then it follows that
Further
Hence IN∗,ODB,N,SP is a fuzzy implication.
Theorem 2.Let IN∗,O,N,G be an (N∗, O, N, G)-implication and denote by neG and neO the neutral elements of an overlap function O and a grouping function G, respectively. Then the following hold:
(i) IN∗,O,N,G (0, u) = 1 for all u∈ [0, 1] ;
(ii) If neG = 0 and neO = 1, then for all u ∈ [0, 1] , IN∗,O,N,G (1, u) = u ⇔ (N∗ ∘ N) (u) = u for all u∈ [0, 1] ;
(iii) If neG = 0, O (e, u) = Omin (e, u) = min{ e, u } for all e, u ∈ [0, 1] and N is a strong negation, then NIN∗,O,N,G = N∗ ;
(iv) If neG = 0, O (e, u) = Omin (e, u) = min{ e, u } for all e, u ∈ [0, 1] and N = N∗ is strong negation, then NIN∗,O,N,G = N ;
(v) If neG = 0 and then
(vi) If IN∗,O,N,G (e, u) = 1 ⇔ e = 0 or u = 1, for all e, u ∈ [0, 1] , then N is non filling.
(vii) If N∗ is non-vanishing and N is frontier, then IN∗,O,N,G (e, u) = 0 ⇔ e = 0 and u = 1, for all e, u∈ [0, 1] ;
(viii) If N∗ is non filling and N is frontier, then IN∗,O,N,G (e, u) = 1 ⇔ e = 0 or u = 1, for all e, u∈ [0, 1] ;
(ie) e ≤ u ⇒ IN∗,O,N,G (e, u) = 1 for all e, u∈ [0, 1] ⇔ N = N⊼ ;
(e) For a fuzzy negation N⊼, IN∗,O,N⊼,G (u, N⊼ (e)) = IN∗,O,N⊼,G (e, N⊼ (u)) , for all e, u ∈ [0, 1] .
Proof. (i) For all u ∈ [0, 1], it follows that
(ii) Let (N∗ ∘ N) (u) = u for all u ∈ [0, 1] . Then
Therefore IN∗,O,N,G (1, u) = u .
Conversely, let u = IN∗,O,N,G (1, u) . We prove that (N∗ ∘ N) (u) = u for all u ∈ [0, 1] . It follows that
Thus (N∗ ∘ N) (u) = u for all u ∈ [0, 1] .
(iii) For neG = 0, O (e, u) = Omin (e, u) = min{ e, u } for all e, u ∈ [0, 1] and N is strong negation, it follows that
(iv) The proof process is straightforward.
(v) Consider
(vi) Let IN∗,O,N,G (e, u) = 1 ⇔ e = 0 or u = 1, for all u ∈ [0, 1] . Then we prove that N is non-filling. Let N be not non filling. Then there exists e ∈ (0, 1) such that N (e) = 1 . Now
But this is a contradiction, because IN∗,O,N,G (e, u) = 1 ⇔ e = 0 or u = 1, for all u ∈ [0, 1] . Hence the only possible value of e = 0 . This contradiction is due to our wrong supposition that is N is not non filling. Therefore we conclude that N is non filling.
(vii) Let IN∗,O,N,G (e, u) = 0 . Then N∗ (O (N (G (N (e) , u)) , e)) = 0 . Since N∗ is non-vanishing, it follows that O (N (G (N (e) , u)) , e) = 1 . This implies that N (G (N (e) , u)) = 1 and e = 1 . Now given that N is frontier, then G (N (e) , u) = 0 . It follows that N (e) = 0 and u = 0 . Thus e = 1 and u = 0 .
Conversely, let e = 1 and u = 0 . Then
(viii) Let IN∗,O,N,G (e, u) = 1 . Then N∗ (O (N (G (N (e) , u)) , e)) = 1 . It follows that O (N (G (N (e) , u)) , e) = 0 . Further N (G (N (e) , u)) = 0 or e = 0 . Now given that N is frontier. Then G (N (e) , u) = 1 . This implies that N (e) = 1 or u = 1 . Thus e = 0 or u = 1 .
Conversely, let e = 0 or u = 1 . Then we discuss the following cases:
(a) If e = 1 and u = 1, then IN∗,O,N,G (1, 1) = 1 .
(b) If e = 0 and u = 0, then IN∗,O,N,G (0, 0) = 1 .
(c) If e = 0 and u = 1, then IN∗,O,N,G (0, 1) = 1 .
(d) If e = 0 and u ∈ (0, 1) , then
(e) If e ∈ (0, 1) and u = 1, then
(ie) Let N ≠ N⊼ . Then there exists e ∈ (0, 1) such that N (e) < 1 . Consider In this case, we have IN∗,O,N,G (e, u) ≠ 1 . Thus e ≤ u ⇒ IN∗,O,N,G (e, u) ≠ 1 for all e, u ∈ [0, 1]. Thus by contraposition, if e ≤ u ⇒ IN∗,O,N,G (e, u) = 1 for all e, u ∈ [0, 1] , then N = N⊼ .
Conversely, let N = N⊼ and e ≤ u . Consider e = u = 1 . Then IN∗,O,N,G (e, u) = 1 . If e < 1, then
(e) Now
From (1) and (2) we have IN∗,O,N⊼,G (e, N⊼ (u)) = IN∗,O,N⊼,G (u, N⊼ (e)) □
Remark 1. Let IN∗,O,N,G be an (N∗, O, N, G)-implication. Then the following do not hold in general:
(i) IN∗,O,N,G (N° (e) , u) = IN∗,O,N,G (N° (u) , e) for any negation N° ;
(ii) IN∗,O,N,G (e, u) = 1nLeftrightarrowe ≤ u, for all e, u ∈ [0, 1] .
Example 2. Consider the overlap function O (e, u) = O2 (e, u) = e2u2, a grouping function G (e, u) = G2 (e, u) = 1 - (1 - e) 2 (1 - u) 2, a fuzzy negation N∗ = any negation N∗ and a fuzzy negation Nz (e) = 1 - e . Then
Hence
Let e, u, v ∈ [0, 1] and e ≤ v . Then e6 ≤ v6 . This implies that e6 (1 - u) 4 ≤ v6 (1 - u) 4 . Thus N∗ (e6 (1 - u) 4) ≥ N∗ (v6 (1 - u) 4) . Therefore IN∗,O2,Nz,G2 (e, u) ≥ IN∗,O2,Nz,G2 (v, u) . Further, let e, u, v ∈ [0, 1] and u ≤ v . Then (1 - u) 4 ≥ (1 - v) 4 . This implies that e6 (1 - u) 4 ≥ e6 (1 - v) 4 . Thus N∗ (e6 (1 - u) 4) ≤ N∗ (e6 (1 - v) 4) . Hence IN∗,O2,Nz,G2 (e, u) ≤ IN∗,O2,Nz,G2 (e, v) . Therefore IN∗,O2,Nz,G2 is a fuzzy implication. If N∗ (e) = Nz (e) = 1 - e, then INz,O2,Nz,G2 (e, u) = 1 - e6 (1 - u) 4 .
(i) If e = 0.4, u = 0.8, then INz,O2,Nz,G2 (Nz (0.4) , 0.8) = 0.99992 and
(ii) If e = 0.4, u = 0.5, then INz,O2,Nz,G2 (0.4, 0.5) = 1 - (0.4) 6 (1 - 0.5) 4 = 0.99974 . Thus INz,O2,Nz,G2 (0.4, 0.5) ≠ 1 .
Distributive Laws and (N∗, O, N, G)-implications
The study of distributive laws involving fuzzy implications has attracted much attention not only due to their theoretical aesthetics but also due to their applicational value. In this section, we present the important distributive laws involving (N∗, O, N, G)-implications with respect to greatest fuzzy negation N = N⊼ .
where O, O(1), O(2), O(3) : [0, 1] 2 → [0, 1] are four overlap functions and G, G(1), G(2), G(3) : [0, 1] 2 → [0, 1] are four grouping functions. Further in this section, we discuss the exchange principle IN∗,O,N,G (e, IN∗,O,N,G (u, v)) = IN∗,O,N,G (u, IN∗,O,N,G (e, v)) under certain conditions.
Theorem 3.LetIN∗,O,N,G be an (N∗, O, N, G)-implication. If neG = 0, neO = 1 and N = N⊼, then the following hold:
(i) IN∗,O,N,G (e, IN∗,O,N,G (u, v)) = IN∗,O,N,G (u, IN∗,O,N,G (e, v)) , for all e, u, v∈ [0, 1] ;
(ii) IN∗,O,N,G (e, IN∗,O,N,G (u, v)) = IN∗,O,N,G (IN∗,O,N,G (e, u) , IN∗,O,N,G (e, v)) , for all e, u, v ∈ [0, 1] .
Proof. (i) Let e, u, v ∈ [0, 1] . We can divide the discussion into the following possible cases:
Case 1 : If e = u = v = 1, then IN∗,O,N,G (1, IN∗,O,N,G (1, 1)) = 1 and
IN∗,O,N,G (1, IN∗,O,N,G (1, 1)) = 1 .
Case 2 : If e = u = v = 0, then IN∗,O,N,G (0, IN∗,O,N,G (0, 0)) = 1 and
IN∗,O,N,G (0, IN∗,O,N,G (0, 0)) = 1 .
Case 3 : If e = u = 1, v = 0, then IN∗,O,N,G (1, IN∗,O,N,G (1, 0)) = 0 and IN∗,O,N,G (1, IN∗,O,N,G (1, 0)) = 0 .
Case 4 : If e = u = 0, v = 1, then IN∗,O,N,G (0, IN∗,O,N,G (0, 1)) = 1 and IN∗,O,N,G (0, IN∗,O,N,G (0, 1)) = 1 .
Case 5 : If e = 0, u = v = 1, then IN∗,O,N,G (0, IN∗,O,N,G (1, 1)) = 1 and IN∗,O,N,G (1, IN∗,O,N,G (0, 1)) = 1 .
Case 6 : If e = v = 1, u = 0, then IN∗,O,N,G (1, IN∗,O,N,G (0, 1)) = 1 and IN∗,O,N,G (0, IN∗,O,N,G (1, 1)) = 1 .
Case 7 : If e = 1, u = v = 0, then IN∗,O,N,G (1, IN∗,O,N,G (0, 0)) = 1 and IN∗,O,N,G (0, IN∗,O,N,G (1, 0)) = 1 .
Case 8 : If e = v = 0, u = 1, then IN∗,O,N,G (0, IN∗,O,N,G (1, 0)) = 1 and IN∗,O,N,G (1, IN∗,O,N,G (0, 0)) = 1 .
Case 9 : If e = u = 1, v ∈ (0, 1) , then
Also
Case 10 : If e = u = 0, v ∈ (0, 1) , then
Also
Case 11 : If e = 1, u = 0, v ∈ (0, 1) , then
Also
Case 12 : If e = 0, u = 1, v ∈ (0, 1) , then
Also
Case 13 : If e = 0, u, v ∈ (0, 1) , then
Further
Case 14 : If e = 1, u, v ∈ (0, 1) , then
Also
Case 15 : If u = 0, e, v ∈ (0, 1) , then IN∗,O,N,G (e, IN∗,O,N,G (0, v)) = 1 and IN∗,O,N,G (0, IN∗,O,N,G (e, v)) = 1 .
Case 16 : If u = 1, e, v ∈ (0, 1) , then IN∗,O,N,G (e, IN∗,O,N,G (1, v)) = 1 and IN∗,O,N,G (1, IN∗,O,N,G (e, v)) = 1 .
Case 17 : If e, u, v ∈ (0, 1) , then
Also
Case 18 : If e, u ∈ (0, 1) , v = 0, then IN∗,O,N,G (e, IN∗,O,N,G (u, 0)) = 1 and IN∗,O,N,G (u, IN∗,O,N,G (e, 0)) = 1 .
Case 19 : If e, u ∈ (0, 1) , v = 1, then IN∗,O,N,G (e, IN∗,O,N,G (u, 1)) = 1 and IN∗,O,N,G (u, IN∗,O,N,G (e, 1)) = 1 .
Case 20 : If e = 0, u ∈ (0, 1) , v = 0, then IN∗,O,N,G (0, IN∗,O,N,G (u, 0)) = 1 and IN∗,O,N,G (u, IN∗,O,N,G (0, 0)) = 1 .
Case 21 : If e = 0, u ∈ (0, 1) , v = 1, then IN∗,O,N,G (0, IN∗,O,N,G (u, 1)) = 1 and IN∗,O,N,G (u, IN∗,O,N,G (0, 1)) = 1 .
Case 22 : If e = 1, u ∈ (0, 1) , v = 1, then IN∗,O,N,G (1, IN∗,O,N,G (u, 1)) = 1 and IN∗,O,N,G (u, IN∗,O,N,G (1, 1)) = 1 .
Case 23 : If e = 1, u ∈ (0, 1) , v = 0, then
Similarly IN∗,O,N,G (u, IN∗,O,N,G (0, 0)) = 1 .
Case 24 : If e ∈ (0, 1) , u = v = 0, then IN∗,O,N,G (e, IN∗,O,N,G (0, 0)) = 1 and IN∗,O,N,G (0, IN∗,O,N,G (e, 0)) = 1 .
Case 25 : If e ∈ (0, 1) , u = 0, v = 1, then IN∗,O,N,G (e, IN∗,O,N,G (0, 1)) = 1 and IN∗,O,N,G (0, IN∗,O,N,G (e, 1)) = 1 .
Case 26 : If e ∈ (0, 1) , u = 1, v = 0, then IN∗,O,N,G (e, IN∗,O,N,G (1, 0)) = 1 and IN∗,O,N,G (1, IN∗,O,N,G (e, 0)) = 1 .
Case 27 : If e ∈ (0, 1) , u = 1, v = 1, then IN∗,O,N,G (e, IN∗,O,N,G (1, 1)) = 1 and IN∗,O,N,G (1, IN∗,O,N,G (e, 1)) = 1 .
(ii) The proof of (ii) is similar to the proof of (i) .
□
Theorem 4.Let IN∗,O,N,G be an (N∗, O, N, G)-implication, G (e, 1) ≥ e and O (e, u) = min { e, u } . Then the following hold:
(i) If N (N (e)) ≤ e, then IN∗,O,N,G (e, 1) ≥ N∗ (e) , for all e∈ [0, 1] ;
(ii) If N is strong fuzzy negation, then IN∗,O,N,G (e, 1) ≥ N∗ (e) , for all e ∈ [0, 1] .
Proof. (i) By hypothesis, G (N (e) , 1) ≥ N (e) . By applying N on both side, N (G (N (e) , 1)) ≤ N (N (e)) . This implies that O (N (G (N (e) , 1)) , e) ≤ O (N (N (e)) , e) . It follows that N∗ (O (N (G (N (e) , 1)) , e)) ≥ N∗ (O (N (N (e)) , e)) . On the other hand, if N (N (e)) ≤ e, then O (N (N (e)) , e) ≤ O (e, e) . This implies that N∗ (O (N (N (e)) , e)) ≥ N∗ (O (e, e)) . It follows that N∗ (O (N (G (N (e) , 1)) , e)) ≥ N∗ (O (e, e)) . Therefore IN∗,O,N,G (e, 1) ≥ N∗ (e) .
(ii) Straightforward. □
Theorem 5.Let IN∗,O,N,G be an (N∗, O, N, G)-implication, where N is frontier and neO = 1 . Then the following hold:
(i) If G (0, u) ≤ u, then IN∗,O,N,G (1, u) ≤ N∗ ∘ N (u) , for all u∈ [0, 1] ;
(ii) If G (0, u) ≥ u, then IN∗,O,N,G (1, u) ≥ N∗ ∘ N (u) , for all u ∈ [0, 1] .
Proof. (i) By hypothesis, G (0, u) ≤ u, then G (N (1) , u) ≤ u . By applying N on both side, N (G (N (1) , u)) ≥ N (u) . This implies that O (N (G (N (1) , u)) , 1) ≥ O (N (u) , 1) . Thus N∗ (O (N (G (N (1) , u)) , 1)) ≤ N∗ (O (N (u) , 1)) . Therefore IN∗,O,N,G (1, u) ≤ N∗ ∘ N (u) .
(ii) Straightforward. □
Theorem 6.Let IN∗,O,N,G be an (N∗, O, N, G)-implication. If neG = 0, neO = 1 and N∗ ∘ N = Id[0,1], then the following hold:
(i) IN∗,O,N,G (e, u) ≥ u, for all e, u∈ [0, 1] ;
(ii) IN∗,O,N,G (e, IN∗,O,N,G (e, u)) ≥ IN∗,O,N,G (e, u) , for all e, u ∈ [0, 1] .
Proof. (i) Since neG = 0, neO = 1 and N∗ ∘ N = Id[0,1], so IN∗,O,N,G (1, u) = u, for all u ∈ [0, 1] . Thus IN∗,O,N,G (e, u) ≥ IN∗,O,N,G (1, u) = u for all e, u ∈ [0, 1] .
(ii) Since neG = 0, neO = 1 and N∗ ∘ N = Id[0,1], so IN∗,O,N,G (e, u) ≥ u, for all e, u ∈ [0, 1] . Hence IN∗,O,N,G (e, IN∗,O,N,G (e, u)) ≥ IN∗,O,N,G (e, u) , for all e, u ∈ [0, 1] . □
Theorem 7.Let IN∗,O,N⊼,G be an (N∗, O, N⊼, G)-implication. If neG = 0, neO = 1 and IN∗,O,N⊼,G (e, IN∗,O,N⊼,G (u, v)) = IN∗,O,N⊼,G (u, IN∗,O,N⊼,G (e, v)) , for all e, u, v ∈ [0, 1] , then N∗ ∘ N⊼ ∘ N∗ = N∗ .
Proof. For all e ∈ [0, 1] , it follows that
Also
Therefore N∗ ∘ N⊼ ∘ N∗ = N∗ .□
Remark 2. Consider the overlap function O (e, u) = O2 (e, u) = e2u2, a grouping function G (e, u) = G2 (e, u) =1 - (1 - e) 2 (1 - u) 2 and fuzzy negations Nz (e) = Nz (e) =1 - e . Then INz,O2,Nz,G2 (e, u) =1 - e6 (1 - u) 4 . For e = 0.4, u = 0.8 and v = 0.5, it follows that INz,O2,Nz,G2 (e, INz,O2,Nz,G2 (u, v)) =0.9999 and INz,O2,Nz,G2 (u, INz,O2,Nz,G2 (e, v)) =1 . This implies that INz,O2,Nz,G2 (e, INz,O2,Nz,G2 (u, v)) ≠ INz,O2,Nz,G2 (u, INz,O2,Nz,G2 (e, v)) . But N ∘ N ∘ N (e) = N (e). Thus, Theorem 7 gives us sufficient, but necessary condition, for the satisfaction of IN∗,O,N,G (e, IN∗,O,N,G (u, v)) = IN∗,O,N,G (u, IN∗,O,N,G (e, v)) , when we have IN∗,O,N,G are an (N∗, O, N, G)-implication.
Theorem 8.LetO, O(1), O(2), O(3) : [0, 1] 2 → [0, 1] be four overlap functions, G, G(1), G(2), G(3) : [0, 1] 2 → [0, 1] be four grouping functions and IN∗,O,N⊼,G be an (N∗, O, N⊼, G)-implication. Then the following hold:
(i) If neG(1) = 0, neO(1) = 1 and O(1) = Omin, then IN∗,O,N⊼,G (O(1) (e, u) , v) = G(1) (IN∗,O,N⊼,G (e, v) , IN∗,O,N⊼,G (u, v)) , for all e, u, v∈ [0, 1] ;
(ii) If neG(1) = 0, neO(1) = 1 and G(1) = Gmax, then IN∗,O,N⊼,G (G(1) (e, u) , v) = O(1) (IN∗,O,N⊼,G (e, v) , IN∗,O,N⊼,G (u, v)) , for all e, u, v∈ [0, 1] ;
(iii) If neO(2) = neO(3) = 1 and O(2) = Omin, then IN∗,O,N⊼,G (e, O(2) (u, v)) = O(3) (IN∗,O,N⊼,G (e, u) , IN∗,O,N⊼,G (u, v)) , for all e, u, v∈ [0, 1] ;
(iv) If neG(2) = neG(3) = 0 and G(2) = Gmax, then IN∗,O,N⊼,G (e, G(2) (u, v)) = G(3) (IN∗,O,N⊼,G (e, u) , IN∗,O,N⊼,G (u, v)) , for all e, u, v ∈ [0, 1] .
Proof. The proofs of (i) , (ii) , (iii) and (iv) are similar to the proof of Theorem (3 (i)). □
Theorem 9.Let O, O(1) : [0, 1] 2 → [0, 1] be two overlap functions with neO = neO(1) = 1, G : [0, 1] 2 → [0, 1] be a grouping function with neG = 0 and IN∗,O(1),N⊼,G be an (N∗, O(1), N⊼, G)-implication. If for all e, u ∈ [0, 1] and e ≤ u, then O (e, IN∗,O(1),N⊼,G (e, u)) ≤ u .
Proof. If N = N⊼, then we discuss the following two cases:
(a) If e = 1, then
(b)If e < 1, then
Therefore O (e, IN∗,O(1),N⊼,G (e, u)) ≤ u . □
Automorphisms and (N∗, O, N, G)-implications
A function β : [0, 1] → [0, 1] is said to be an automorphism if β is bijective and increasing. Another equivalent definition considers β : [0, 1] → [0, 1] is an automorphism if it is a continuous and strictly increasing function such that β (0) = 0 and β (1) = 1 . Automorphisms are closed under composition of function and the inverse of an automorphism is also automorphism.
Given a function f : [0, 1] n → [0, 1] and an automorphism β . The action of β on f is the function f
β : [0, 1] n → [0, 1] defined by
where f
β is the conjugate of f . Define the function f(β) : [0, 1] n → [0, 1] , given by f(β) = β ∘ f
β, that is
Theorem 10.Let β : [0, 1] → [0, 1] be an automorphism and IN∗,O,N,G an (N∗, O, N, G)-implication. Then is an (N∗, O, N, G)-implication constructed from the tuple (N∗, O, N, G) .
Proof. Consider
Hence is also (N∗, O, N, G)-implication. □
Theorem 11.Let β : [0, 1] → [0, 1] be an automorphism and IN∗,O,N,G an (N∗, O, N, G)-implication. Then is an (N∗, O, N, G)-implication constructed from the tuple (N∗, O, N, G) .
Proof. The proof process is similar to the proof of Theorem 10. □
Theorem 12.Let β : [0, 1] → [0, 1] be an automorphism and IN∗,O,N,G an (N∗, O, N, G)-implication. Then is an (N∗, O, N, G)-implication constructed from the tuple (N∗, O, N, G) .
Proof. The proof process is similar to the proof of Theorem 10. □
⊚G,N-compositions on (N∗, O, N, G)-implications
The main aim of this section is to introduce the concept of ⊚G,N-compositions on (N∗, O, N, G)-implications. Given that or are two fuzzy implications constructed from the tuples or which satisfy certain property P . We now investigate whether ⊚G,N-composition denoted by satisfies the same property or not? If not, then we attempt to characterize those implications satisfy the property P such that also satisfies the same property.
Definition 6. Let and be two fuzzy implications constructed for the tuples and We define as
Theorem 13.The is a fuzzy implication.
Proof. We prove that is a fuzzy implication. For this consider
Further let e, e1, u, u1 ∈ [0, 1] and e ≤ e1 . Then and Also and Furthermore This implies that
Thus
Similarly we can prove that if u ≤ u1, then Therefore is a fuzzy implication. □
Theorem 14.Let and be two fuzzy implications constructed for the tuples and and neG = 0 . Then for all u ∈ [0, 1] if and only if
Proof. Let for all u ∈ [0, 1] . Then it follows
Conversely, let . Then
□
Theorem 15.Let and be two fuzzy implications constructed from the tuples and where O(1) = Omin, neG = neG(1) = 0 and N1is strong negation. Then
Proof. Given that and are two fuzzy implications constructed from the tuples and where O(1) = Omin, neG = neG(1) = 0 and N1 is strong negation. Then it follows that
□
Theorem 16.Let and be two fuzzy implications constructed from the tuples and where O(1) = Omin, neG = neG(1) = 0 and N1is strong negation. Then
Proof. The proof is similar to the proof of Theorem 15. □
Theorem 17.Let I(N⊼)∗,O(1),N1,G(1) and be two fuzzy implications constructed from the tuples and where O(1) = Omin, neG = neG(1) = 0 and N1is strong negation. Then
Proof. The proof is similar to the proof of Theorem 15. □
Theorem 18.Let and be two fuzzy implications constructed from the tuples and Let
if and only if e = 0 or u = 1 . Then N1is non filling.
Proof. Let if and only if e = 0 or u = 1, for all u ∈ [0, 1] . Then we prove that N1 is non filling. Let N1 be not non filling. Then there exists e ∈ (0, 1) such that N1 (e) = 1 . It follows that
But this is a contradiction, because or u = 1, for all u ∈ [0, 1] . Hence the only possible value of e = 0 . This contradiction is due to our wrong supposition that is N1 is not non filling. Therefore we conclude that N1 is non filling. □
Theorem 19.Let and be two fuzzy implications constructed from the tuples and with neG(2) = 0 and neO(2) = 1 . Then for all e, u ∈ [0, 1] if and only if N1 = N2 = N⊼ .
Proof. Let N1 ≠ N2 ≠ N⊼ . Then there exist e, u ∈ (0, 1) such that N1 (e) < 1 and N2 (u) < 1 . Consider In this case, we have and This implies that Thus for all e, u ∈ [0, 1]. Thus by contraposition, if for all e, u ∈ [0, 1] , then N1 = N2 = N⊼ .
Conversely, let N1 = N2 = N⊼ and e ≤ u . Then we can divide the discussion into the following possible cases:
Case 1 : If e = u = 1, then it follows that
Case 2 : If e < u = 1, then it follows that
Case 3 : If e < u < 1, then it follows that
□
Theorem 20.Let and be two fuzzy implications constructed from the tuples and Then for a fuzzy negation N⊼,
for all e, u ∈ [0, 1] .
Proof. Consider
If e < 1, then
If u < 1, then
If e = 1 and u = 1, then
Thus
Similarly
From (1) and (2) we have
□
Theorem 21.Let β : [0, 1] → [0, 1] be an automorphism and let and be two fuzzy implications constructed from the tuples and Then
is a fuzzy implication constructed from the tuples and
Proof. Let β : [0, 1] → [0, 1] be an automorphism and let and be two fuzzy implications constructed from the tuples and For e, u ∈ [0, 1] , it follows that
□
Theorem 22.Let β : [0, 1] → [0, 1] be an automorphism and let be two fuzzy implications constructed from the tuples and Then
is a fuzzy implication constructed from the tuples and
Proof.Let β : [0, 1] → [0, 1] be an automorphism and let be two fuzzy implications constructed from the tuples and For e, u ∈ [0, 1] , it follows that
□
sup-⊚O-Composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G)
Sup- min composition of fuzzy relations was the first to be introduced by Zadeh [37], the min can be replaced by any overlap t-norm T . We introduced sup-⊚O-Composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Furthermore, derive that under which condition sup-⊚O-Composition of (N∗, O, N, G)-implications are fuzzy implication.
Definition 7. Let and be two fuzzy implications constructed from the tuples and The sup-⊚O-composition of and is given by:
where O is an overlap function with neO = 1 and e, u ∈ [0, 1] .
Theorem 23.Let and be two fuzzy implications constructed from the tuples and
Then satisfy the following properties, for all e, u, v ∈ [0, 1]:
(i)
(ii)
(iii) If e ≤ e1, then
(iv) If e ≤ e1, then
Proof. (i) Now
(ii)
(iii) Let e, e1, v ∈ [0, 1] and e ≤ e1 . Then by definition of and the monotonicity of O, we have for all t ∈ [0, 1] . This implies that Now
(iv) Let e, e1, v ∈ [0, 1] and e ≤ e1 . Then by definition of and the monotonicity of O, we have for all t ∈ [0, 1] . This implies that Now
□
Theorem 24.Let and be two fuzzy implications constructed from the tuples and
Then is a fuzzy implication if and only if
Proof. Let be a fuzzy implication. Then clearly from the definition of fuzzy implication
Conversely, let Based on Theorem 23, and Further, if e ≤ e1, then
hold. Similarly, if e ≤ e1, then
hold. Thus is a fuzzy implication. □
Theorem 25.Let and be three fuzzy implications constructed from the tuples and If O is an associative overlap function, then
Proof. Let be three fuzzy implications constructed for the tuples and and let O be an associative overlap function. For any e, u, v ∈ [0, 1] , it follows that
□
Theorem 26.Let β : [0, 1] → [0, 1] be an automorphism and letandbe two fuzzy implications constructed from the tuplesandThen
Proof. Let be two fuzzy implications constructed from the tuples and Since β is a bijection on [0, 1] , for each u ∈ [0, 1] such that β (t) = u . It follows that
□
Intersections between families of fuzzy implications
This section establishes the intersections between families of fuzzy implications, including RO-implications (residual implication), (G, N)-implications, (O, G, N)-implications, D-implications and (N∗, O, N, G)-implications.
At first, the intersection between interval RO- and (N∗, O, N, G)-implications is discussed.
Theorem 27.The RO-implication IO : [0, 1] 2 → [0, 1] is not an (N∗, O, N, G)-implication for any grouping function G : [0, 1] 2 → [0, 1] , and any overlap function O : [0, 1] 2 → [0, 1] and any fuzzy negations N∗, N : [0, 1] → [0, 1] .
Proof. Suppose that IO = I(N∗,O,N,G) for some grouping functions G, some overlap functions O and the fuzzy negations N∗, N . Then we have
which contradicts IO = I(N∗,O,N,G) . Hence IO is not an I(N∗,O,N,G) implication constructed from the tuple (N∗, O, N, G) . □
Next, we study the intersection between (G, N)- and (N∗, O, N, G)-implications.
Theorem 28.The (G, N)-implication I(G,N) : [0, 1] 2 → [0, 1] is not an (N∗, O, N, G)-implication for any grouping function G : [0, 1] 2 → [0, 1] , and any overlap function O : [0, 1] 2 → [0, 1] and any fuzzy negations N∗, N : [0, 1] → [0, 1] .
Proof. Suppose that I(G,N) = I(N∗,O,N,G) for some grouping functions G, some overlap functions O and the fuzzy negations N∗, N . Then we have
which contradicts I(G,N) = I(N∗,O,N,G) . Hence I(G,N) is not an I(N∗,O,N,G) implication constructed from the tuple (N∗, O, N, G) . □
Further, we study the intersection between (O, G, N)- and (N∗, O, N, G)-implications.
Theorem 29.The (O, G, N)-implication I(O,G,N) : [0, 1] 2 → [0, 1] is not an (N∗, O, N, G)-implication for any grouping function G : [0, 1] 2 → [0, 1] with identity 0, any overlap function O : [0, 1] 2 → [0, 1] and any fuzzy negations N∗, N : [0, 1] → [0, 1] .
Proof. Suppose that I(O,G,N) = I(N∗,O,N,G) for some grouping functions G, some overlap functions O and the fuzzy negations N∗, N . Then we have
which contradicts I(O,G,N) = I(N∗,O,N,G) . Hence I(O,G,N) is not an I(N∗,O,N,G) implication constructed from the tuple (N∗, O, N, G) . □
The following example shows that the construction of (N∗, O, N, G)-implications from grouping function G, overlap function O and a fuzzy negations N∗, N . But may or may not construct a QL-implications from the same grouping function G, overlap function O and a fuzzy negation N .
Example 3. Consider the overlap function O (e, u) = Tmin = min { e, u } , a grouping function G (e, u) = Smax = max{ e, u } and a fuzzy negations Nz (e) = Nz (e) = 1 - e . Then
Hence
Let e, u, v ∈ [0, 1] and e ≤ u . Then 1 - e ≥ 1 - u . This implies that {1 - e, v } ⊇ { 1 - u, v } . Which implies that 1 - max { 1 - e, v } ≤ 1 - max { 1 - u, v } . This implies that {1 - max { 1 - e, v } , e } ⊆ { 1 - max { 1 - u, v } , u } . Thus 1 - { min { 1 - max { 1 - e, v } , e }} ≥ 1 - { min { 1 - max { 1 - u, v } , u }} . Therefore INz,Tmin,Nz,Smax (e, v) ≥ INz,Tmin,Nz,Smax (u, v) . Further, let e, u, v ∈ [0, 1] and u ≤ v . Then {1 - e, u } ⊆ { 1 - e, v } . This implies that max { 1 - e, u } ≤ max { 1 - e, v } . Which implies that max { 1 - e, u } ≤ max { 1 - e, v } . Also {1 - max { 1 - e, u } , e } ⊇ { 1 - max { 1 - e, v } , e } . Thus 1 - { min { 1 - { max { 1 - e, u }} , e }} ≤ 1 - { min { 1 - { max { 1 - e, v }} , e }} . Hence
INz,Tmin,Nz,Smax (e, u) ≤ INz,Tmin,Nz,Smax (e, v) . Therefore INz,Tmin,Nz,Smax is a fuzzy implication. Now consider a QL-operation ITmin,Nz,Smax (e, u) = max { 1 - e, min { e, u }} . We show that ITmin,Nz,Smax is not a QL-implication. For this
Let e, u, v ∈ [0, 1] and e ≤ u . Then min { e, v } ≤ min { u, v } . But 1 - e ≥ 1 - u . This shows that there are no relation between max { 1 - e, min { e, v }} = ITmin,Nz,Smax (e, v) and max { 1 - u, min { u, v }} = ITmin,Nz,Smax (u, v) . Thus
ITmin,Nz,Smax (e, v) ngeqITmin,Nz,Smax (u, v) . Therefore ITmin,Nz,Smax is not a QL-implication.
We study the intersection between D-implications and (N∗, O, N, G)-implications.
Theorem 30.The D-implication constructed from the tuple (O, G, N) , is not an (N∗, O, N, G)-implication for any grouping function G : [0, 1] 2 → [0, 1] with identity 0, any overlap function O : [0, 1] 2 → [0, 1] and any fuzzy negations N∗, N : [0, 1] → [0, 1] .
Proof. Suppose that for some grouping functions G, some overlap functions O and the fuzzy negations N∗, N . Then we have
which contradicts Hence is not an I(N∗,O,N,G) implication constructed from the tuple (N∗, O, N, G) . □
Further, the upcoming example shows that the construction of (N∗, O, N, G)-implications from grouping function G, overlap function O and a fuzzy negations N∗, N . But may or may not construct a D-implications from the same grouping function G, overlap function O and a fuzzy negation N .
Example 4. Consider the overlap function a grouping function G (e, u) = Sp (e, u) = e + u - eu and a fuzzy negations Nz (e) = Nz (e) = 1 - e . Then
Hence
Let e, u, v ∈ [0, 1] and e ≤ u . Then This implies that Which implies that Thus Therefore Further, let e, u, v ∈ [0, 1] and u ≤ v . Then 1 - u ≥ 1 - v . This implies that Which implies that Thus Hence Therefore is a fuzzy implication. Now consider a D-operation
We show that is not a D-implication. For this
Let e, u, v ∈ [0, 1] and e ≤ u . Then This implies that
. Which implies that
But e ≤ u . This shows that their are no relation between and Thus Therefore is not a D-implication.
Conclusion
In this paper, we have introduced the concept of (N∗, O, N, G)-implication derived from non associative structures, overlap function O, grouping function G while two different fuzzy negations N∗ and N are used for the generalization of the implication p → q ≡ ¬ [p ∧ ¬ (¬ p ∨ q)] . Further, we have showed that (N∗, O, N, G)-implication are fuzzy implication without any restricted conditions. Furthermore we presented the important distributive laws involving (N∗, O, N, G)-implications with respect to greatest fuzzy negation N = N⊼ . Subsequently, we presented the concept of ⊚G,N-compositions on (N∗, O, N, G)-implications. If -implication or -implication constructed from the tuples or satisfy a certain property P, then we investigated whether ⊚G,N-composition of -implication and -implication satisfies the same property or not. Further we have introduced sup-⊚O-composition of (N∗, O, N, G)-implications constructed from tuples (N∗, O, N, G) . Furthermore, we have demonstrated that under which condition sup-⊚O-composition of (N∗, O, N, G)-implications are fuzzy implication. Finally, we discussed the intersections between families of fuzzy implications, including RO-implications, (G, N)-implications, (O, G, N)-implications, D-implications and (N∗, O, N, G)-implications.
References
1.
BaczynskiM. and JayaramB., On the characterization of(S, N)-implications, Fuzzy Sets Syst.158 (2007), 1713–1727.
2.
BaczynskiM. and JayaramB., Fuzzy Implications, Berlin, Springer, 2008.
3.
BaczynskiM. and JayaramB., (S, N)- and R-implications: a state-of-the-art survey, Fuzzy Sets Syst.159 (2008), 1836–1859.
4.
BaczynskiM. and JayaramB., QL-implications: some properties and intersections, Fuzzy Sets Syst.161 (2010), 158–188.
5.
BaczynskiM., BeliakovG., SolaH. Bustince and PraderaA. (Eds.), Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, vol. 300, Springer, Berlin, 2013.
6.
BaldwinJ.F., A new approach to approximate reasoning using a fuzzylogic, Fuzzy Sets Syst.2 (1979), 309–325.
7.
BaldwinJ.F. and PilsworthB.W., Axiomatic approach to implicationfor approximate reasoning with fuzzy logic, Fuzzy Sets Syst.3 (1980), 193–219.
8.
BarrenecheaE., FernandezJ., PagolaM., ChiclanaF. and BustinceH., Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Applicationto decision making, Knowl.-Based Syst.58 (2014), 33–44.
9.
BustinceH., FernandezJ., MesiarR., MonteroJ. and OrdunaOverlapR., index, overlap functions and migrativity, in: Proceedings of IFSA/EUSFLAT Conference (2009), 300–305.
10.
BustinceH., FernandezJ., MesiarR., MonteroJ. and OrdunaR., Overlap functions, Nonlinear Anal. Theory Methods Appl.72(3–4) (2010), 1488–1499.
11.
BustinceH., PagolaM., MesiarR., HüllermeierE. and HerreraF., Grouping, overlaps, and generalized bientropic functions for fuzzy modeling of pairwise comparisons, IEEE Trans. Fuzzy Syst.20(3) (2012), 405–415.
12.
BustinceH., BarrenecheaE., CalvoT., JamesS. and BeliakovConsensusG., in multi-expert decision making problems using penaltyfunctions defined over a cartesian product of lattices, Inf. Fusion17 (2014), 56–64.
13.
DeerL., VerbiestN., CornelisC. and GodoL., A comprehensive studyof implicator–conjunctor-based and noise-tolerant fuzzy rough sets:definitions, properties and robustness analysis, Fuzzy SetsSyst.275 (2015), 1–38.
14.
DengT.Q. and HeijmansH.J.A.M., Grey-scale morphology based onfuzzy logic, J. Math. Imaging Vis.16 (2002), 155–171.
15.
DimuroG.P. and BedregalB., Archimedean overlap functions: Theordinal sum and the cancellation, idempotency and limitingproperties, Fuzzy Sets Syst.252 (2014), 39–54.
16.
DimuroG.P., BedregalB. and SantiagoR.H.N., On (G, N)-implications derived from grouping functions, Inform. Sci.279 (2014), 1–17.
17.
DimuroG.P. and BedregalB., On residual implications derived from overlap functions, Inform. Sci.312 (2015), 78–88.
18.
DimuroG.P., BedregalB., BustinceH., JurioA., BaczynskiM. and MisK., QL-operations and QL-implication functions constructed from tuples (O, G, N) and the generation of fuzzy subsethood and entropy measures, Int. J. Approx. Reason.82 (2017), 170–192.
19.
DimuroG.P., BedregalB., FernandezJ., Sesma-SaraM., PintorandJ.M. and BustinceH., The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions, Int. J. Approx. Reason.105 (2019), 27–48.
20.
DombiJ. and BaczynskiM., General characterization of implication’s distributivity properties: the preference implication, IEEE Trans. Fuzzy Syst.28 (2020), 2982–2995.
21.
DuboisD. and PradeH., Fuzzy sets in approximate reasoning. Part 1: inference with possibility distributions, Fuzzy Sets Syst.40 (1991), 143–202.
22.
DuranteF., KlementE.P., MesiarR. and SempiC., Conjunctors and their residual implicators: characterizations and construction methods, Mediterr. J. Math.4 (2007), 343–356.
23.
ElkanoM., GalarM., SanzJ., FernándezA., BarrenecheaE.HerreraF. and BustinceH., Enhancing multi-class classification in FARC-HD fuzzy classifier: on the synergy between n-dimensional overlap functions and decomposition strategies, IEEE Trans. Fuzzy Syst.23(5) (2015), 1562–1580.
24.
ElkanoM., GalarM., SanzJ. and BustinceH., Fuzzy rule-based classification systems for multi-class problems using binary decomposition strategies: on the influence of n-dimensional overlap functions in the fuzzy reasoning method, Inf. Sci.332(2016), 94–114.
25.
FangB.W. and HuB.Q., Granular fuzzy rough sets based on fuzzyimplicators and complicators, Fuzzy Sets Syst.359(2019), 112–139.
26.
FodorJ.C. and KeresztfalviT., Nonstandard conjunctions and implications in fuzzy logic, Int. J. Approx. Reason.12(2) (1995), 69–84.
27.
JurioA., BustinceH., PagolaM., PraderaA. and YagerR., Some properties of overlap and grouping functions and their applicationto image thresholding, Fuzzy Sets Syst.229 (2013), 69–90.
28.
KerreE.E. and NachtegaelM., Fuzzy Techniques in Image Processing, Physica Verlag, Heidelberg, 2000.
29.
LuccaG., DimuroG.P., MattosV., BedregalB., BustinceH., and SanzJ.A., A family of Choquet-based non-associative aggregation functions for application in fuzzy rule-based classification systems, in: 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), IEEE, Los Alamitos (2015), pp. 1–8.
30.
MassanetS. and TorrensJ., On a new class of fuzzy implications: h-implications and generalizations, Inform. Sci.181(2011) (2011), 2111–2127.
31.
MassanetS. and TorrensJ., Threshold generation method of construction of a new implication from two given ones, FuzzySets Syst.205 (2012), 50–75.
32.
MassanetS. and TorrensJ., On the vertical threshold generation method of fuzzy implication and its properties, Fuzzy SetsSyst.226 (2013), 232–252.
33.
PinheiroJ., BedregalB., SantiagoR.H. and SantosH., (T, N)-implications, in: 2017 IEEE International Conference on Fuzzy Systems (2017), 1–6.
34.
Ruiz-AguileraD. and TorrensJ., Distributivity of residualimplications over conjunctive and disjunctive uninorms, FuzzySets Syst.158(1) (2007), 23–37.
35.
YagerR.R., On some new classes of implication operators and theirrole in approximate reasoning, Inform. Sci.167 (2004), 193–216.
36.
YanP. and ChenG.Q., Discovering a cover set of ARsi with hierarchy from quantitative databases, Inform. Sci.173 (2005), 319–336.
37.
ZadehL.A., Outline of a new approach to the analysis of complexsystems and decision processes, IEEE Trans. Syst. Man Cyber.3 (1973), 28–44.
