1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia
2University of Ostrava, Institute for Research and Applications of Fuzzy Modelling, 30. Dubna 22, 701 03 Ostrava, Czech Republic
3Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology, Radlinského 9, 812 37 Bratislava, Slovakia
In this paper, we study relations between fuzzy implicators and some kinds of fuzzy conjunctors, in particular, quasi-copulas and copulas. We show that there is a one-to-one correspondence between the classes of all quasi-copulas and 1-Lipschitz fuzzy implicators. A similar relation holds for copulas and 2-increasing 1-Lipschitz fuzzy implicators. Based on these relations, we introduce, discuss and exemplify several new construction methods for fuzzy implicators, and also discuss several dualities on the class of all 1-Lipschitz fuzzy implicators. In addition, we introduce a construction method for fuzzy implicators based on any two copulas and a Lebesgue integrable fuzzy implicator, and discuss some of its special cases.
Boolean functions {0,1}2→{0,1} determining the truth values of the Boolean negation, conjunction, disjunction and implication, denoted below by n, AND, OR and IMPL, respectively, are linked by the rules which exclude their independence. For example, for all x,y∈{0,1} we have
n(x)=IMPL(x,0)=1−x,(1)
AND(x,y)=1−IMPL(x,1−y),(2)
OR(x,y)=1−IMPL(1−x,y),(3)
etc. Any sound extension of these Boolean logical operators should extend the relations given in (1)–(3). This is, in particular, the case of fuzzy logical operators, where the Boolean range {0,1} of the truth values has been extended into the real unit interval [0,1], see, e.g., [1,2]. As a general approach to the axiomatic characterization of fuzzy logical operators one can consider a coordinate-wise extension of the corresponding Boolean operators. Let us note that as far as terminology is concerned, it is not unified, and in this paper, fuzzy extensions of the Boolean operators AND, OR, IMPL and n will be called fuzzy conjunctors, disjunctors, implicators, and fuzzy negators, respectively (similarly to, e.g., [3,4]). The main aim of our contribution1 is to study some new constructions of fuzzy implicators and their connections with particular fuzzy conjunctors. To avoid superfluous repetitions of the well-known notions, in this paper, we only recall the notions necessary for developing the intended topic.
Definition 1.
A function CON:[0,1]2→[0,1] is called a fuzzy conjunctor if it is monotone, 0 is its annihilator and 1 an idempotent element (i.e., CON(x,0)=CON(0,x)=0 for each x∈[0,1] and CON(1,1)=1).
The class of all fuzzy conjunctors will be denoted by CON.
Obviously, a function CON:[0,1]2→[0,1] is a fuzzy conjunctor if and only if it is a monotone extension of the Boolean conjunction operator AND, CON|{0,1}2=AND. Though Zadeh in [2] originally proposed as a fuzzy conjunctor the minimum operator, there were also mentioned some other possible fuzzy conjunctors, such as the product Π, or a partial sum x+y−1 defined for all x,y∈[0,1] satisfying x+y⩾1. This partial sum is related to the fuzzy conjunctor W:[0,1]2→[0,1] given by W(x,y)=max{0,x+y−1}. Note that in the framework of t-norms, W is called the Łukasiewicz t-norm and usually denoted by TL [5], whereas in the framework of copulas W is called the Fréchet-Hoeffding lower bound [6]. A distinguished subclass S of fuzzy conjunctors is constrained by the existence of neutral element e=1. Such fuzzy conjunctors are called semicopulas [7]. Fuzzy implicators based on semicopulas were exhaustively studied in [8].
In this contribution we will primarily work with two particular subclasses of fuzzy conjunctors, namely, with the class C of all copulas and the class Q of all quasi-copulas.
Definition 2.
A fuzzy conjunctor F:[0,1]2→[0,1] is a copula, if
e=1 is a neutral element of F, i.e., F(x,1)=F(1,x)=x for each x∈[0,1],
F is 2-increasing, i.e.,
F(x′,y′)−F(x′,y)⩾F(x,y′)−F(x,y)(4)
for all x,x′,y,y′∈[0,1] with x⩽x′,y⩽y′.
Definition 3.
A fuzzy conjunctor F:[0,1]2→[0,1] is a quasi-copula, if
e=1 is a neutral element of F,
F is 1-Lipschitz, i.e.,
|F(x′,y′)−F(x,y)|⩽|x′−x|+|y′−y|(5)
for all x,x′,y,y′∈[0,1].
Note that each copula satisfies the 1-Lipschitz property, thus it is also a quasi-copula. In general, we have C⫋Q⫋S⫋CON. Many of our results will be related to the basic three quasi-copulas which are also copulas, namely, W,Π and M, where M(x,y)=min{x,y}.
Example 1.
Consider any a∈[0,1].
Let Fa:[0,1]2→[0,1] be given by Fa(x,y)=med(min{x,y},a,x+y−1), where med denotes the classical median. Then Fa is a copula, F0=W, F1=M. Observe that Fa is in fact an ordinal sum copula with a unique Archimedean summand if a∈]0,1[, Fa=(〈a,1,W〉).
If Fa is given by Fa(x,y)=med(xy,a,x+y−1), then Fa is for each a∈]0,1[ a proper quasi-copula, i.e., Fa∈Q\C.
Note that throughout the paper, from the class of all fuzzy negators, we will only use the standard fuzzy negator introduced by Zadeh in [2], i.e., Ns:[0,1]→[0,1], Ns(x)=1−x.
Definition 4.
A function I:[0,1]2→[0,1] is called a fuzzy implicator if
I is decreasing in the first variable and increasing in the second one.
I(0,0)=I(1,1)=1 and I(1,0)=0.
We will denote the class of all fuzzy implicators by ℐ. From the hybrid monotonicity of fuzzy implicators it can be deduced that for each I∈ℐ and each x∈[0,1] we have I(0,x)=1 and I(x,1)=1. Each fuzzy implicator I is a coordinate-wise monotone extension of the Boolean implication operator IMPL, I|{0,1}2=IMPL.
Some other properties of fuzzy implicators which are often of interest, are the following ones:
The left neutrality principle (NP),
I(1,y)=y,for eachy∈[0,1].(NP)
The identity principle (IP),
I(x,x)=1,for eachx∈[0,1].(IP)
The ordering property(OP),
x⩽y⇔I(x,y)=1,for allx,y∈[0,1].(OP)
The law of contraposition (CP) (with respect to the standard fuzzy negator),
I(1−y,1−x)=I(x,y),for allx,y∈[0,1].(CP)
The exchange principle(EP),
I(x,I(y,z))=I(y,I(x,z)),for allx,y,z∈[0,1].(EP)
Let I be a fuzzy implicator. If for each x∈[0,1], I(x,0)=Ns(x), we will say that Ns is the natural negator corresponding to I.
Finally, let us mention several typical fuzzy implicators which will be used throughout the paper.
Example 2.
The following [0,1]2→[0,1] functions are distinguished examples of fuzzy implicators:
the Łukasiewicz fuzzy implicator IL, IL(x,y)=min{1,1−x+y};
the Reichenbach fuzzy implicator IR, IR(x,y)=1−x+xy;
the Kleene-Dienes fuzzy implicator, IKD(x,y)=max{1−x,y};
the Gödel fuzzy implicator,
IG(x,y)=1ifx⩽y,yotherwise;
the Goguen fuzzy implicator,
IGG(x,y)=1ifx⩽y,yxotherwise.
2. FUZZY IMPLICATORS AND QUASI-COPULAS
Observe that the relation between the Boolean conjunction operator AND and the Boolean implication operator IMPL given in (2), is also preserved in the case of their fuzzy extensions CON and I, see [9,10].
Proposition 1.
For all CON∈CON and I∈ℐ we have
ICON:[0,1]2→[0,1], given by
ICON(x,y)=1−CON(x,1−y),(6)
is a fuzzy implicator.
CONI:[0,1]2→[0,1], given by
CONI(x,y)=1−I(x,1−y),(7)
is a fuzzy conjunctor.
CONICON=CON and ICONI=I.
If CON∈CON then ICON∈ℐ, and vice-versa, I∈ℐ implies that CONI∈CON. Due to Proposition 1, there is a one-to-one correspondence between the classes CON and ℐ, expressed by Eqs. (6) and (7). Such relations can also be studied for particular subclasses of CON and ℐ. For example, focusing on the class of all quasi-copulas, we obtain the following characterization of the subclass Q⊂CON.
Lemma 2.
Let CON∈CON. Then the following are equivalent:
CON∈Q.
CON is 1-Lipschitz.
Proof.
(i) ⇒ (ii) is a trivial consequence of the definition of a quasi-copula. For proving (ii) ⇒ (i) we have to show that e=1 is a neutral element of CON. This fact follows from the 1-Lipschitz property of the considered CON and the boundary conditions valid for any CON. Namely, CON(1,1)−CON(1,x)⩽1−x implies CON(1,x)⩾x, and CON(1,x)−CON(1,0)⩽x implies CON(1,x)⩽x, i.e., CON(1,x)=x for each x∈[0,1]. A similar proof can be done for the property CON(x,1)=x for each x∈[0,1].
If we denote by ℐ1−Lip the class of all 1-Lipschitz fuzzy implicators, we can formulate the following interesting result.
Theorem 3.
LetI∈ℐ. Then the following are equivalent:
I∈ℐ1−Lip.
CONI∈Q.
Proof.
(i) ⇒ (ii): Let I∈ℐ1−Lip. Then, due to the 1-Lipschitz property and the boundary conditions I(1,1)=1 and I(1,0)=0, for each y∈[0,1] we obtain I(1,y)=I(1,y)−I(1,0)⩽y and 1−I(1,y)=I(1,1)−I(1,y)⩽1−y, which yields I(1,y)⩾y, and thus, we have I(1,y)=y, i.e., I satisfies the left NP.
Similarly, we can show that for each x∈[0,1] we have I(x,0)=1−x. Therefore, for Q=CONI we can write Q(1,y)=1−I(1,1−y)=1−(1−y)=y for each y∈[0,1], and similarly, Q(x,1)=1−I(x,0)=1−(1−x)=x for all x∈[0,1], which means that e=1 is a neutral element of Q.
Moreover, for all x,x′,y,y′∈[0,1] we have
|Q(x′,y′)−Q(x,y)|
=|1−I(x′,1−y′)−1+I(x,1−y)|⩽|x′−x|+|y′−y|,
i.e., Q is also 1-Lipschitz, thus a quasi-copula.
(ii) ⇒ (i): Let I∈ℐ and suppose that CONI is a quasi-copula. Then for all x,x′,y,y′∈[0,1] we have
It is known that the class Q of all quasi-copulas is closed under aggregation by means of n-ary 1-Chebyshev aggregation functions [11]. The same result holds for the class ℐ1−Lip of all 1-Lipschitz fuzzy implicators. Thus, for example, convex combinations, suprema or infima of I1,…,In∈ℐ1−Lip also belong to ℐ1−Lip.
In the first part of the previous proof we have shown that each I∈ℐ1−Lip satisfies the left NP. Similarly, if I∈ℐ1−Lip we have x=CONI(x,1)=1−I(x,0) which implies that for each I∈ℐ1−Lip, Ns is its natural negator.
Theorem 3 and Lemma 2 show the existence of a one-to-one correspondence between the classes of all 1-Lipschitz fuzzy implicators and 1-Lipschitz fuzzy conjunctors. If we considered some additional properties of fuzzy implicators, we could obtain more specific subclasses of fuzzy conjunctors. Have a look at the following examples:
Example 3.
Let I∈ℐ1−Lip satisfy the OP. Then the related quasi-copula Q=CONI satisfies the equality Q(x,y)=0 if and only if x⩽1−y. The only quasi-copula satisfying this property is Q=W, and thus
I(x,y)=1−W(x,1−y)=min{1,1−x+y}=IL(x,y)
for each (x,y)∈[0,1]2, and we can conclude that
The only 1-Lipschitz fuzzy implicator satisfying the OP is the Łukasiewicz fuzzy implicator.
Similarly, one can show that the only I∈ℐ1−Lip satisfying the IP is the Łukasiewicz fuzzy implicator IL.
Example 4.
Let I∈ℐ1−Lip satisfy the EP and the law of CP. These properties of I are equivalent to the associativity of the corresponding quasi-copula Q=CONI, i.e., Q is a 1-Lipschitz triangular norm, or equivalently, associative copula [5,12]. Thus Q=(〈ak,bk,fk〉|k∈K) is an ordinal sum of Archimedean copulas,
where (]ak,bk[|k∈K) is a system of disjoint open subintervals of [0,1] (possibly empty), and for each k∈K, fk:[ak,bk]→[0,∞] is a continuous convex strictly decreasing function with fk(bk)=0.
In particular, if we consider K={1}, ]a1,b1[=]0,1[ and f1(x)=1x−1, for all (x,y)≠(0,0) we obtain the function
Q(x,y)=xyx+y−xy,
which is known as a member of the Ali-Haq-Mikhail copula family, or Clayton copula family, or as a member of the Hamacher family of t-norms, see [5,6]. The corresponding fuzzy implicator is given by
I(x,y)=1−x−y+2xy1−y+xy
(with convention 00=1).
Now, we recall a well-known result concerning a transformation of quasi-copulas called flipping [13]. Let Q∈Q be any quasi-copula. Then the function Q−:[0,1]2→[0,1] given by
Q−(x,y)=x−Q(x,1−y)
is also a quasi-copula. Based on this knowledge, the following result for 1-Lipschitz fuzzy implicators can be proved.
Theorem 4.
LetI∈ℐ1−Lip. Then the functionI−:[0,1]2→ℝdefined by
I−(x,y)=2−x−I(x,1−y)(8)
is also a1-Lipschitz fuzzy implicator.
Proof.
Let I∈ℐ1−Lip. Then, by Theorem 3, the function QI given by QI(x,y)=1−I(x,1−y) is a quasi-copula. This ensures that (QI)− is a quasi-copula whose values are
(QI)−(x,y)=x−QI(x,1−y)=I(x,y)+x−1.
Denote I(QI)− simply by I−. Then I− is a 1-Lipschitz fuzzy implicator, and for all (x,y)∈[0,1]2 we have
I−(x,y)=1−(QI)−(x,1−y)=2−x−I(x,1−y).
Note that the previous steps ensure that the range of I− is [0,1].
It is not difficult to check that the flipping of 1-Lipschitz fuzzy implicators is a duality on the class ℐ1−Lip, because, for each I∈ℐ1−Lip, we have (I−)−=I.
Example 5.
As mentioned above, the Łukasiewicz implicator IL belongs to ℐ1−Lip, and thus its flipping IL− is, for each (x,y)∈[0,1]2, given by
i.e., it results in the Kleene-Dienes fuzzy implicator.
On the other hand, for the Reichenbach fuzzy implicator IR we have IR−=IR, i.e., IR is invariant with respect to flipping. Both the exemplified facts are connected with the well-known results for quasi-copulas, namely, W−=M and Π−=Π.
Arguments similar to those used in the proof of Theorem 4 enable us to introduce two other dualities on the class ℐ1−Lip, namely the mappings φ,τ:ℐ1−Lip→ℐ1−Lip which assign to a fuzzy implicator I∈ℐ1−Lip fuzzy implicators φ(I) and τ(I), respectively, given by
φ(I)(x,y)=1+y−I(1−x,y),τ(I)(x,y)=y−x+I(1−x,1−y).
If we denote I− introduced in (8) for a moment by α(I), and also take into consideration the identity mapping id on ℐ1−Lip, the obtained set of four mappings P={id,α,φ,τ} equipped with the operation of composition (denoted by ∘) is an Abelian group which is isomorphic to the standard (ℤ2,+)2 group, see Table 1.
∘
id
α
φ
τ
id
id
α
φ
τ
α
α
id
τ
φ
φ
φ
τ
id
α
τ
τ
φ
α
id
Table 1
Description of the group (P,∘).
Example 6.
Consider the proper quasi-copula Q=med(W,0.5,Π)∈Q\C, where med is the standard median. Then the corresponding 1-Lipschitz fuzzy implicator IQ is given by med(IL,0.5,IR),
All four discussed fuzzy implicators coincide on the greatest parts of their domain [0,1]2 with the Reichenbach fuzzy implicator IR, but they differ from IR on some corner areas which, for IQ, αIQ, τIQ, φIQ, are situated around the corners [1,0], [1,1], [0,1] and [0,0] of the unit square, respectively, as can be seen in Figures 1–4.
Example 7.
Let us construct a function J:[0,1]2→[0,1] by means of fuzzy implicators IQ, α(IQ), τ(IQ) and φ(IQ) as follows (see Figure 5):
Then J is a fuzzy implicator which is invariant with respect to the transformations α, τ and φ.
The fact that J is a fuzzy implicator immediately follows from the way of its construction and the properties of the fuzzy implicators IQ, α(IQ), τ(IQ) and φ(IQ). The proof of the invariance of J w.r.t. the transformations α, τ and φ is only a matter of computation. We only outline several steps for proving α(J)=J. At the points in the middle part of [0,1]2 (Figure 5), where the values of J are given by J(x,y)=IR(x,y), the invariance α(J)(x,y)=J(x,y) follows from the invariance of IR w.r.t. α.
The rest part of [0,1]2 consists of 4 pairs of subsets (Ai,Bi) such that there is a one-to-one correspondence between the points of Ai and Bi, (x,y)⇄(x,1−y), see Figure 5. Consider, e.g., (x,y)∈A1,
A1={(x,y)∈[0,1]2|0.5⩽x⩽1,0⩽y⩽x−0.5},
with J(x,y)=1−x+y. Then the corresponding point (x,1−y)∈B1,
B1={(u,v)∈[0,1]2|0.5⩽u⩽1,1.5−u⩽v⩽1},
with J(u,v)=v. Thus
α(J)(x,y)
=2−x−J(x,1−y)=2−x−(1−y)=1−x+y=J(x,y).
Conversely, if (x,y)∈B1, then its corresponding point is in A1, and thus
α(J)(x,y)
=2−x−J(x,1−y)=2−x−(1−x−(1−y))=y=J(x,y).
It can be proved in a similar way that for each (x,y)∈[0,1]2 we have
α(J)(x,y)=J(x,y),τ(J)(x,y)=J(x,y),
and also
φ(J)(x,y)=J(x,y).
Note that the values J(x,y) can simply be written as
see Figure 6. Note that the fuzzy conjunctor corresponding to K is a copula CK given in Figure 7. This singular copula CK has also been mentioned in [6] (for more details see p.66, Example 3.10).
Observe that fuzzy implicators introduced in Examples 7 and 8 are, in fact, rectangular patchworks of four fuzzy implicators. Note that the patchwork method was successfully applied in the case of copulas in several papers, see, e.g., [14].
As already mentioned, for each 1-Lipschitz fuzzy implicator I there is a unique quasi-copula Q such that I(x,y)=1−Q(x,1−y). For the transforms αI, φI and τI of I, mentioned above, we have
α(I)(x,y)=1−x+Q(x,y),
φ(I)(x,y)=y+Q(1−x,1−y),
τ(I)(x,y)=y+1−x−Q(1−x,y).
Based on these constructions, it is possible to introduce several constructions for fuzzy implicators (some of them have already been known, see [15]).
Proposition 5.
Let f,g:[0,1]→[0,1] be unary aggregation functions, i.e., increasing functions satisfying the boundary conditions f(0)=g(0)=0 and f(1)=g(1)=1. Then:
For any semicopula S, the function JS:[0,1]2→[0,1] given by
JS(x,y)=1−S(f(x),1−g(y))
is a fuzzy implicator.
For any semicopula H which is 1-Lipschitz in the first coordinate, the function JH:[0,1]2→[0,1] given by
JH(x,y)=1−f(x)+H(f(x),g(y))
is a fuzzy implicator.
For any semicopula G which is 1-Lipschitz in the second coordinate, the function JG:[0,1]2→[0,1] given by
JG(x,y)=g(y)+G(1−f(x),1−g(y))
is a fuzzy implicator.
For any quasi-copula Q, the function JQ:[0,1]2→[0,1] given by
JQ(x,y)=g(y)+1−f(x)−Q(1−f(x),g(y))
is a fuzzy implicator.
Note that there are also some other relations between quasi-copulas and fuzzy implicators, for example, if they are related by Galois connections (adjointness or residuation property). More details about these relations can be found in [4,16].
3. FUZZY IMPLICATORS AND COPULAS
We briefly mention some recent constructions of fuzzy implicators by means of copulas and relations between them, and also propose some new ones. Besides the relations between copulas and fuzzy implicators based on Galois connections discussed in [17], several other connections between them were studied, e.g., in [8,18]. For example, Grzegorzewski in [18] studied construction of fuzzy implicators by means of copulas given by the formula IC(x,y)=C(x,y)x (x>0), valid for a subclass of copulas satisfying the inequality C(x′,y)⋅x⩽C(x,y)⋅x′ for all x<x′ and y∈[0,1]. Note that copulas M and Π satisfy the given constraint, and result in IM=IGG and IΠ, i.e., the smallest fuzzy implicator satisfying the left NP(NP), IΠ(x,y)=1 if x=0, otherwise, IΠ(x,y)=y. However, this approach is not appropriate for W (IW is not a fuzzy implicator). From approaches discussed in [8], we recall a construction yielding for any copula C the function IC:[0,1]2→[0,1], IC(x,y)=1−x+C(x,y), which belongs to ℐ1−Lip. We have IM=IL, IΠ=IR, IW=IKD. Some other constructions of copulas by means of fuzzy implicators (and vice-versa) have also been studied, e.g., in [19,20].
Now, motivated by the so-called Darsow product of copulas ∗:C×C→C introduced and studied in [21], we propose a new construction method for fuzzy implicators based on a fixed I∈ℐ and a couple of copulas C1,C2∈C.
Theorem 6.
LetI∈ℐbe a Lebesgue integrable function, andC1,C2∈C. Then the functionJI,C1,C2:[0,1]2→[0,1], given by
We first observe that for each t∈[0,1] and all C1,C2∈C we have
f1(t)=g1(t)=1andf0(t)=g0(t)=0.
These properties ensure the validity of the boundary conditions: JI,C1,C2(0,0)=JI,C1,C2(1,1)=1 and JI,C1,C2(1,0)=0. The 2-increasing property of copulas ensures that the functions fx and gy are increasing, i.e., which implies the decreasingness of JI,C1,C2 in the first variable and increasingness in the second one. Summarizing, JI,C1,C2 is in ℐ.
Observe that any continuous fuzzy implicator is Lebesgue integrable. On the other hand, if we follow the ideas of Klement [22], see also Example 3.75 in [5], and consider a subset C⊂]0,1[ which is not Lebesgue measurable, then the function I:[0,1]2→[0,1] given by
I(x,y)=0ifx=y∈Cor(x>0andx>y),1otherwise,
is a fuzzy implicator which is not Lebesgue measurable and thus not Lebesgue integrable.
It is only a matter of computation to show the validity of the following consequence of Theorem 6.
Corollary 7.
Let I∈ℐ be a Lebesgue integrable fuzzy implicator satisfying the (NP) property, and let Ns be its natural negator. Then for any C1,C2∈C, the function JI,C1,C2 given by (9) is a fuzzy implicator with natural negator Ns, satisfying the (NP) property.
Observe that if I is a fuzzy implicator with natural negator Ns, satisfying the (NP) property, and such that range Ranfx⊆{0,1} for all x∈[0,1], or Rangy⊆{0,1} for all y∈[0,1], respectively (see Theorem 6), then JI,C1,C2 does not depend on I.
Example 9.
Consider a copula Ca given by
Ca(x,y)=med(0,x+ay−a,y)
(see Figure 8), an arbitrary Lebesgue integrable fuzzy implicator I satisfying I(x,0)=1−x for each x∈[0,1], and an arbitrary copula C. Then for each y∈[0,1[, gy=1]a(1−y),a+(1−a)y[ and g1=1[0,1] (see Figure 9), and further,
As we can see, the resulting function JI,C,Ca in Example 9 depends on C∈C and a∈[0,1] only. Hence, we can introduce a new copula-based construction of fuzzy implicators, assigning to each C∈C and any a∈[0,1] directly a fuzzy implicator Ia,C=JI,C,Ca.
Proposition 8.
LetCbe any copula anda∈[0,1]. Then the functionIa,C:[0,1]2→[0,1]given by
Ia,C(x,y)=1−x−C(x,a(1−y))+C(x,a+(1−a)y)
is a fuzzy implicator.
Note that for each C∈C, I0,C(x,y)=1−x+C(x,y), i.e., I0,C=IC, which means that our construction Ia,C also covers the construction IC from [8] mentioned above.
The other boundary case for a=1 also leads to a well-known relation between copulas and fuzzy implicators, namely,
I1,C(x,y)=1−C(x,1−y).
Finally, for each (x,y)∈[0,1]2 we have
I0.5,C(x,y)=1−x−Cx,1−y2+Cx,1+y2.
Example 10.
Consider the three basic copulas W,Π and M. Then
Ia,Π=IR for each a∈[0,1].
Ia,Wa∈[0,1] is an increasing family with respect to parameter a of 1-Lipschitz 2-increasing fuzzy implicators, see Figure 10. Note that I0,W=IKD and I1,W=IL, and for each a∈]0,1[ we have
Ia,W(x,y)=max{1−x,a+(1−a)y}−max{0,x+a−1−ay}.
The family Ia,Ma∈[0,1] is a decreasing family with respect to parameter a of 1-Lipschitz 2-increasing fuzzy implicators with extremal elements I0,M=IL and I1,M=IKD. Observe that, for any a∈[0,1], Ia,M=Ia,W−. This case is illustrated in Figure 11.
Note that the 2-increasing property of fuzzy implicators is of the form as given in (4).
The following result for fuzzy implicators related to copulas is an analogue of Theorem 3 for fuzzy implicators related to quasi-copulas.
Theorem 9.
LetIbe a fuzzy implicator. Then the following are equivalent.
Iis1-Lipschitz and2-increasing.
Iis2-increasing, satisfies the left NP andNsis the natural negator corresponding toI.
CONIis a copula.
Proof.
(i) ⇒ (ii) It is enough to recall that due to the assumption I∈ℐ1−Lip, I satisfies the left NP, and Ns is the natural negator of I.
To see (ii) ⇒ (iii), we first note that if I satisfies the left NP, so does CONI, i.e., CONI(1,y)=y for each y∈[0,1]. As for each x∈[0,1], I(x,0)=Ns(x)=1−x, we obtain CONI(x,1)=1−I(x,0)=x. So, we have shown that CONI is a fuzzy conjunctor with neutral element e=1, and thus a semicopula. This implies that 0 is an annihilator of CONI. Finally, suppose that x,x′,y,y′ are in [0, 1] such that x⩽x′ and y⩽y′. Then 1−y′⩽1−y, and due to the 2-increasing property of I, we can write
which shows that CONI is 2-increasing. Summarizing, CONI is a copula.
The validity of (iii) ⇒ (i) is evident.
In what follows, we are presenting two construction methods for fuzzy implicators which are based on the previous theorem, Proposition 1 and the so-called (a,b)-constructions of copulas introduced in [6] and deeply studied in [23]. This construction of copulas is defined for any pair (a,b)∈[0,1]2 of parameters transforming an original copula C into the copula Ca,b:[0,1]2→[0,1] given by
Leta,b∈[0,1]. LetIbe a2-increasing fuzzy implicator satisfying the left NP and letNsbe its natural negator. Then the functionsIa, Ia,b:[0,1]2→[0,1]given by (10) and (11), respectively,
We have discussed the relations between fuzzy implicators and some particular classes of fuzzy conjunctors, namely the classes of quasi-copulas and copulas. We have shown the existence of a one-to-one correspondence between the class of all quasi-copulas and the class of 1-Lipschitz fuzzy implicators. Based on some well-known dualities on the class of all quasi-copulas, we have introduced and discussed some related dualities on the class ℐ1−Lip of all 1-Lipschitz fuzzy implicators. Exploiting unary aggregation functions, these dualities have then been extended to the other four construction methods for fuzzy implicators. Further, we have proposed several new construction methods leading to parametric families of particular fuzzy implicators. In Theorem 6, we have proposed a powerful tool for constructing fuzzy implicators based on a Lebesgue integrable fuzzy implicator I and any two copulas C1,C2. Moreover, some particular cases have been exemplified. We believe that our approach opens a new look at constructions of fuzzy implicators. In particular, we expect that Theorem 6 can be further generalized by considering more general systems of unary functions fxx∈[0,1], gyy∈[0,1].
CONFLICT OF INTEREST
Both authors have participated equally on the work on this paper.
ACKNOWLEDGMENTS
Both authors are grateful for the support of the project APVV–18–0052. R. Mesiar was also supported by the NPUII project LQ1602 and the project VEGA 1/0006/19. A. Kolesárová thanks to the support of the project VEGA 1/0614/18.
12.B. Schweizer and A. Sklar, Probabilistc Metric Spaces, Dover Books on Mathematics, North Holland Series in Probability and Applied Mathematics, Courier Corporation, New York, U.S.A, 2005.
TY - JOUR
AU - Radko Mesiar
AU - Anna Kolesárová
PY - 2020
DA - 2020/06/16
TI - Quasi-Copulas, Copulas and Fuzzy Implicators
JO - International Journal of Computational Intelligence Systems
SP - 681
EP - 689
VL - 13
IS - 1
SN - 1875-6883
UR - https://doi.org/10.2991/ijcis.d.200527.004
DO - 10.2991/ijcis.d.200527.004
ID - Mesiar2020
ER -
, Anna Kolesárová3
