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On the composition of intuitionistic fuzzy relations
Fuzzy Sets and Systems 136 (2003) 333 – 361 www.elsevier.com/locate/fss
On the composition of intuitionistic fuzzy relations Glad Deschrijver ∗ , Etienne E. Kerre Fuzziness and Uncertainty Modelling, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium Received 22 August 2001; received in revised form 23 April 2002; accepted 3 May 2002
Abstract Fuzzy relations are able to model vagueness, in the sense that they provide the degree to which two objects are related to each other. However, they cannot model uncertainty: there is no means to attribute reliability information to the membership degrees. Intuitionistic fuzzy sets, as de1ned by Atanassov (Instuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999), give us a way to incorporate uncertainty in an additional degree. Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a cartesian product of universes. One of the main concepts in relational calculus is the composition of two relations. Burillo and Bustince (Fuzzy Sets and Systems 78 (1996) 293; Soft Comput. 2 (1995) 5) have extended the sup-T composition of fuzzy relations to a composition of intuitionistic fuzzy relations. In this paper, we present an intuitionistic fuzzy version of the triangular compositions of Bandler and Kohout (in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, p. 341) and the variants of these compositions given by De Baets and Kerre (Adv. Electron. Electron Phys. 89 (1994) 255). Some properties of these compositions are investigated: containment, convertibility, monotonicity, interaction c 2002 Elsevier Science B.V. All rights reserved. with union and intersection. Keywords: Fuzzy relation; Intuitionistic fuzzy set; Intuitionistic fuzzy relation; Triangular composition
1. Introductory remarks Relations are a suitable tool for describing correspondences between objects. Crisp relations like ∈, ⊆, =, : : : have served well in developing mathematical theories. The use of fuzzy relations originated from the observation that real-life objects can be related to each other to a certain degree. However, in real-life situations, a person may assume that a certain object A is in relation R with another object B to a certain degree, but it is possible that he is not so sure about it. In other words, there may ∗
Corresponding author. E-mail addresses: [email protected] (G. Deschrijver), [email protected] (Etienne E. Kerre).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 2 6 9 - 5
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be a hesitation or uncertainty about the degree that is assigned to the relationship between A and B. In fuzzy set theory there is no means to incorporate that hesitation in the membership degrees. A possible solution is to use intuitionistic fuzzy sets, de1ned by Atanassov in 1983 [1]. Intuitionistic fuzzy sets give us the possibility to model hesitation and uncertainty by using an additional degree. Each intuitionistic fuzzy set A assigns to each element x of the universe X a membership degree A (x) (∈ [0; 1]) and a non-membership degree A (x) (∈ [0; 1]) such that A (x) + A (x)61. For all x ∈ X , the number A (x) = 1 − A (x) − A (x) is called the hesitation degree or the intuitionistic index of x to A. In fuzzy set theory, the non-membership degree of an element x of the universe is de1ned as one minus the membership degree (using the standard negation) and thus it is 1xed. In intuitionistic fuzzy set theory, the non-membership degree is a more-or-less independent degree: the only condition is that it is smaller than one minus the membership degree. Note that both A and A can be seen as fuzzy sets on X (that are not completely independent from each other, because of the condition that the sum of the two degrees should be less than or equal to 1). In this way, the negation of the nonmembership degree w.r.t. the standard fuzzy negation can be seen as a degree of membership. So for each element x ∈ X there exist two degrees that model the membership of x in the intuitionistic fuzzy set A, namely A (x) and 1 − A (x). The length of the interval [A (x); 1 − A (x)], which is given by A (x), can then be seen as degree modelling the hesitation between the two membership degrees. An intuitionistic fuzzy relation between two universes X and Y is de1ned as an intuitionistic fuzzy set in X × Y [3,4]. In a similar way as above, if R is a relation between X and Y , x ∈ X and y ∈ Y , then R (x; y) denotes the degree to which x is in relation R with y and R (x; y) denotes the uncertainty degree to which x and y are in relation R with each other. So the “real” degree to which x is in relation R with y lies somewhere between R (x; y) and R (x; y) + R (x; y) = 1 − R (x; y). 1.1. The composition of relations One of the main concepts in relational calculus is the composition of relations. Let us 1rst consider the crisp case. A (crisp) relation R from a universe X to a universe Y is a subset of X × Y . When (x; y) ∈ R, we say that x is in relation R with y and write shortly xRy. The classical composition R ◦ S of a relation R from X to Y and a relation S from Y to Z is de1ned as R ◦ S = {(x; z) ∈ X × Z | (∃y ∈ Y )(xRy ∧ ySz)} = {(x; z) ∈ X × Z | xR ∩ Sz = ∅}; where the afterset xR of x w.r.t. R is de1ned as xR = {y ∈ Y | (x; y) ∈ R} and the foreset Sz of z w.r.t S is de1ned as Sz = {y ∈ Y | (y; z) ∈ S}. The relation R can be identi1ed with its characteristic mapping, namely R : X ×Y → {0; 1} (x; y) → 1
if (x; y) ∈ R;
(x; y) → 0
else:
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Then the composition R ◦ S of R and S can also be written as R ◦ S : X × Z → [0; 1] (x; z) → sup R(x; y) ∧B S(y; z); y ∈Y
where ∧B denotes the Boolean conjunction. Denote by Rt the converse relation of R de1ned as Rt = {(y; x) ∈ Y × X | (x; y) ∈ R}. The composition R ◦ S is a relation from X to Y , consisting of those couples (x; z) for which there exists at least one element of Y that is in relation Rt with x and that is in relation S with z. Example 1.1 (See also De Baets and Kerre [11]). Let X be a set of patients, Y a set of symptoms and Z a set of illnesses. De1ne the relation R ⊆ X × Y as R(x; y) = 1 if patient x shows symptom y, and R(x; y) = 0 if patient x does not show symptom y. De1ne S ⊆ Y × Z as S(y; z) = 1 if symptom y is a symptom of illness z, and S(y; z) = 0 else. Then R ◦ S(x; z) = 1 if patient x shows at least one symptom of illness z. However, therapists may also want to know whether the symptoms shown by patient x are all symptoms of illness z, or whether all the symptoms of illness z are shown by patient x. Bandler and Kohout have de1ned the triangular compositions, in order to be able to model situations as described in the example. The subcomposition of R and S is de1ned as R /bk S = {(x; z) ∈ X × Z | xR ⊆ Sz} and the supercomposition of R and S as R .bk S = {(x; z) ∈ X × Z | Sz ⊆ xR}: Using the characteristic mappings, these compositions are also given by R /bk S(x; z) = inf R(x; y) ⇒B S(y; z); y ∈Y
R .bk S(x; z) = inf S(y; z) ⇒B R(x; y); y ∈Y
where ⇒B denotes the Boolean implication. Example 1.2. In the previous example, the pair (x; z) belongs to R / bk S if all symptoms shown by patient x are symptoms of illness z, and it belongs to R .bk S if patient x shows all symptoms of illness z. The domain of a (crisp) relation R is de1ned as dom(R) = {x ∈ X | xR = ∅}, and the range of R is de1ned as rng(R) = {y ∈ Y | Ry = ∅}. Using these de1nitions, it is clear that co(dom(R)) × Z ⊆ R /bk S; X × co(rng(S)) ⊆ R .bk S:
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The 1rst expression means that if x is not in the domain of R, then x is in relation R / bk S with all elements of Z, even if there is no element of Y that is in relation Rt with x. A similar remark holds for the second expression. De Baets and Kerre [10,11,12] have therefore introduced the following modi1ed de1nitions: the subcomposition R / S = {(x; z) ∈ X × Z | ∅ ⊂ xR ⊆ Sz} and the supercomposition R . S = {(x; z) ∈ X × Z | ∅ ⊂ Sz ⊆ xR}: They also showed that the sub- and the supercomposition can be written in the following equivalent ways: R / S = (R /bk S) ∩ (dom(R) × rng(S)) = (R /bk S) ∩ (R ◦ S) for the subcomposition, and R . S = (R .bk S) ∩ (dom(R) × rng(S)) = (R .bk S) ∩ (R ◦ S) for the supercomposition. In [11], De Baets and Kerre have extended these de1nitions to the fuzzy case and investigated their properties. In this paper we extend these compositions to intuitionistic fuzzy relations and investigate whether the results of De Baets and Kerre still hold. In Section 2 we 1rst recall the de1nitions of the fuzzy relational compositions. In Section 3 we give the de1nition of intuitionistic fuzzy sets, show how intuitionistic fuzzy sets can be seen as L-fuzzy sets for some lattice L, and de1ne some operators which we will need later on. In Section 4 we give the de1nition of intuitionistic fuzzy relation and introduce the extension to intuitionistic fuzzy relations of the diMerent types of relational composition. Finally in Section 5 we investigate the properties of these compositions.
2. The composition of fuzzy relations A fuzzy relation from X to Y is a fuzzy set in X × Y . The afterset xR is the fuzzy set in Y de1ned by xR(y) = R(x; y), ∀y ∈ Y . The foreset Ry is the fuzzy set in X de1ned by Ry(x) = R(x; y), ∀x ∈ X . The domain of R is the fuzzy set in X de1ned by dom(R)(x) = hgt(xR) = supy∈Y R(x; y), where for an arbitrary fuzzy set A in a universe X the height of A is de1ned as hgt(A) = supx∈X A(x). The range of R is the fuzzy set in Y de1ned by rng(R)(y) = hgt(Ry) = supx∈X R(x; y). The classical composition of relations has been extended to fuzzy relations by Zadeh [15]. Let T be a triangular norm, then the sup −T composition R ◦T S of a fuzzy relation R from X to Y and a
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fuzzy relation S from Y to Z is the fuzzy relation from X to Z de1ned by R ◦T S(x; z) = sup T (R(x; y); S(y; z)) = hgt T (xR; Sz): y ∈Y
Bandler and Kohout have extended their triangular compositions to fuzzy relations in the following way: I R /bk S(x; z) = inf I(R(x; y); S(y; z)) = plt I(xR; Sz); y ∈Y
I R .bk S(x; z) = inf I(S(y; z); R(x; y)) = plt I(Sz; xR); y ∈Y
where I is a fuzzy implicator, i.e. a [0; 1]2 → [0; 1] mapping that satis1es the boundary conditions I(0; 0) = I(0; 1) = I(1; 1) = 1 and I(1; 0) = 0 and where the plinth of a fuzzy set A in a universe X is de1ned as plt(A) = inf x∈X A(x). De Baets and Kerre have introduced two classes of improved de1nitions based on the improved versions in the crisp case [11]: R /bI S(x; z) = min(plt I(xR; Sz); hgt(xR); hgt(Sz)); R .bI S(x; z) = min(plt I(Sz; xR); hgt(xR); hgt(Sz)); and R /kT;I S(x; z) = min(plt I(xR; Sz); hgt T (xR; Sz)); R .kT;I S(x; z) = min(plt I(Sz; xR); hgt T (xR; Sz)): 3. Intuitionistic fuzzy sets and the lattice L∗ We give the de1nition of intuitionistic fuzzy sets and some operators on them. We introduce the lattice L∗ which will be very useful in the sequel. De nition 3.1 (Atanassov [1–3]). An intuitionistic fuzzy set (shortly IFS) on a universe X is an object of the form A = {(x; A (x); A (x)) | x ∈ X }; where A (x) (∈ [0; 1]) is called the “degree of membership of x in A”, A (x) (∈ [0; 1]) is called the “degree of non-membership of x in A”, and where A and A satisfy the following condition: (∀x ∈ X )
(A (x) + A (x) 6 1):
The class of IFSs on a universe X will be denoted IFS(X ).
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Fig. 1. Graphical representation of the lattice L∗ .
An IFS A is said to be contained in an IFS B (notation A ⊆ B) if and only if, for all x ∈ X : A (x)6 B (x) and A (x)¿ B (x). The intersection (resp. the union) of two IFSs A and B on X is de1ned as the IFS A ∩ B = {(x; min (A (x); B (x)); max( A (x); B (x))) | x ∈ X } (resp. A ∪ B = {(x; max(A (x); B (x)); min( A (x); B (x))) | x ∈ X }). Starting from these de1nitions, a generalized intersection ∩T; S and union ∪S; T can be de1ned by replacing min by an arbitrary t-norm T and max by an arbitrary t-conorm S. The generalized intersection and union yield IFSs if (∀(x; y) ∈ [0; 1]2 )(T (x; y)6S ∗ (x; y)) (shortly T 6S ∗ ) where S ∗ denotes the dual t-norm of S de1ned by S ∗ (x; y) = 1 − S(1 − x; 1 − y), ∀(x; y) ∈ [0; 1]2 [13]. In this paper we will often use the following lattice and operators de1ned on it. De1ne a set L∗ and an operation 6L∗ such that L∗ = {(x1 ; x2 ) ∈ [0; 1]2 | x1 + x2 6 1}; (x1 ; x2 ) 6L∗ (y1 ; y2 ) ⇔ x1 6 y1 ∧ x2 ¿ y2 ; then (L∗ ; 6L∗ ) is a complete lattice [13]. For each A ⊆ L∗ we have sup A = (sup{x1 ∈ [0; 1] | x1 A = ∅}; inf {x2 ∈ [0; 1] | Ax2 = ∅}); inf A = (inf {x1 ∈ [0; 1] | x1 A = ∅}; sup{x2 ∈ [0; 1] | Ax2 = ∅}): The shaded area in Fig. 1 is the set of elements x = (x1 ; x2 ) belonging to L∗ . Equivalently, this lattice can also be de1ned as an algebraic structure (L∗ ; ∧; ∨) where the meet operator ∧ and the join operator ∨ are de1ned as follows, for (x1 ; x2 ); (y1 ; y2 ) ∈ L∗ : (x1 ; x2 ) ∧ (y1 ; y2 ) = (min(x1 ; y1 ); max(x2 ; y2 )); (x1 ; x2 ) ∨ (y1 ; y2 ) = (max(x1 ; y1 ); min(x2 ; y2 )):
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For our purposes, we will also consider a generalized meet operator ∧T; S and join operator ∨S; T on (L∗ ; 6L∗ ): (x1 ; x2 ) ∧T;S (y1 ; y2 ) = (T (x1 ; y1 ); S(x2 ; y2 )); (x1 ; x2 ) ∨S;T (y1 ; y2 ) = (S(x1 ; y1 ); T (x2 ; y2 )) for a given t-norm T and t-conorm S satisfying T 6S ∗ . It has been shown that the condition T 6S ∗ is necessary and suOcient for these operators to be well de1ned and that they are increasing in both components [9]. Furthermore, we de1ne an order-reversing operator Ns by Ns (x1 ; x2 ) = (x2 ; x1 ), ∀(x1 ; x2 ) ∈ L∗ . Using this lattice, it can be easily seen that with every IFS A = {(x; A (x); A (x)) | x ∈ X } corresponds an L∗ -fuzzy set in the sense of Goguen [14], i.e. a mapping A : X → L∗ : x → (A (x); A (x)) [13]. In the sequel we will use the same notation for an IFS and its associated L∗ -fuzzy set. So, for the IFS A we will also use the notation A(x) = (A (x); A (x)). From now on, we will assume that if a ∈ L∗ , then a1 and a2 denote, respectively, the 1rst and the second coordinate of a, i.e. a = (a1 ; a2 ). Interpretating intuitionistic fuzzy sets as L∗ -fuzzy sets gives way to a greater Pexibility in calculating with membership and non-membership degrees, since the pair formed by the two degrees is an element of L∗ , and often allows to obtain signi1cantly more compact formulas. Moreover, some operators that are de1ned in the fuzzy case, such as fuzzy implicators, can be extended to the intuitionistic fuzzy case by using the lattice (L∗ ; 6L∗ ). For instance, fuzzy implicators can be extended to intuitionistic fuzzy implicators as follows. De nition 3.2. An intuitionistic fuzzy implicator is any (L∗ )2 → L∗ mapping I satisfying the border conditions I(0L∗ ; 0L∗ ) = I(0L∗ ; 1L∗ ) = I(1L∗ ; 1L∗ ) = 1L∗ ;
I(1L∗ ; 0L∗ ) = 0L∗ ;
where 0L∗ = (0; 1) and 1L∗ = (1; 0) are the identities of (L∗ ; 6L∗ ). In some cases we will require an intuitionistic fuzzy implicator to be decreasing in its 1rst, and increasing in its second component, i.e. (∀y ∈ L∗ )(∀(x; x ) ∈ (L∗ )2 )(x 6L∗ x ⇒ I(x; y) ¿L∗ I(x ; y));
(M.1)
(∀x ∈ L∗ )(∀(y; y ) ∈ (L∗ )2 )(y 6L∗ y ⇒ I(x; y) 6L∗ I(x; y )):
(M.2)
These properties will be referred to in the sequel as the hybrid monotonicity properties.
4. The composition of intuitionistic fuzzy relations De nition 4.1 (Atanassov [4] and Burillo and Bustince [8]). An intuitionistic fuzzy relation R (IFR, for short) from a universe X to a universe Y is an intuitionistic fuzzy set in X × Y , i.e.
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a set R = {((x; y); R (x; y); R (x; y)) | x ∈ X; y ∈ Y }; where R : X × Y → [0; 1] and R : X × Y → [0; 1] satisfy the condition (R (x; y) + R (x; y) 6 1):
(∀(x; y) ∈ X × Y )
The class of IFRs from a universe X to a universe Y will be denoted IFR(X × Y ). In the sequel we will consider an IFR R also as a mapping R : X × Y → L∗ : (x; y) → R(x; y) = (R (x; y); R (x; y)):
(1)
The afterset xR of an IFR R is the IFS in Y de1ned by xR = {(y; R (x; y); R (x; y)) | y ∈ Y }. The foreset Ry of an IFR R is the IFS in X de1ned by Ry = {(x; R (x; y); R (x; y)) | x ∈ X }. The height of an IFS A in X is de1ned as hgt(A) = supx∈X A(x) = (supx∈X A (x); inf x∈X A (x)) (notice that hgt(A) ∈ L∗ ). The plinth of an IFS A in X is de1ned as plt(A) = inf x∈X A(x) = (inf x∈X A (x); supx∈X
A (x)). The domain of an IFR R is the IFS dom(R) in X de1ned by dom(R)(x) = hgt(xR) = (supy∈Y R (x; y); inf y∈Y R (x; y)). The range of an IFR R is the IFS rng(R) in Y de1ned by rng(R)(y) = hgt(Ry) = (supx∈X R (x; y); inf x∈X R (x; y)). The converse relation Rt of R is the IFR from Y to X de1ned as Rt = {((y; x); R (x; y); R (x; y)) | y ∈ Y; x ∈ X }. Bustince and Burillo have extended the sup −T composition to a composition of IFRs, in the case that X , Y and Z are 1nite, as follows [7,8]: De nition 4.2. Let ; ; ; be t-norms or t-conorms, R ∈ IFR(X × Y ) and S ∈ IFR(Y × Z). Then the composition of R and S is the IFR from X to Z de1ned as R ◦ S = {((x; z); ◦ (x; z); ◦ (x; z)) | x ∈ X; z ∈ Z};
where ◦ (x; z) = ((R (x; y); S (y; z))); y ∈Y
◦ (x; z) = (( R (x; y); S (y; z))); y ∈Y
whenever 0 6 ◦ (x; z) + ◦ (x; z) 6 1;
∀(x; z) ∈ X × Z:
The properties of this composition and the choice of ; ; and for which this composition ful1lls a maximal number of properties are investigated in [7]. We now extend the triangular compositions to intuitionistic triangular compositions as follows. De nition 4.3. Let R be an IFR from X to Y , S an IFR from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm, then the triangular sub- and supercomposition and their
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improved versions are de1ned as IFRs from X to Z de1ned by I S(x; z) = inf I(R(x; y); S(y; z)) = plt I(xR; Sz); R /bk y ∈Y
I R .bk S(x; z) = inf I(S(y; z); R(x; y)) = plt I(Sz; xR); y ∈Y
I
R /b S(x; z) = inf
inf I(R(x; y); S(y; z)); sup R(x; y); sup S(y; z)
y ∈Y
y ∈Y
y ∈Y
= inf {plt I(xR; Sz); hgt(xR); hgt(Sz)}; R .bI S(x; z) = inf {plt I(Sz; xR); hgt(xR); hgt(Sz)}; R /kT
;S ;I
S(x; z) = inf
inf I(R(x; y); S(y; z)); sup(R(x; y) ∧T ;S S(y; z))
y ∈Y
y ∈Y
= inf {plt I(xR; Sz); hgt(xR ∩T ;S Sz)}; R .kT
;S ;I
S(x; z) = inf {plt I(Sz; xR); hgt(xR ∩T ;S Sz)}:
In the above de1nition we used the notation (1) to represent the IFRs. For instance R /I bk S(x; z) ∗ is an element of L given by inf pr 1 I((R (x; y); R (x; y)); (S (y; z); S (y; z)));
y ∈Y
sup pr 2 I((R (x; y); R (x; y)); (S (y; z); S (y; z))) ;
y ∈Y
where pri : L∗ → [0; 1] : (a1 ; a2 ) → ai , i = 1; 2. Then the IFR R /I bk S is given by I I I S = {((x; z); pr 1 R /bk S(x; z); pr 2 R /bk S(x; z)) | x ∈ X; z ∈ Z}: R /bk
Example 4.1. Consider a set of patients X , a set of symptoms Y and a set of illnesses Z. Let R be the IFR from X to Y de1ned by R (x; y) = the degree to which patient x shows symptom y;
R (x; y) = the degree to which patient x does not show symptom y
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and S the IFR from Y to Z de1ned by S (y; z) = the degree to which y is a symptom of illness z;
S (y; z) = the degree to which y is not a symptom of illness z: Let x ∈ X be a patient, Y = {y1 ; : : : ; y5 } a set of symptoms, and z ∈ Z an illness. Assume furthermore that xR = {(y1 ; 0; 1); (y2 ; 0; 1); (y3 ; 0; 1); (y4 ; 0:1; 0:85); (y5 ; 0; 1)}; Sz = {(y1 ; 1; 0); (y2 ; 0:7; 0:2); (y3 ; 0:6; 0:2); (y4 ; 0:7; 0:1); (y5 ; 0:3; 0:6)}: Consider, for instance, the following intuitionistic fuzzy implicator: I(a; b) = (min(1; a2 + b1 ); max(0; a1 + b2 − 1));
∀a; b ∈ L∗ :
Then R /I bk S(x; z) = (1; 0). This means that the degree to which all symptoms shown by patient x are symptoms of illness z is equal to 1, although patient x is only showing symptom y4 to degree 0:1, and with only an uncertainty given by 0:05. Such surprising results are the consequence of the fact that the Bandler and Kohout compositions do not take into account the degree of emptiness or non-emptyness of the foresets and the aftersets involved. For the improved versions we get R /I b S(x; z) = inf {(1; 0); (0:1; 0:85); (1; 0)} = (0:1; 0:85) and, if we use for example the Lukasiewicz t-norm and its dual t-conorm, de1ned by, for all a; b ∈ [0; 1], a ∩W b = max(0; a + b − 1) and a +b b = min(1; a + b), respectively, R /∩k W ;+b ;I S(x; z) = inf {(1; 0); (0; 0:95)} = (0; 0:95). These results correspond better to intuition. Example 4.2. Let X , Y , Z, R and S be as in the previous example. Assume that a therapist wants to 1nd out to which degree a patient x suMers from illness z. The therapist determines to which degree the patient shows the symptoms y1 ; : : : ; y5 and for each symptom he gives a degree of how sure he is about that degree. Assume he gets the following table (in the 1rst row is written to which degree patient x shows each symptom, in the second row is written the degree of how sure the therapist is when attributing the degree in the 1rst row in the same column): y1
y2
y3
y4
y5
Degree of showing symptom
0.9
0.5
0.4
0.7
0.5
Degree of being sure
1.0
0.8
0.9
0.9
0.6
The 1rst row contains the membership degrees of (x; yi ) to R, for each i ∈ {1; : : : ; 5}, while, if we take the fuzzy negation of each element in the second row, then we get the degree of uncertainty to which (x; yi ) ∈ R, for each i ∈ {1; : : : ; 5}. For instance, for y2 , the membership degree of (x; y2 ) to R is given by R (x; y2 ) = 0:5 and the degree of uncertainty is given by R (x; y2 ) = 1−0:8 = 0:2. We obtain immediately the degree of non-membership as R (x; y2 ) = 1 − R (x; y2 ) − R (x; y2 ) = 0:3. Applying
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the same method for the other symptoms y1 ; y3 ; : : : ; y5 , we obtain the following IFS modelling the degree to which patient x shows the symptoms in Y : xR = {(y1 ; 0:9; 0:1); (y2 ; 0:5; 0:3); (y3 ; 0:4; 0:5); (y4 ; 0:7; 0:2); (y5 ; 0:5; 0:1)}: Similarly an IFS Sz can be constructed, modelling the degree to which the symptoms in Y are characteristic for illness z: Sz = {(y1 ; 1; 0); (y2 ; 0:7; 0:2); (y3 ; 0:2; 0:6); (y4 ; 0:7; 0:1); (y5 ; 0:3; 0:6)}: min Let T = min, S = max, I(a; b) = sup(Ns (a); b). Then we obtain for instance, R sup inf ◦max S(x; z) = I I (0:9; 0:1), R / b S(x; z) = (0:3; 0:5), and R .b S(x; z) = (0:4; 0:5). This means that the degree to which patient x shows at least one symptom of illness z is equal to 0:9 with no hesitation, the degree to which all symptoms shown by x are symptoms of z is equal to 0:3 with uncertainty 1−0:3−0:5 = 0:2, and the degree to which x shows all symptoms of z equals 0:4 with uncertainty 0:1. If we only min consider the sup inf ◦max composition to determine if a patient suMers from a certain disease, then we obtain a high degree, namely 0:9, with no uncertainty. If we also take into account the sub- and supercomposition, then we get a more nuanced view.
5. Properties of the intuitionistic triangular compositions In this section we will investigate the following properties of the intuitionistic triangular compositions: containment in the composition de1ned by Burillo and Bustince, convertibility, monotonicity, interaction with union and intersection. 5.1. On the containment of the intuitionistic compositions Theorem 5.1. Let R ∈ IFR(X × Y ), S ∈ IFR(Y × Z), I be an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Then R /kT
;S ;I
T S ⊆ R sup inf ◦S S;
R .kT
;S ;I
T S ⊆ R sup inf ◦S S;
R /kT
;S ;I
S ⊆ R /bI S;
R .kT
;S ;I
S ⊆ R .bI S:
sup T A general containment rule between R /I b S and R inf ◦S S does not exist. Consider for instance the IFRs R = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:5; 0:5); ((x2 ; y1 ); 0:9; 0:1); ((x2 ; y2 ); 0:9; 0:1)} and S = {((y1 ; z); 0:2; 0:8); ((y2 ; z); 0:1; 0:9)} from X = {x1 ; x2 } to Y = {y1 ; y2 } and from Y to Z = {z} respectively. Let the intuitionistic fuzzy implicator I be such that I(a; b) = Ns (a) ∨ b, ∀a; b ∈ L∗ and let T = min
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and S = max. Then we obtain T R /bI S(x1 ; z) = (0:2; 0:8) ¿L∗ R sup inf ◦S S(x1 ; z) = (0:1; 0:9)
and T R /bI S(x2 ; z) = (0:1; 0:9) ¡L∗ R sup inf ◦S S(x2 ; z) = (0:2; 0:8):
sup T I Similarly there is no containment rule between R /I bk S and R inf ◦S S(x1 ; z), since R / bk S(x1 ; z) = (0:5; sup T sup T I 0:5)¿L∗ R inf ◦S S(x1 ; z) and R .bk S(x2 ; z) = (0:1; 0:9)¡L∗ R inf ◦S S(x1 ; z). In the crisp case we have, using / as composition, that the number of pairs (x; z) ∈ X × Z for which all the y ∈ Y that are in relation Rt with x are also in relation S with z, is smaller than the number of pairs (x; z) for which there exists at least one y that is in relation Rt with x and in relation S with z. The theorem shows that this intuitively natural property still holds for intuitionistic fuzzy relations, but only for /Tk ; S ; I . We then have that for any (x; z) ∈ X × Z, the degree to which all the y ∈ Y that are in relation Rt with x are also in relation with z, is smaller than the degree to which there exist at least one y that is in relation Rt with x and in relation S with z.
5.2. On the convertibility of the intuitionistic triangular compositions The formulas in the following theorem allow us to write the converse of the triangular compositions in terms of the converse relations of the composing relations. Theorem 5.2. Let R ∈ IFR(X × Y ), S ∈ IFR(Y × Z), I be an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Then: I I t S)t = S t .bk R; (R /bk I I t (R .bk S)t = S t /bk R;
(R /bI S)t = S t .bI Rt ; (R .bI S)t = S t /bI Rt ; (R /kT
; S; I
S)t = S t .kT
; S; I
Rt ;
(R .kT
; S; I
S)t = S t /kT
; S; I
Rt :
Proof. Notice that (R /Tk (R /kT
; S; I
; S; I
S)t ∈ IFR(Z × X ). Hence we calculate, for x ∈ X and z ∈ Z:
S)t (z; x) = R /kT ; S ; I S(x; z)
= inf
inf I(R(x; y); S(y; z)); sup(R(x; y) ∧T ;S S(y; z))
y ∈Y
y ∈Y
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= inf = S t .kT
345
t
t
t
t
inf I(R (y; x); S (z; y)); sup(S (z; y) ∧T ;S R (y; x))
y ∈Y
;S ;I
y ∈Y
Rt (z; x):
The other equalities are proved in a similar way. 5.3. On the monotonicity of the intuitionistic triangular compositions We assume in this subsection that R; R1 ; R2 ∈ IFR(X × Y ), S; S1 ; S2 ∈ IFR(Y × Z), I is an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Lemma 5.1. If R1 ⊆ R2 , then, for all (x; z) ∈ X × Z, hgt(xR1 ) 6L∗ hgt(xR2 ); hgt(xR1 ∩T ;S Sz) 6L∗ hgt(xR2 ∩T ;S Sz):
Theorem 5.3. If I satis7es (M.1), then: dom(R1 ) = dom(R2 ) ∧ R1 ⊆ R2 ⇒ R1 /bI S ⊇ R2 /bI S: If I satis7es (M.2), then: R1 ⊆ R2 ⇒ R1 .bI S ⊆ R2 .bI S; R1 ⊆ R2 ⇒ R1 .kT
;S ;I
S ⊆ R2 .kT
;S ;I
S:
In the case of /kT ; S ; I , from dom(R1 ) = dom(R2 ) a monotonicity rule cannot be deduced. Consider, for instance, the relations R1 and R2 from X = {x1 } to Y = {y1 ; y2 } and S from Y to Z = {z1 ; z2 }, and the intuitionistic fuzzy implicator I de1ned as follows:
R1 = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:6; 0:4)}; R2 = {((x1 ; y1 ); 0:5; 0:5); ((x1 ; y2 ); 0:6; 0:4)}; S = {((y1 ; z1 ); 0:2; 0:8); ((y2 ; z1 ); 0:1; 0:9); ((y1 ; z2 ); 0:2; 0:8); ((y2 ; z2 ); 0:9; 0:1)}; I(a; b) = Ns (a) ∨ b = (max(a2 ; b1 ); min(a1 ; b2 ));
∀a = (a1 ; a2 ); b = (b1 ; b2 ) ∈ L∗ :
Then R1 /kmin; max; I S(x1 ; z1 ) = (0:1; 0:9)¡L∗ R2 /kmin; max; I S(x1 ; z1 ) = (0:2; 0:8) and R1 /kmin; max; I S(x1 ; z2 ) = (0:6; 0:4)¿L∗ R2 /kmin; max; I S(x1 ; z2 ) = (0:5; 0:5).
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The reason for this is the fact that from dom(R1 ) = dom(R2 ) it does not follow hgt(xR1 ∩T ; S Sz) = hgt(xR2 ∩T ; S Sz), ∀(x; z) ∈ X × Z. In the above example we have dom(R1 ) = dom(R2 ), but hgt(x1 R1 ∩ Sz1 ) = (0:1; 0:9)¡L∗ (0:2; 0:8) = hgt(x1 R2 ∩ Sz1 ). Similarly, there is a monotonicity property for the second component of the diMerent compositions.
Theorem 5.4. If I satis7es (M.1), then dom(S1 ) = dom(S2 ) ∧ S1 ⊆ S2 ⇒ R .bI S1 ⊇ R .bI S2 : If I satis7es (M.2), then S1 ⊆ S2 ⇒ R /bI S1 ⊆ R /bI S2 ; S1 ⊆ S2 ⇒ R /kT
; S; I
S1 ⊆ R /kT
; S; I
S2 :
5.4. On the interaction of the intuitionistic triangular compositions with the union In this subsection we will assume that R; R1 ; R2 ∈ IFR(X × Y ), S; S1 ; S2 ∈ IFR(Y × Z), I is an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. We start with some lemmas. Lemma 5.2. For arbitrary R1 ; R2 ∈ IFR(X × Y ), it holds dom(R1 ∪ R2 ) = dom(R1 ) ∪ dom(R2 ): As a corollary of this lemma it follows that, if dom(R1 ) = dom(R2 ) then dom(R1 ∪ R2 ) = dom(R1 ) = dom(R2 ). Eq. (2) in the following theorem will be a useful tool for proving some of the interactions between the triangular compositions and the union. Lemma 5.3. If I satis7es the following condition: I(sup(a; b); c) = inf (I(a; c); I(b; c)):
(2)
then it satis7es (M.1). If I satis7es (M.1), then I(sup(a; b); c) 6L∗ inf (I(a; c); I(b; c)):
(3)
Note that in fuzzy set theory, Eq. (2) holds for any fuzzy implicator which is decreasing in its 1rst component. For intuitionistic fuzzy implicators, this is not the case. Let, for instance, I be the intuitionistic fuzzy implicator de1ned by, for all x; y ∈ L∗ , (Ns x ∨+;ˆ · y) ∧·;+ˆ y if x1 ¿ x2 ; I(x; y) = Ns x ∨+;ˆ · y else;
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347
ˆ the probabilistic sum de1ned as, for m; n ∈ [0; 1], m+n ˆ = m + n − m · n. where · is the product, and + Let a = (0:3; 0:2), b = (0:4; 0:5) and c = (0:2; 0:8), then sup(a; b) = (0:4; 0:2) and I(a; c) = ((0:2; 0:3) ∨+;ˆ · (0:2; 0:8)) ∧·;+ˆ (0:2; 0:8) = (0:36; 0:24) ∧·;+ˆ (0:2; 0:8) = (0:072; 0:848); I(b; c) = (0:5; 0:4) ∨+;ˆ · (0:2; 0:8) = (0:6; 0:32); I(sup(a; b); c) = ((0:2; 0:4) ∨+;ˆ · (0:2; 0:8)) ∧·;+ˆ (0:2; 0:8) = (0:36; 0:32) ∧·;+ˆ (0:2; 0:8) = (0:072; 0:864): Hence I(sup(a; b); c)¡L∗ inf {I(a; c); I(b; c)} = I(a; c). Two important classes of intuitionistic fuzzy implicators satisfy the condition (2), the S-implicators and the R-implicators, both de1ned in [9]. Theorem 5.5. Let T be a t-norm, S a t-conorm and N a unary involutive order reversing operator on L∗ . The S-implicator IS; T; N , de7ned as [9] IS;T;N (x; y) = N(x) ∨S;T y;
∀(x; y) ∈ (L∗ )2 ;
satis7es formula (2). Theorem 5.6. Let T be a t-norm and S a t-conorm. The R-implicator IT; S , de7ned as [9] IT;S (x; y) = sup{# ∈ L∗ | x ∧T;S # 6L∗ y} satis7es formula (2). Now we present the diMerent interactions between the triangular compositions and the union. It will be seen that only some semi-distributivity properties hold. Theorem 5.7. If dom R1 = dom R2 and I satis7es (M.1), then (R1 /bI S) ∩ (R2 /bI S) ⊇ (R1 ∪ R2 ) /bI S: If I satis7es formula (2) and not necessarily dom R1 = dom R2 , then (R1 /bI S) ∩ (R2 /bI S) ⊆ (R1 ∪ R2 ) /bI S; (R1 /kT
;S ;I
S) ∩ (R2 /kT
;S ;I
S) ⊆ (R1 ∪ R2 ) /kT
;S ;I
S:
Theorem 5.8. If dom S1 = dom S2 and I satis7es (M.1), then (R .bI S1 ) ∩ (R .bI S2 ) ⊇ R .bI (S1 ∪ S2 ):
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If I satis7es formula (2) and not necessarily dom S1 = dom S2 , then (R .bI S1 ) ∩ (R .bI S2 ) ⊆ R .bI (S1 ∪ S2 ); (R .kT
;S ;I
S1 ) ∩ (R .kT
;S ;I
S2 ) ⊆ R .kT
;S ;I
(S1 ∪ S2 ):
Lemma 5.4. For arbitrary R1 ; R2 ∈ IFR(X × Y ), S ∈ IFR(Y × Z) we have sup(hgt(xR1 ∩T ;S Sz); hgt(xR2 ∩T ;S Sz)) = hgt(x(R1 ∪ R2 ) ∩T ;S Sz): Proof. This property is equivalent to sup T sup T T (R1 sup inf ◦S S) ∪ (R2 inf ◦S S)(x; z) = (R1 ∪ R2 ) inf ◦S S(x; z);
which is proved in [7, Theorem 7]. Theorem 5.9. If I satis7es (M.1) and not necessarily dom R1 = dom R2 , then (R1 ∪ R2 ) /bI S ⊆ (R1 /bI S) ∪ (R2 /bI S); (R1 ∪ R2 ) /kT
;S ;I
S ⊆ (R1 /kT
;S ;I
S) ∪ (R2 /kT
;S ;I
S):
Theorem 5.10. If I satis7es (M.1) and not necessarily dom S1 = dom S2 , then R .bI (S1 ∪ S2 ) ⊆ (R .bI S1 ) ∪ (R .bI S2 ); R .kT
;S ;I
(S1 ∪ S2 ) ⊆ (R .kT
;S ;I
S1 ) ∪ (R .kT
;S ;I
S2 ):
Using the above theorems, we may also conclude the following: • if I satis1es formula (2), then (R1 /bI S) ∩ (R2 /bI S) ⊆ (R1 ∪ R2 ) /bI S ⊆ (R1 /bI S) ∪ (R2 /bI S) and similarly for /kT ; S ; I ; • if I satis1es formula (2) and dom(R1 ) = dom(R2 ), then
(R1 /bI S) ∩ (R2 /bI S) = (R1 ∪ R2 ) /bI S; • if I satis1es formula (2), then (R .bI S1 ) ∩ (R .bI S2 ) ⊆ R .bI (S1 ∪ S2 ) ⊆ (R .bI S1 ) ∪ (R .bI S2 ) and similarly for .kT
; S; I
;
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349
• if I satis1es formula (2) and dom(S1 ) = dom(S2 ), then (R .bI S1 ) ∩ (R .bI S2 ) = R .bI (S1 ∪ S2 ): 5.5. On the interaction of the intuitionistic triangular compositions with the intersection Also in this section it will become clear that the intuitionistic triangular compositions only satisfy some kind of semi-distributivity w.r.t. the intuitionistic fuzzy intersection. Theorem 5.11. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. If I satis7es (M.2) and not necessarily dom R1 = dom R2 , then (R1 ∩ R2 ) .bI S ⊆ (R1 .bI S) ∩ (R2 .bI S); (R1 ∩ R2 ) .kT
;S ;I
S ⊆ (R1 .kT
;S ;I
S) ∩ (R2 .kT
;S ;I
S):
Theorem 5.12. Let R be an IFR from X to Y , S1 and S2 be IFRs from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. If I satis7es (M.2) and not necessarily dom S1 = dom S2 , then R /bI (S1 ∩ S2 ) ⊆ (R /bI S1 ) ∩ (R /bI S2 ); R /kT
;S ;I
(S1 ∩ S2 ) ⊆ (R /kT
;S ;I
S1 ) ∩ (R /kT
;S ;I
S2 ):
In a similar way as Lemma 5.3, the following lemma is proved. Lemma 5.5. If I satis7es the following condition: I(inf (a; b); c) = sup(I(a; c); I(b; c))
(4)
then it satis7es (M.1). If I satis7es (M.1), then I(inf (a; b); c) ¿L∗ sup(I(a; c); I(b; c)):
(5)
In a similar way as above it is proved that S-implicators satisfy condition (4). However R-implicators do in general not satisfy condition (4). Consider Imin; max (x; y) = sup{# ∈ L∗ | (min(x1 ; #1 ); max(x2 ; #2 ))6L∗ y}, and let a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ such that a1 6c1 ¡b1 and a2 ¡c2 6b2 . Then inf (a; b) = (a1 ; b2 ), so we obtain Imin; max (a; c) = (1−c2 ; c2 ), Imin; max (b; c) = (c1 ; 0) and Imin; max ((a1 ; b2 ); c) = (1; 0). Hence Imin; max (inf (a; b); c)¿L∗ sup(Imin; max (a; c); Imin; max (b; c)). Theorem 5.13. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z and I an intuitionistic fuzzy implicator. If I satis7es (M.1) then (R1 /bk S) ∪ (R2 /bk S) ⊆ (R1 ∩ R2 ) /bk S
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Even if dom(R1 ) = dom(R2 ) and if I satis1es condition (4), there exists in general no relation beI I I I tween (R1 /I b S) ∩ (R2 / b S), (R1 / b S) ∪ (R2 / b S) and (R1 ∩ R2 ) / b S. See appendix for more details. Appendix: Proofs and examples A.1. The containment of the intuitionistic compositions Theorem 5.1. Let R ∈ IFR(X × Y ), S ∈ IFR(Y × Z), I be an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Then R /kT
;S ;I
T S ⊆ R sup inf ◦S S;
R .kT
;S ;I
T S ⊆ R sup inf ◦S S;
R /kT
;S ;I
S ⊆ R /bI S;
R .kT
;S ;I
S ⊆ R .bI S:
Proof. Let x ∈ X , z ∈ Z, then R /kT
;S ;I
S(x; z) = inf {plt I(xR; Sz); hgt(xR ∩T ;S Sz)} 6L∗ hgt(xR ∩T ;S Sz) =
sup T (R (x; y); S (y; z)); inf S ( R (x; y); S (y; z)) y ∈Y
y ∈Y
T = R sup inf ◦S S(x; z):
The second inclusion is proved in an analogous way. To prove the last two inclusions, we observe that xR ∩T ; S Sz ⊆ xR, and thus hgt(xR ∩T ; S Sz)6L∗ hgt(xR). Analogously we obtain hgt(xR ∩T ; S Sz)6L∗ hgt(Sz). Hence hgt(xR ∩T ; S Sz)6L∗ inf {hgt (xR); hgt(Sz)}. Now the inclusions follow easily. sup T To prove that there does not exist a general containment rule between R /I b S and R inf ◦S S, we consider the IFRs R = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:5; 0:5); ((x2 ; y1 ); 0:9; 0:1); ((x2 ; y2 ); 0:9; 0:1)} and S = {((y1 ; z); 0:2; 0:8); ((y2 ; z); 0:1; 0:9)} from X = {x1 ; x2 } to Y = {y1 ; y2 } and from Y to Z = {z} respectively. Let the intuitionistic fuzzy implicator I be such that I(a; b) = Ns (a) ∨ b, ∀a; b ∈ L∗ and let T = min and S = max. Then
R /bI S(x1 ; z) = inf {inf ((0:9; 0:1); (0:5; 0:5)); sup((0:1; 0:9); (0:5; 0:5)); sup((0:2; 0:8); (0:1; 0:9))} = inf {(0:5; 0:5); (0:5; 0:5); (0:2; 0:8)} = (0:2; 0:8);
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351
T R sup inf ◦S S(x1 ; z) = sup((0:1; 0:9); (0:1; 0:9)) = (0:1; 0:9);
R /bI S(x2 ; z) = inf {inf ((0:2; 0:8); (0:1; 0:9)); sup((0:9; 0:1); (0:9; 0:1)); sup((0:2; 0:8); (0:1; 0:9))} = inf {(0:1; 0:9); (0:9; 0:1); (0:2; 0:8)} = (0:1; 0:9); T R sup inf ◦S S(x2 ; z) = sup((0:2; 0:8); (0:1; 0:9)) = (0:2; 0:8):
A.2. On the monotonicity of the intuitionistic triangular compositions Lemma 5.1. If R1 ⊆ R2 , then, for all (x; z) ∈ X × Z, hgt(xR1 ) 6L∗ hgt(xR2 ); hgt(xR1 ∩T ;S Sz) 6L∗ hgt(xR2 ∩T ;S Sz): Proof. We prove the second inequality. Let x ∈ X , z ∈ Z. From R1 ⊆ R2 it follows R1 (x; y)6L∗ R2 (x; y), ∀y ∈ Y , and so R1 (x; y) ∧T ; S S(y; z)6L∗ R2 (x; y) ∧T ; S S(y; z), ∀y ∈ Y . From the monotonicity of sup we obtain sup(R1 (x; y) ∧T ;S S(y; z)) 6L∗ sup(R2 (x; y) ∧T ;S S(y; z)): Theorem 5.3. If I satis7es (M.1), then dom(R1 ) = dom(R2 ) ∧ R1 ⊆ R2 ⇒ R1 /bI S ⊇ R2 /bI S: If I satis7es (M.2), then: R1 ⊆ R2 ⇒ R1 .bI S ⊆ R2 .bI S; R1 ⊆ R2 ⇒ R1 .kT
;S ;I
S ⊆ R2 .kT
;S ;I
S:
Proof. Choose arbitrary x ∈ X and z ∈ Z. From dom(R1 ) = dom(R2 ) it follows hgt(xR1 ) = hgt(xR2 ), so from R1 ⊆ R2 and (M.1) we obtain consequently R1 (x; y) 6L∗ R2 (x; y);
∀y ∈ Y;
I(R1 (x; y); S(y; z)) ¿L∗ I(R2 (x; y); S(y; z));
∀y ∈ Y;
inf I(R1 (x; y); S(y; z)) ¿L∗ inf I(R2 (x; y); S(y; z));
y ∈Y
y ∈Y
R1 /bI S ⊇ R2 /bI S: Analogously, using Lemma 5.1, the second statement is proved.
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In the case of /kT ; S ; I , from dom(R1 ) = dom(R2 ) a monotonicity rule cannot be deduced. Consider, for instance, the relations R1 and R2 from X = {x1 } to Y = {y1 ; y2 } and S from Y to Z = {z1 ; z2 }, and the intuitionistic fuzzy implicator I de1ned as follows:
R1 = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:6; 0:4)}; R2 = {((x1 ; y1 ); 0:5; 0:5); ((x1 ; y2 ); 0:6; 0:4)}; S = {((y1 ; z1 ); 0:2; 0:8); ((y2 ; z1 ); 0:1; 0:9); ((y1 ; z2 ); 0:2; 0:8); ((y2 ; z2 ); 0:9; 0:1)}; I(a; b) = Ns (a) ∨ b = (max(a2 ; b1 ); min(a1 ; b2 ));
∀a = (a1 ; a2 ); b = (b1 ; b2 ) ∈ L∗ :
Then R1 /kmin;max;I S(x1 ; z1 ) = inf {inf ((0:9; 0:1); (0:4; 0:6)); sup((0:1; 0:9); (0:1; 0:9))} = (0:1; 0:9); R2 /kmin;max;I S(x1 ; z1 ) = inf {inf ((0:5; 0:5); (0:4; 0:6)); sup((0:2; 0:8); (0:1; 0:9))} = (0:2; 0:8); R1 /kmin;max;I S(x1 ; z2 ) = inf {inf ((0:9; 0:1); (0:9; 0:1)); sup((0:1; 0:9); (0:6; 0:4))} = (0:6; 0:4); R2 /kmin;max;I S(x1 ; z2 ) = inf {inf ((0:5; 0:5); (0:9; 0:1)); sup((0:2; 0:8); (0:6; 0:4))} = (0:5; 0:5): A.3. On the interaction of the intuitionistic triangular compositions with the union Lemma 5.2. For arbitrary R1 ; R2 ∈ IFR(X × Y ), it holds dom(R1 ∪ R2 ) = dom(R1 ) ∪ dom(R2 ): Proof. Let x ∈ X . Then we obtain: dom(R1 ∪ R2 )(x) = hgt x(R1 ∪ R2 ) = sup(R1 ∪ R2 )(x; y) y ∈Y
= sup(sup(R1 (x; y); R2 (x; y))) y ∈Y
= sup(sup(R1 (x; y); R2 (x; y)); inf ( R1 (x; y); R2 (x; y))) y ∈Y
= sup[sup(R1 (x; y); R2 (x; y))]; inf [inf ( R1 (x; y); R2 (x; y))] y ∈Y
y ∈Y
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353
= sup sup R1 (x; y); sup R2 (x; y) ; inf inf R1 (x; y); inf R2 (x; y) y ∈Y
y ∈Y
y ∈Y
y ∈Y
= sup sup R1 (x; y); sup R2 (x; y) y ∈Y
y ∈Y
= sup(hgt xR1 ; hgt xR2 ) = sup(dom(R1 )(x); dom(R2 )(x)): Lemma 5.3. If I satis7es the following condition: I(sup(a; b); c) = inf (I(a; c); I(b; c)):
(2)
then it satis7es (M.1). If I satis7es (M.1), then I(sup(a; b); c) 6L∗ inf (I(a; c); I(b; c)):
(3)
Proof. Let I be an intuitionistic fuzzy implicator which satis1es formula (2). Then, for all x; x ; y ∈ L∗ , from x6L∗ x follows I(x ; y) = I(sup(x; x ); y) = inf (I(x; y); I(x ; y)), i.e. I(x ; y)6L∗ I(x; y). Hence (M.1) is veri1ed. Let now I satisfy (M.1). Then, for all a; b; c ∈ L∗ , from a6L∗ sup(a; b) follows I(a; c)¿L∗ I(sup (a; b); c). Similarly we obtain I(b; c)¿L∗ I(sup(a; b); c). Hence inf (I(a; c); I(b; c))¿L∗ I(sup (a; b); c). Theorem 5.5. Let T be a t-norm, S a t-conorm and N a unary involutive order reversing operator on L∗ . The S-implicator IS; T; N , de7ned as [9] IS;T;N (x; y) = N(x) ∨S;T y;
∀(x; y) ∈ (L∗ )2 ;
satis7es formula (2). Proof. Let a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ . De1ne a = (a1 ; a2 ) = N(a) and b = (b1 ; b2 ) = N(b). Since by de1nition inf (N(a); N(b)) is the largest element in L∗ that is smaller than both N(a) and N(b), we have (∀d ∈ L∗ )
(d 6L∗ N(a) ∧ d 6L∗ N(b) ⇒ d 6L∗ inf (N(a); N(b))):
Since N is order reversing, this is equivalent to (∀d ∈ L∗ )
(N(d) ¿L∗ a ∧ N(d) ¿L∗ b ⇒ N(d) ¿L∗ N(inf (N(a); N(b)))):
Since N is involutive (and thus bijective), this is equivalent to (∀d ∈ L∗ )
(d ¿L∗ a ∧ d ¿L∗ b ⇒ d ¿L∗ N(inf (N(a); N(b)))):
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Hence sup(a; b) = N(inf (N(a); N(b))), or N(sup(a; b)) = inf (N(a); N(b)). Using this equality, we obtain successively: IS;T;N (sup(a; b); c) = N(sup(a; b)) ∨S;T c = inf (N(a); N(b)) ∨S;T c = (S(inf (a1 ; b1 ); c1 ); T (sup(a2 ; b2 ); c2 )) = (S(min(a1 ; b1 ); c1 ); T (max(a2 ; b2 ); c2 )) = (min(S(a1 ; c1 ); S(b1 ; c1 )); max(T (a2 ; c2 ); T (b2 ; c2 ))) = inf (N(a) ∨S;T c; N(b) ∨S;T c) = inf (IS;T;N (a; c); IS;T;N (b; c)): Theorem 5.6. Let T be a t-norm and S a t-conorm. The R-implicator IT; S , de7ned as [9] IT;S (x; y) = sup{# ∈ L∗ | x ∧T;S # 6L∗ y} satis7es formula (2). Proof. Let a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ . {# = (#1 ; #2 ) ∈ L∗ | (T (max(a1 ; b1 ); #1 ); S(min(a2 ; b2 ); #2 )) 6L∗ c} = {# ∈ L∗ | (max(T (a1 ; #1 ); T (b1 ; #1 )); min(S(a2 ; #2 ); S(b2 ; #2 ))) 6L∗ c} = {# ∈ L∗ | T (a1 ; #1 ) 6 c1 ∧ T (b1 ; #1 ) 6 c1 ∧ S(a2 ; #2 ) ¿ c2 ∧ S(b2 ; #2 ) ¿ c2 } = {# ∈ L∗ | T (a1 ; #1 ) 6 c1 ∧ S(a2 ; #2 ) ¿ c2 } ∩ {# ∈ L∗ | T (b1 ; #1 ) 6 c1 ∧ S(b2 ; #2 ) ¿ c2 }: In the last formula, the symbol ∩ represents the intersection of two crisp sets. De1ne A1 = {# ∈ L∗ | T (a1 ; #1 )6c1 ∧ S(a2 ; #2 )¿c2 }, A2 = {# ∈ L∗ | T (b1 ; #1 )6c1 ∧ S(b2 ; #2 )¿c2 }, then IT;S (sup(a; b); c) = sup(A1 ∩ A2 ): From # ∈ A1 and # 6L∗ # follows # ∈ A1 , since ∧T; S is increasing. On the other hand, from # ∈ A1 and # ∈ A1 follows T (a1 ; #1 )6c1 , T (a1 ; #1 )6c1 , S(a2 ; #2 )¿c2 and S(a2 ; #2 )¿c2 , which implies T (a1 ; max(#1 ; #1 )) = max(T (a1 ; #1 ); T (a1 ; #1 ))6c1 and S(a2 ; min(#2 ; #2 )) = min(S(a2 ; #2 ); S(a2 ; #2 ))¿c2 . Hence sup(#; # ) ∈ A1 . It is now easy to see that A1 must have the form as in Fig. 2. Let = (1 ; 2 ) = sup(A1 ) and = (1 ; 2 ) = sup(A2 ). It is easily seen that ∧ is an upper bound for A1 ∩ A2 . Since {# ∈ L∗ | #1 ¡1 ∧ #2 ¿2 } ⊆ A1 and similarly {# ∈ L∗ | #1 ¡1 ∧ #2 ¿2 } ⊆ A2 , we
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355
Fig. 2. The set A1 = {# ∈ L∗ | T (a1 ; #1 )6c1 ∧ S(a2 ; #2 )¿c2 }.
obtain that A3 = {# ∈ L∗ | #1 ¡ min(1 ; 1 ) ∧ #2 ¿ max(2 ; 2 )} ⊆ A1 ∩ A2 . Hence it is clear that ∧ is the smallest upper bound, i.e. the supremum, of A1 ∩ A2 . It follows that IT;S (sup(a; b); c) = sup(A1 ∩ A2 ) = sup(A1 ) ∧ sup(A2 ) = inf (IT;S (a; c); IT;S (b; c)): Theorem 5.7. If dom R1 = dom R2 and I satis7es (M.1), then (R1 /bI S) ∩ (R2 /bI S) ⊇ (R1 ∪ R2 ) /bI S: If I satis7es formula (2) and not necessarily dom R1 = dom R2 , then (R1 /bI S) ∩ (R2 /bI S) ⊆ (R1 ∪ R2 ) /bI S; (R1 /kT
;S ;I
S) ∩ (R2 /kT
;S ;I
S) ⊆ (R1 ∪ R2 ) /kT
;S ;I
S:
Proof. Since R1 ⊆ R1 ∪ R2 and the same holds for R2 , we obtain by the corollary of Lemma 5.2 and I the monotonicity of the triangular subcomposition that R1 /I b S ⊇ (R1 ∪ R2 ) / b S and similarly for R2 . The 1rst statement follows easily. Let I be an intuitionistic fuzzy implicator satisfying formula (2). Then we obtain successively: inf (plt I(xR1 ; Sz); plt I(xR2 ; Sz)) = inf
inf I(R1 (x; y); S(y; z)); inf I(R2 (x; y); S(y; z))
y ∈Y
y ∈Y
= inf (inf (I(R1 (x; y); S(y; z)); I(R2 (x; y); S(y; z)))) y ∈Y
= inf (I(sup(R1 (x; y); R2 (x; y)); S(y; z))) y ∈Y
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= inf (I(R1 ∪ R2 (x; y); S(y; z))) y ∈Y
= plt I(x(R1 ∪ R2 ); Sz): Using the previous equality, we obtain (R1 /bI S) ∩ (R2 /bI S)(x; z) = inf (inf {plt I(xR1 ; Sz); plt I(xR2 ; Sz)}; inf {hgt(xR1 ); hgt(xR2 )}; hgt(Sz)); 6L∗ inf (plt I(x(R1 ∪ R2 ); Sz); sup{hgt(xR1 ); hgt(xR2 )}; hgt(Sz)) = inf (plt I(x(R1 ∪ R2 ); Sz); hgt(x(R1 ∪ R2 )); hgt(Sz)) = (R1 ∪ R2 ) /bI S(x; z): Since xR1 ⊆ x(R1 ∪ R2 ) and the t-norm T , the t-conorms S and sup are increasing, we obtain inf {hgt(xR1 ∩T ;S Sz); hgt(xR2 ∩T ;S Sz)} 6L∗ hgt(xR1 ∩T ;S Sz) 6L∗ hgt(x(R1 ∪ R2 ) ∩T ;S Sz): In a similar way as above, the last inequality can now be proved. Theorem 5.9. If I satis7es (M.1) and not necessarily dom R1 = dom R2 , then (R1 ∪ R2 ) /bI S ⊆ (R1 /bI S) ∪ (R2 /bI S); (R1 ∪ R2 ) /kT
;S ;I
S ⊆ (R1 /kT
;S ;I
S) ∪ (R2 /kT
;S ;I
S):
Proof. For all a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ we have sup(inf (a; b); inf (a; c)) = (max(min(a1 ; b1 ); min(a1 ; c1 )); min(max(a2 ; b2 ); max(a2 ; c2 ))) = (min(a1 ; max(b1 ; c1 )); max(a2 ; min(b2 ; c2 ))) = inf (a; sup(b; c)): Now we can prove the 1rst inequality: (R1 /bI S) ∪ (R2 /bI S)(x; z) = sup(inf {plt I(xR1 ; Sz); hgt(xR1 ); hgt(Sz)}; inf {plt I(xR2 ; Sz); hgt(xR2 ); hgt(Sz)})
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357
¿L∗ sup(inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR1 ); hgt(Sz)}; inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR2 ); hgt(Sz)}) = sup(inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; hgt(xR1 )}; inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; hgt(xR2 )}) = inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; sup[hgt(xR1 ); hgt(xR2 )]} = inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; hgt x(R1 ∪ R2 )} = (R1 ∪ R2 ) /bI S(x; z): The second inequality is proved similarly in the following way: (R1 /kT
; S; I
S) ∪ (R2 /kT
; S; I
S)(x; z)
= sup(inf {plt I(xR1 ; Sz); hgt(xR1 ∩T ;S Sz)}; inf {plt I(xR2 ; Sz); hgt(xR2 ∩T ;S Sz)}) ¿L∗ sup(inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR1 ∩T ;S Sz)}; inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR2 ∩T ;S Sz)}) = inf (plt I(x(R1 ∪ R2 ); Sz); sup[hgt(xR1 ∩T ;S Sz); hgt(xR2 ∩T ;S Sz)]) = inf {plt I(x(R1 ∪ R2 ); Sz); hgt(x(R1 ∪ R2 ) ∩T ;S Sz)} = (R1 ∪ R2 ) /kT
; S; I
S(x; z):
A.4. On the interaction of the intuitionistic triangular compositions with the intersection Theorem 5.11. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. If I satis7es (M.2) and not necessarily
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dom R1 = dom R2 , then (R1 ∩ R2 ) .bI S ⊆ (R1 .bI S) ∩ (R2 .bI S); (R1 ∩ R2 ) .kT
;S ;I
S ⊆ (R1 .kT
;S ;I
S) ∩ (R2 .kT
;S ;I
S):
Proof. Notice that inf (I(a; b); I(a; c))¿L∗ I(a; inf (b; c)), for all a; b; c ∈ L∗ , if I satis1es (M.2). Let (x; z) ∈ X × Z. Then (R1 .bI S) ∩ (R2 .bI S)(x; z) = inf {inf [plt I(Sz; xR1 ); hgt(xR1 ); hgt(Sz)]; inf [plt I(Sz; xR2 ); hgt(xR2 ); hgt(Sz)]} = inf {inf [plt I(Sz; xR1 ); plt I(Sz; xR2 )]; inf [hgt(xR1 ); hgt(xR2 )]; hgt(Sz)} ¿L∗ inf {plt I(Sz; x(R1 ∩ R2 )); hgt(x(R1 ∩ R2 )); hgt(Sz)} = (R1 ∩ R2 ) .bI S(x; z): In a similar way we can prove the second inequality, since hgt(x(R1 ∩ R2 ) ∩T ; S S)6L∗ inf (hgt(xR1 ∩T ; S S); hgt(xR2 ∩T ; S S)). Theorem 5.13. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z and I an intuitionistic fuzzy implicator. If I satis7es (M.1) then (R1 /bk S) ∪ (R2 /bk S) ⊆ (R1 ∩ R2 ) /bk S: Proof. If I satis1es (M.1), then it satis1es condition (5). Hence ((R1 /bk S) ∪ (R2 /bk S))(x; z) = sup{plt I(xR1 ; Sz); plt I(xR2 ; Sz)} = sup inf I(R1 (x; y); S(y; z)); inf I(R2 (x; y); S(y; z)) y ∈Y
y ∈Y
6L∗ inf (sup{I(R1 (x; y); S(y; z)); I(R2 (x; y); S(y; z))}) y ∈Y
6L∗ inf I(inf (R1 (x; y); R2 (x; y)); S(y; z)) y ∈Y
= plt I(x(R1 ∩ R2 ); Sz) = (R1 ∩ R2 ) /bk S(x; z):
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Even if dom(R1 ) = dom(R2 ) and if I satis1es condition (4), there exists in general no relation I I I I between (R1 /I b S) ∩ (R2 / b S), (R1 / b S) ∪ (R2 / b S) and (R1 ∩ R2 ) / b S. Consider the IFRs R1 and R2 from X = {x} to Y = {y1 ; y2 ; y3 } and the IFR S from Y to Z = {z} de1ned as R1 = {((x; y1 ); 0:3; 0:2); ((x; y2 ); 0:3; 0:4); ((x; y3 ); 0:1; 0:9)}; R2 = {((x; y1 ); 0:2; 0:3); ((x; y2 ); 0:3; 0:2); ((x; y3 ); 0:1; 0:9)}; R1 ∩ R2 = {((x; y1 ); 0:2; 0:3); ((x; y2 ); 0:3; 0:4); ((x; y3 ); 0:1; 0:9)}; S = {((y1 ; z); 0:2; 0:3); ((y2 ; z); 0:2; 0:3); ((y3 ; z); 0:9; 0:1)} and the intuitionistic fuzzy implicator I de1ned as I(a; b) = Ns (a) ∨ b, ∀a; b ∈ L∗ . Then R1 /bI S(x; z) = inf {(0:2; 0:3); (0:3; 0:2); (0:9; 0:1)} = (0:2; 0:3); R2 /bI S(x; z) = inf {(0:2; 0:3); (0:3; 0:2); (0:9; 0:1)} = (0:2; 0:3); (R1 ∩ R2 ) /bI S(x; z) = inf {(0:3; 0:3); (0:3; 0:3); (0:9; 0:1)} = (0:3; 0:3): Hence (R1 /bI S) ∪ (R2 /bI S)(x; z) ¡L∗ (R1 ∩ R2 ) /bI S(x; z): Consider now the IFRs R1 and R2 from X to Y and the IFR S from Y to Z de1ned as R1 = {((x; y1 ); 0:2; 0:3); ((x; y2 ); 0:4; 0:5); ((x; y3 ); 0:4; 0:5)}; R2 = {((x; y1 ); 0:4; 0:6); ((x; y2 ); 0:1; 0:3); ((x; y3 ); 0:1; 0:3)}; R1 ∩ R2 = {((x; y1 ); 0:2; 0:6); ((x; y2 ); 0:1; 0:5); ((x; y3 ); 0:1; 0:5)}; S = {((y1 ; z); 0:6; 0:3); ((y2 ; z); 0:6; 0:3); ((y3 ; z); 0:6; 0:3)}: Then R1 /bI S (x; z) = inf {(0:6; 0:3); (0:4; 0:3); (0:6; 0:3)} = (0:4; 0:3); R2 /bI S (x; z) = inf {(0:6; 0:3); (0:4; 0:3); (0:6; 0:3)} = (0:4; 0:3); (R1 ∩ R2 ) /bI S (x; z) = inf {(0:6; 0:2); (0:2; 0:5); (0:6; 0:3)} = (0:2; 0:5):
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Hence (R1 /bI S ) ∩ (R2 /bI S )(x; z) ¿L∗ (R1 ∩ R2 ) /bI S (x; z): as
Consider at last the IFRs R1 and R2 from X to Y = {y1 ; y2 } and the IFR S from Y to Z de1ned R1 = {((x; y1 ); 0:6; 0:2); ((x; y2 ); 0:3; 0:6)}; R2 = {((x; y1 ); 0:2; 0:5); ((x; y2 ); 0:6; 0:2)}; R1 ∩ R2 = {((x; y1 ); 0:2; 0:5); ((x; y2 ); 0:3; 0:6)}; S = {((y1 ; z); 0:2; 0:6); ((y2 ; z); 0:4; 0:5)}:
Then R1 /bI S (x; z) = inf {(0:2; 0:6); (0:6; 0:2); (0:4; 0:5)} = (0:2; 0:6); R2 /bI S (x; z) = inf {(0:4; 0:5); (0:6; 0:2); (0:4; 0:5)} = (0:4; 0:5); (R1 ∩ R2 ) /bI S (x; z) = inf {(0:5; 0:3); (0:3; 0:5); (0:4; 0:5)} = (0:3; 0:5): Hence (R1 /bI S ) ∩ (R2 /bI S )(x; z) ¡L∗ (R1 ∩ R2 ) /bI S (x; z) ¡L∗ (R1 /bI S ) ∪ (R2 /bI S )(x; z): References [1] K.T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, So1a, June 1983 (Deposed in Central Science— Technical Library of Bulg. Academy of Science, 1697=84) (in Bulgarian). [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. [3] K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37–45. [4] K.T. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999. [5] W. Bandler, B. Kohout, Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and arti1cial systems, in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, pp. 341–367. [6] G. BirkhoM, Lattice Theory, AMS Colloquium Publications, Vol. 25, AMS Providence, RI, 1973, pp. 1–19, 111–112. [7] P. Burillo, H. Bustince, Intuitionistic fuzzy relations (Part I), Mathware Soft Comput. 2 (1995) 5–38. [8] H. Bustince, P. Burillo, Structures on intuitionistic fuzzy relations, Fuzzy Sets and Systems 78 (1996) 293–303. [9] C. Cornelis, G. Deschrijver, The compositional rule of inference in an intuitionistic fuzzy logic setting, in: K. Striegnitz (Ed.), Proc. 6th ESSLLI Student Session, Helsinki, August 2001. [10] B. De Baets, E.E. Kerre, Fuzzy relational compositions, Fuzzy Sets and Systems 60 (1993) 109–120.
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[11] B. De Baets, E.E. Kerre, Fuzzy relations and applications, Adv. Electron. Electron Phys. 89 (1994) 255–324. [12] B. De Baets, E.E. Kerre, The cutting of compositions, Fuzzy Sets and Systems 62 (1994) 295–309. [13] G. Deschrijver, E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 133 (2003) 227–235. [14] J. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. [15] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338–353.
On the composition of intuitionistic fuzzy relations Glad Deschrijver ∗ , Etienne E. Kerre Fuzziness and Uncertainty Modelling, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium Received 22 August 2001; received in revised form 23 April 2002; accepted 3 May 2002
Abstract Fuzzy relations are able to model vagueness, in the sense that they provide the degree to which two objects are related to each other. However, they cannot model uncertainty: there is no means to attribute reliability information to the membership degrees. Intuitionistic fuzzy sets, as de1ned by Atanassov (Instuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999), give us a way to incorporate uncertainty in an additional degree. Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a cartesian product of universes. One of the main concepts in relational calculus is the composition of two relations. Burillo and Bustince (Fuzzy Sets and Systems 78 (1996) 293; Soft Comput. 2 (1995) 5) have extended the sup-T composition of fuzzy relations to a composition of intuitionistic fuzzy relations. In this paper, we present an intuitionistic fuzzy version of the triangular compositions of Bandler and Kohout (in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, p. 341) and the variants of these compositions given by De Baets and Kerre (Adv. Electron. Electron Phys. 89 (1994) 255). Some properties of these compositions are investigated: containment, convertibility, monotonicity, interaction c 2002 Elsevier Science B.V. All rights reserved. with union and intersection. Keywords: Fuzzy relation; Intuitionistic fuzzy set; Intuitionistic fuzzy relation; Triangular composition
1. Introductory remarks Relations are a suitable tool for describing correspondences between objects. Crisp relations like ∈, ⊆, =, : : : have served well in developing mathematical theories. The use of fuzzy relations originated from the observation that real-life objects can be related to each other to a certain degree. However, in real-life situations, a person may assume that a certain object A is in relation R with another object B to a certain degree, but it is possible that he is not so sure about it. In other words, there may ∗
Corresponding author. E-mail addresses: [email protected] (G. Deschrijver), [email protected] (Etienne E. Kerre).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 2 6 9 - 5
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be a hesitation or uncertainty about the degree that is assigned to the relationship between A and B. In fuzzy set theory there is no means to incorporate that hesitation in the membership degrees. A possible solution is to use intuitionistic fuzzy sets, de1ned by Atanassov in 1983 [1]. Intuitionistic fuzzy sets give us the possibility to model hesitation and uncertainty by using an additional degree. Each intuitionistic fuzzy set A assigns to each element x of the universe X a membership degree A (x) (∈ [0; 1]) and a non-membership degree A (x) (∈ [0; 1]) such that A (x) + A (x)61. For all x ∈ X , the number A (x) = 1 − A (x) − A (x) is called the hesitation degree or the intuitionistic index of x to A. In fuzzy set theory, the non-membership degree of an element x of the universe is de1ned as one minus the membership degree (using the standard negation) and thus it is 1xed. In intuitionistic fuzzy set theory, the non-membership degree is a more-or-less independent degree: the only condition is that it is smaller than one minus the membership degree. Note that both A and A can be seen as fuzzy sets on X (that are not completely independent from each other, because of the condition that the sum of the two degrees should be less than or equal to 1). In this way, the negation of the nonmembership degree w.r.t. the standard fuzzy negation can be seen as a degree of membership. So for each element x ∈ X there exist two degrees that model the membership of x in the intuitionistic fuzzy set A, namely A (x) and 1 − A (x). The length of the interval [A (x); 1 − A (x)], which is given by A (x), can then be seen as degree modelling the hesitation between the two membership degrees. An intuitionistic fuzzy relation between two universes X and Y is de1ned as an intuitionistic fuzzy set in X × Y [3,4]. In a similar way as above, if R is a relation between X and Y , x ∈ X and y ∈ Y , then R (x; y) denotes the degree to which x is in relation R with y and R (x; y) denotes the uncertainty degree to which x and y are in relation R with each other. So the “real” degree to which x is in relation R with y lies somewhere between R (x; y) and R (x; y) + R (x; y) = 1 − R (x; y). 1.1. The composition of relations One of the main concepts in relational calculus is the composition of relations. Let us 1rst consider the crisp case. A (crisp) relation R from a universe X to a universe Y is a subset of X × Y . When (x; y) ∈ R, we say that x is in relation R with y and write shortly xRy. The classical composition R ◦ S of a relation R from X to Y and a relation S from Y to Z is de1ned as R ◦ S = {(x; z) ∈ X × Z | (∃y ∈ Y )(xRy ∧ ySz)} = {(x; z) ∈ X × Z | xR ∩ Sz = ∅}; where the afterset xR of x w.r.t. R is de1ned as xR = {y ∈ Y | (x; y) ∈ R} and the foreset Sz of z w.r.t S is de1ned as Sz = {y ∈ Y | (y; z) ∈ S}. The relation R can be identi1ed with its characteristic mapping, namely R : X ×Y → {0; 1} (x; y) → 1
if (x; y) ∈ R;
(x; y) → 0
else:
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Then the composition R ◦ S of R and S can also be written as R ◦ S : X × Z → [0; 1] (x; z) → sup R(x; y) ∧B S(y; z); y ∈Y
where ∧B denotes the Boolean conjunction. Denote by Rt the converse relation of R de1ned as Rt = {(y; x) ∈ Y × X | (x; y) ∈ R}. The composition R ◦ S is a relation from X to Y , consisting of those couples (x; z) for which there exists at least one element of Y that is in relation Rt with x and that is in relation S with z. Example 1.1 (See also De Baets and Kerre [11]). Let X be a set of patients, Y a set of symptoms and Z a set of illnesses. De1ne the relation R ⊆ X × Y as R(x; y) = 1 if patient x shows symptom y, and R(x; y) = 0 if patient x does not show symptom y. De1ne S ⊆ Y × Z as S(y; z) = 1 if symptom y is a symptom of illness z, and S(y; z) = 0 else. Then R ◦ S(x; z) = 1 if patient x shows at least one symptom of illness z. However, therapists may also want to know whether the symptoms shown by patient x are all symptoms of illness z, or whether all the symptoms of illness z are shown by patient x. Bandler and Kohout have de1ned the triangular compositions, in order to be able to model situations as described in the example. The subcomposition of R and S is de1ned as R /bk S = {(x; z) ∈ X × Z | xR ⊆ Sz} and the supercomposition of R and S as R .bk S = {(x; z) ∈ X × Z | Sz ⊆ xR}: Using the characteristic mappings, these compositions are also given by R /bk S(x; z) = inf R(x; y) ⇒B S(y; z); y ∈Y
R .bk S(x; z) = inf S(y; z) ⇒B R(x; y); y ∈Y
where ⇒B denotes the Boolean implication. Example 1.2. In the previous example, the pair (x; z) belongs to R / bk S if all symptoms shown by patient x are symptoms of illness z, and it belongs to R .bk S if patient x shows all symptoms of illness z. The domain of a (crisp) relation R is de1ned as dom(R) = {x ∈ X | xR = ∅}, and the range of R is de1ned as rng(R) = {y ∈ Y | Ry = ∅}. Using these de1nitions, it is clear that co(dom(R)) × Z ⊆ R /bk S; X × co(rng(S)) ⊆ R .bk S:
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The 1rst expression means that if x is not in the domain of R, then x is in relation R / bk S with all elements of Z, even if there is no element of Y that is in relation Rt with x. A similar remark holds for the second expression. De Baets and Kerre [10,11,12] have therefore introduced the following modi1ed de1nitions: the subcomposition R / S = {(x; z) ∈ X × Z | ∅ ⊂ xR ⊆ Sz} and the supercomposition R . S = {(x; z) ∈ X × Z | ∅ ⊂ Sz ⊆ xR}: They also showed that the sub- and the supercomposition can be written in the following equivalent ways: R / S = (R /bk S) ∩ (dom(R) × rng(S)) = (R /bk S) ∩ (R ◦ S) for the subcomposition, and R . S = (R .bk S) ∩ (dom(R) × rng(S)) = (R .bk S) ∩ (R ◦ S) for the supercomposition. In [11], De Baets and Kerre have extended these de1nitions to the fuzzy case and investigated their properties. In this paper we extend these compositions to intuitionistic fuzzy relations and investigate whether the results of De Baets and Kerre still hold. In Section 2 we 1rst recall the de1nitions of the fuzzy relational compositions. In Section 3 we give the de1nition of intuitionistic fuzzy sets, show how intuitionistic fuzzy sets can be seen as L-fuzzy sets for some lattice L, and de1ne some operators which we will need later on. In Section 4 we give the de1nition of intuitionistic fuzzy relation and introduce the extension to intuitionistic fuzzy relations of the diMerent types of relational composition. Finally in Section 5 we investigate the properties of these compositions.
2. The composition of fuzzy relations A fuzzy relation from X to Y is a fuzzy set in X × Y . The afterset xR is the fuzzy set in Y de1ned by xR(y) = R(x; y), ∀y ∈ Y . The foreset Ry is the fuzzy set in X de1ned by Ry(x) = R(x; y), ∀x ∈ X . The domain of R is the fuzzy set in X de1ned by dom(R)(x) = hgt(xR) = supy∈Y R(x; y), where for an arbitrary fuzzy set A in a universe X the height of A is de1ned as hgt(A) = supx∈X A(x). The range of R is the fuzzy set in Y de1ned by rng(R)(y) = hgt(Ry) = supx∈X R(x; y). The classical composition of relations has been extended to fuzzy relations by Zadeh [15]. Let T be a triangular norm, then the sup −T composition R ◦T S of a fuzzy relation R from X to Y and a
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fuzzy relation S from Y to Z is the fuzzy relation from X to Z de1ned by R ◦T S(x; z) = sup T (R(x; y); S(y; z)) = hgt T (xR; Sz): y ∈Y
Bandler and Kohout have extended their triangular compositions to fuzzy relations in the following way: I R /bk S(x; z) = inf I(R(x; y); S(y; z)) = plt I(xR; Sz); y ∈Y
I R .bk S(x; z) = inf I(S(y; z); R(x; y)) = plt I(Sz; xR); y ∈Y
where I is a fuzzy implicator, i.e. a [0; 1]2 → [0; 1] mapping that satis1es the boundary conditions I(0; 0) = I(0; 1) = I(1; 1) = 1 and I(1; 0) = 0 and where the plinth of a fuzzy set A in a universe X is de1ned as plt(A) = inf x∈X A(x). De Baets and Kerre have introduced two classes of improved de1nitions based on the improved versions in the crisp case [11]: R /bI S(x; z) = min(plt I(xR; Sz); hgt(xR); hgt(Sz)); R .bI S(x; z) = min(plt I(Sz; xR); hgt(xR); hgt(Sz)); and R /kT;I S(x; z) = min(plt I(xR; Sz); hgt T (xR; Sz)); R .kT;I S(x; z) = min(plt I(Sz; xR); hgt T (xR; Sz)): 3. Intuitionistic fuzzy sets and the lattice L∗ We give the de1nition of intuitionistic fuzzy sets and some operators on them. We introduce the lattice L∗ which will be very useful in the sequel. De nition 3.1 (Atanassov [1–3]). An intuitionistic fuzzy set (shortly IFS) on a universe X is an object of the form A = {(x; A (x); A (x)) | x ∈ X }; where A (x) (∈ [0; 1]) is called the “degree of membership of x in A”, A (x) (∈ [0; 1]) is called the “degree of non-membership of x in A”, and where A and A satisfy the following condition: (∀x ∈ X )
(A (x) + A (x) 6 1):
The class of IFSs on a universe X will be denoted IFS(X ).
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Fig. 1. Graphical representation of the lattice L∗ .
An IFS A is said to be contained in an IFS B (notation A ⊆ B) if and only if, for all x ∈ X : A (x)6 B (x) and A (x)¿ B (x). The intersection (resp. the union) of two IFSs A and B on X is de1ned as the IFS A ∩ B = {(x; min (A (x); B (x)); max( A (x); B (x))) | x ∈ X } (resp. A ∪ B = {(x; max(A (x); B (x)); min( A (x); B (x))) | x ∈ X }). Starting from these de1nitions, a generalized intersection ∩T; S and union ∪S; T can be de1ned by replacing min by an arbitrary t-norm T and max by an arbitrary t-conorm S. The generalized intersection and union yield IFSs if (∀(x; y) ∈ [0; 1]2 )(T (x; y)6S ∗ (x; y)) (shortly T 6S ∗ ) where S ∗ denotes the dual t-norm of S de1ned by S ∗ (x; y) = 1 − S(1 − x; 1 − y), ∀(x; y) ∈ [0; 1]2 [13]. In this paper we will often use the following lattice and operators de1ned on it. De1ne a set L∗ and an operation 6L∗ such that L∗ = {(x1 ; x2 ) ∈ [0; 1]2 | x1 + x2 6 1}; (x1 ; x2 ) 6L∗ (y1 ; y2 ) ⇔ x1 6 y1 ∧ x2 ¿ y2 ; then (L∗ ; 6L∗ ) is a complete lattice [13]. For each A ⊆ L∗ we have sup A = (sup{x1 ∈ [0; 1] | x1 A = ∅}; inf {x2 ∈ [0; 1] | Ax2 = ∅}); inf A = (inf {x1 ∈ [0; 1] | x1 A = ∅}; sup{x2 ∈ [0; 1] | Ax2 = ∅}): The shaded area in Fig. 1 is the set of elements x = (x1 ; x2 ) belonging to L∗ . Equivalently, this lattice can also be de1ned as an algebraic structure (L∗ ; ∧; ∨) where the meet operator ∧ and the join operator ∨ are de1ned as follows, for (x1 ; x2 ); (y1 ; y2 ) ∈ L∗ : (x1 ; x2 ) ∧ (y1 ; y2 ) = (min(x1 ; y1 ); max(x2 ; y2 )); (x1 ; x2 ) ∨ (y1 ; y2 ) = (max(x1 ; y1 ); min(x2 ; y2 )):
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For our purposes, we will also consider a generalized meet operator ∧T; S and join operator ∨S; T on (L∗ ; 6L∗ ): (x1 ; x2 ) ∧T;S (y1 ; y2 ) = (T (x1 ; y1 ); S(x2 ; y2 )); (x1 ; x2 ) ∨S;T (y1 ; y2 ) = (S(x1 ; y1 ); T (x2 ; y2 )) for a given t-norm T and t-conorm S satisfying T 6S ∗ . It has been shown that the condition T 6S ∗ is necessary and suOcient for these operators to be well de1ned and that they are increasing in both components [9]. Furthermore, we de1ne an order-reversing operator Ns by Ns (x1 ; x2 ) = (x2 ; x1 ), ∀(x1 ; x2 ) ∈ L∗ . Using this lattice, it can be easily seen that with every IFS A = {(x; A (x); A (x)) | x ∈ X } corresponds an L∗ -fuzzy set in the sense of Goguen [14], i.e. a mapping A : X → L∗ : x → (A (x); A (x)) [13]. In the sequel we will use the same notation for an IFS and its associated L∗ -fuzzy set. So, for the IFS A we will also use the notation A(x) = (A (x); A (x)). From now on, we will assume that if a ∈ L∗ , then a1 and a2 denote, respectively, the 1rst and the second coordinate of a, i.e. a = (a1 ; a2 ). Interpretating intuitionistic fuzzy sets as L∗ -fuzzy sets gives way to a greater Pexibility in calculating with membership and non-membership degrees, since the pair formed by the two degrees is an element of L∗ , and often allows to obtain signi1cantly more compact formulas. Moreover, some operators that are de1ned in the fuzzy case, such as fuzzy implicators, can be extended to the intuitionistic fuzzy case by using the lattice (L∗ ; 6L∗ ). For instance, fuzzy implicators can be extended to intuitionistic fuzzy implicators as follows. De nition 3.2. An intuitionistic fuzzy implicator is any (L∗ )2 → L∗ mapping I satisfying the border conditions I(0L∗ ; 0L∗ ) = I(0L∗ ; 1L∗ ) = I(1L∗ ; 1L∗ ) = 1L∗ ;
I(1L∗ ; 0L∗ ) = 0L∗ ;
where 0L∗ = (0; 1) and 1L∗ = (1; 0) are the identities of (L∗ ; 6L∗ ). In some cases we will require an intuitionistic fuzzy implicator to be decreasing in its 1rst, and increasing in its second component, i.e. (∀y ∈ L∗ )(∀(x; x ) ∈ (L∗ )2 )(x 6L∗ x ⇒ I(x; y) ¿L∗ I(x ; y));
(M.1)
(∀x ∈ L∗ )(∀(y; y ) ∈ (L∗ )2 )(y 6L∗ y ⇒ I(x; y) 6L∗ I(x; y )):
(M.2)
These properties will be referred to in the sequel as the hybrid monotonicity properties.
4. The composition of intuitionistic fuzzy relations De nition 4.1 (Atanassov [4] and Burillo and Bustince [8]). An intuitionistic fuzzy relation R (IFR, for short) from a universe X to a universe Y is an intuitionistic fuzzy set in X × Y , i.e.
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a set R = {((x; y); R (x; y); R (x; y)) | x ∈ X; y ∈ Y }; where R : X × Y → [0; 1] and R : X × Y → [0; 1] satisfy the condition (R (x; y) + R (x; y) 6 1):
(∀(x; y) ∈ X × Y )
The class of IFRs from a universe X to a universe Y will be denoted IFR(X × Y ). In the sequel we will consider an IFR R also as a mapping R : X × Y → L∗ : (x; y) → R(x; y) = (R (x; y); R (x; y)):
(1)
The afterset xR of an IFR R is the IFS in Y de1ned by xR = {(y; R (x; y); R (x; y)) | y ∈ Y }. The foreset Ry of an IFR R is the IFS in X de1ned by Ry = {(x; R (x; y); R (x; y)) | x ∈ X }. The height of an IFS A in X is de1ned as hgt(A) = supx∈X A(x) = (supx∈X A (x); inf x∈X A (x)) (notice that hgt(A) ∈ L∗ ). The plinth of an IFS A in X is de1ned as plt(A) = inf x∈X A(x) = (inf x∈X A (x); supx∈X
A (x)). The domain of an IFR R is the IFS dom(R) in X de1ned by dom(R)(x) = hgt(xR) = (supy∈Y R (x; y); inf y∈Y R (x; y)). The range of an IFR R is the IFS rng(R) in Y de1ned by rng(R)(y) = hgt(Ry) = (supx∈X R (x; y); inf x∈X R (x; y)). The converse relation Rt of R is the IFR from Y to X de1ned as Rt = {((y; x); R (x; y); R (x; y)) | y ∈ Y; x ∈ X }. Bustince and Burillo have extended the sup −T composition to a composition of IFRs, in the case that X , Y and Z are 1nite, as follows [7,8]: De nition 4.2. Let ; ; ; be t-norms or t-conorms, R ∈ IFR(X × Y ) and S ∈ IFR(Y × Z). Then the composition of R and S is the IFR from X to Z de1ned as R ◦ S = {((x; z); ◦ (x; z); ◦ (x; z)) | x ∈ X; z ∈ Z};
where ◦ (x; z) = ((R (x; y); S (y; z))); y ∈Y
◦ (x; z) = (( R (x; y); S (y; z))); y ∈Y
whenever 0 6 ◦ (x; z) + ◦ (x; z) 6 1;
∀(x; z) ∈ X × Z:
The properties of this composition and the choice of ; ; and for which this composition ful1lls a maximal number of properties are investigated in [7]. We now extend the triangular compositions to intuitionistic triangular compositions as follows. De nition 4.3. Let R be an IFR from X to Y , S an IFR from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm, then the triangular sub- and supercomposition and their
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improved versions are de1ned as IFRs from X to Z de1ned by I S(x; z) = inf I(R(x; y); S(y; z)) = plt I(xR; Sz); R /bk y ∈Y
I R .bk S(x; z) = inf I(S(y; z); R(x; y)) = plt I(Sz; xR); y ∈Y
I
R /b S(x; z) = inf
inf I(R(x; y); S(y; z)); sup R(x; y); sup S(y; z)
y ∈Y
y ∈Y
y ∈Y
= inf {plt I(xR; Sz); hgt(xR); hgt(Sz)}; R .bI S(x; z) = inf {plt I(Sz; xR); hgt(xR); hgt(Sz)}; R /kT
;S ;I
S(x; z) = inf
inf I(R(x; y); S(y; z)); sup(R(x; y) ∧T ;S S(y; z))
y ∈Y
y ∈Y
= inf {plt I(xR; Sz); hgt(xR ∩T ;S Sz)}; R .kT
;S ;I
S(x; z) = inf {plt I(Sz; xR); hgt(xR ∩T ;S Sz)}:
In the above de1nition we used the notation (1) to represent the IFRs. For instance R /I bk S(x; z) ∗ is an element of L given by inf pr 1 I((R (x; y); R (x; y)); (S (y; z); S (y; z)));
y ∈Y
sup pr 2 I((R (x; y); R (x; y)); (S (y; z); S (y; z))) ;
y ∈Y
where pri : L∗ → [0; 1] : (a1 ; a2 ) → ai , i = 1; 2. Then the IFR R /I bk S is given by I I I S = {((x; z); pr 1 R /bk S(x; z); pr 2 R /bk S(x; z)) | x ∈ X; z ∈ Z}: R /bk
Example 4.1. Consider a set of patients X , a set of symptoms Y and a set of illnesses Z. Let R be the IFR from X to Y de1ned by R (x; y) = the degree to which patient x shows symptom y;
R (x; y) = the degree to which patient x does not show symptom y
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and S the IFR from Y to Z de1ned by S (y; z) = the degree to which y is a symptom of illness z;
S (y; z) = the degree to which y is not a symptom of illness z: Let x ∈ X be a patient, Y = {y1 ; : : : ; y5 } a set of symptoms, and z ∈ Z an illness. Assume furthermore that xR = {(y1 ; 0; 1); (y2 ; 0; 1); (y3 ; 0; 1); (y4 ; 0:1; 0:85); (y5 ; 0; 1)}; Sz = {(y1 ; 1; 0); (y2 ; 0:7; 0:2); (y3 ; 0:6; 0:2); (y4 ; 0:7; 0:1); (y5 ; 0:3; 0:6)}: Consider, for instance, the following intuitionistic fuzzy implicator: I(a; b) = (min(1; a2 + b1 ); max(0; a1 + b2 − 1));
∀a; b ∈ L∗ :
Then R /I bk S(x; z) = (1; 0). This means that the degree to which all symptoms shown by patient x are symptoms of illness z is equal to 1, although patient x is only showing symptom y4 to degree 0:1, and with only an uncertainty given by 0:05. Such surprising results are the consequence of the fact that the Bandler and Kohout compositions do not take into account the degree of emptiness or non-emptyness of the foresets and the aftersets involved. For the improved versions we get R /I b S(x; z) = inf {(1; 0); (0:1; 0:85); (1; 0)} = (0:1; 0:85) and, if we use for example the Lukasiewicz t-norm and its dual t-conorm, de1ned by, for all a; b ∈ [0; 1], a ∩W b = max(0; a + b − 1) and a +b b = min(1; a + b), respectively, R /∩k W ;+b ;I S(x; z) = inf {(1; 0); (0; 0:95)} = (0; 0:95). These results correspond better to intuition. Example 4.2. Let X , Y , Z, R and S be as in the previous example. Assume that a therapist wants to 1nd out to which degree a patient x suMers from illness z. The therapist determines to which degree the patient shows the symptoms y1 ; : : : ; y5 and for each symptom he gives a degree of how sure he is about that degree. Assume he gets the following table (in the 1rst row is written to which degree patient x shows each symptom, in the second row is written the degree of how sure the therapist is when attributing the degree in the 1rst row in the same column): y1
y2
y3
y4
y5
Degree of showing symptom
0.9
0.5
0.4
0.7
0.5
Degree of being sure
1.0
0.8
0.9
0.9
0.6
The 1rst row contains the membership degrees of (x; yi ) to R, for each i ∈ {1; : : : ; 5}, while, if we take the fuzzy negation of each element in the second row, then we get the degree of uncertainty to which (x; yi ) ∈ R, for each i ∈ {1; : : : ; 5}. For instance, for y2 , the membership degree of (x; y2 ) to R is given by R (x; y2 ) = 0:5 and the degree of uncertainty is given by R (x; y2 ) = 1−0:8 = 0:2. We obtain immediately the degree of non-membership as R (x; y2 ) = 1 − R (x; y2 ) − R (x; y2 ) = 0:3. Applying
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the same method for the other symptoms y1 ; y3 ; : : : ; y5 , we obtain the following IFS modelling the degree to which patient x shows the symptoms in Y : xR = {(y1 ; 0:9; 0:1); (y2 ; 0:5; 0:3); (y3 ; 0:4; 0:5); (y4 ; 0:7; 0:2); (y5 ; 0:5; 0:1)}: Similarly an IFS Sz can be constructed, modelling the degree to which the symptoms in Y are characteristic for illness z: Sz = {(y1 ; 1; 0); (y2 ; 0:7; 0:2); (y3 ; 0:2; 0:6); (y4 ; 0:7; 0:1); (y5 ; 0:3; 0:6)}: min Let T = min, S = max, I(a; b) = sup(Ns (a); b). Then we obtain for instance, R sup inf ◦max S(x; z) = I I (0:9; 0:1), R / b S(x; z) = (0:3; 0:5), and R .b S(x; z) = (0:4; 0:5). This means that the degree to which patient x shows at least one symptom of illness z is equal to 0:9 with no hesitation, the degree to which all symptoms shown by x are symptoms of z is equal to 0:3 with uncertainty 1−0:3−0:5 = 0:2, and the degree to which x shows all symptoms of z equals 0:4 with uncertainty 0:1. If we only min consider the sup inf ◦max composition to determine if a patient suMers from a certain disease, then we obtain a high degree, namely 0:9, with no uncertainty. If we also take into account the sub- and supercomposition, then we get a more nuanced view.
5. Properties of the intuitionistic triangular compositions In this section we will investigate the following properties of the intuitionistic triangular compositions: containment in the composition de1ned by Burillo and Bustince, convertibility, monotonicity, interaction with union and intersection. 5.1. On the containment of the intuitionistic compositions Theorem 5.1. Let R ∈ IFR(X × Y ), S ∈ IFR(Y × Z), I be an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Then R /kT
;S ;I
T S ⊆ R sup inf ◦S S;
R .kT
;S ;I
T S ⊆ R sup inf ◦S S;
R /kT
;S ;I
S ⊆ R /bI S;
R .kT
;S ;I
S ⊆ R .bI S:
sup T A general containment rule between R /I b S and R inf ◦S S does not exist. Consider for instance the IFRs R = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:5; 0:5); ((x2 ; y1 ); 0:9; 0:1); ((x2 ; y2 ); 0:9; 0:1)} and S = {((y1 ; z); 0:2; 0:8); ((y2 ; z); 0:1; 0:9)} from X = {x1 ; x2 } to Y = {y1 ; y2 } and from Y to Z = {z} respectively. Let the intuitionistic fuzzy implicator I be such that I(a; b) = Ns (a) ∨ b, ∀a; b ∈ L∗ and let T = min
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and S = max. Then we obtain T R /bI S(x1 ; z) = (0:2; 0:8) ¿L∗ R sup inf ◦S S(x1 ; z) = (0:1; 0:9)
and T R /bI S(x2 ; z) = (0:1; 0:9) ¡L∗ R sup inf ◦S S(x2 ; z) = (0:2; 0:8):
sup T I Similarly there is no containment rule between R /I bk S and R inf ◦S S(x1 ; z), since R / bk S(x1 ; z) = (0:5; sup T sup T I 0:5)¿L∗ R inf ◦S S(x1 ; z) and R .bk S(x2 ; z) = (0:1; 0:9)¡L∗ R inf ◦S S(x1 ; z). In the crisp case we have, using / as composition, that the number of pairs (x; z) ∈ X × Z for which all the y ∈ Y that are in relation Rt with x are also in relation S with z, is smaller than the number of pairs (x; z) for which there exists at least one y that is in relation Rt with x and in relation S with z. The theorem shows that this intuitively natural property still holds for intuitionistic fuzzy relations, but only for /Tk ; S ; I . We then have that for any (x; z) ∈ X × Z, the degree to which all the y ∈ Y that are in relation Rt with x are also in relation with z, is smaller than the degree to which there exist at least one y that is in relation Rt with x and in relation S with z.
5.2. On the convertibility of the intuitionistic triangular compositions The formulas in the following theorem allow us to write the converse of the triangular compositions in terms of the converse relations of the composing relations. Theorem 5.2. Let R ∈ IFR(X × Y ), S ∈ IFR(Y × Z), I be an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Then: I I t S)t = S t .bk R; (R /bk I I t (R .bk S)t = S t /bk R;
(R /bI S)t = S t .bI Rt ; (R .bI S)t = S t /bI Rt ; (R /kT
; S; I
S)t = S t .kT
; S; I
Rt ;
(R .kT
; S; I
S)t = S t /kT
; S; I
Rt :
Proof. Notice that (R /Tk (R /kT
; S; I
; S; I
S)t ∈ IFR(Z × X ). Hence we calculate, for x ∈ X and z ∈ Z:
S)t (z; x) = R /kT ; S ; I S(x; z)
= inf
inf I(R(x; y); S(y; z)); sup(R(x; y) ∧T ;S S(y; z))
y ∈Y
y ∈Y
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= inf = S t .kT
345
t
t
t
t
inf I(R (y; x); S (z; y)); sup(S (z; y) ∧T ;S R (y; x))
y ∈Y
;S ;I
y ∈Y
Rt (z; x):
The other equalities are proved in a similar way. 5.3. On the monotonicity of the intuitionistic triangular compositions We assume in this subsection that R; R1 ; R2 ∈ IFR(X × Y ), S; S1 ; S2 ∈ IFR(Y × Z), I is an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Lemma 5.1. If R1 ⊆ R2 , then, for all (x; z) ∈ X × Z, hgt(xR1 ) 6L∗ hgt(xR2 ); hgt(xR1 ∩T ;S Sz) 6L∗ hgt(xR2 ∩T ;S Sz):
Theorem 5.3. If I satis7es (M.1), then: dom(R1 ) = dom(R2 ) ∧ R1 ⊆ R2 ⇒ R1 /bI S ⊇ R2 /bI S: If I satis7es (M.2), then: R1 ⊆ R2 ⇒ R1 .bI S ⊆ R2 .bI S; R1 ⊆ R2 ⇒ R1 .kT
;S ;I
S ⊆ R2 .kT
;S ;I
S:
In the case of /kT ; S ; I , from dom(R1 ) = dom(R2 ) a monotonicity rule cannot be deduced. Consider, for instance, the relations R1 and R2 from X = {x1 } to Y = {y1 ; y2 } and S from Y to Z = {z1 ; z2 }, and the intuitionistic fuzzy implicator I de1ned as follows:
R1 = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:6; 0:4)}; R2 = {((x1 ; y1 ); 0:5; 0:5); ((x1 ; y2 ); 0:6; 0:4)}; S = {((y1 ; z1 ); 0:2; 0:8); ((y2 ; z1 ); 0:1; 0:9); ((y1 ; z2 ); 0:2; 0:8); ((y2 ; z2 ); 0:9; 0:1)}; I(a; b) = Ns (a) ∨ b = (max(a2 ; b1 ); min(a1 ; b2 ));
∀a = (a1 ; a2 ); b = (b1 ; b2 ) ∈ L∗ :
Then R1 /kmin; max; I S(x1 ; z1 ) = (0:1; 0:9)¡L∗ R2 /kmin; max; I S(x1 ; z1 ) = (0:2; 0:8) and R1 /kmin; max; I S(x1 ; z2 ) = (0:6; 0:4)¿L∗ R2 /kmin; max; I S(x1 ; z2 ) = (0:5; 0:5).
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The reason for this is the fact that from dom(R1 ) = dom(R2 ) it does not follow hgt(xR1 ∩T ; S Sz) = hgt(xR2 ∩T ; S Sz), ∀(x; z) ∈ X × Z. In the above example we have dom(R1 ) = dom(R2 ), but hgt(x1 R1 ∩ Sz1 ) = (0:1; 0:9)¡L∗ (0:2; 0:8) = hgt(x1 R2 ∩ Sz1 ). Similarly, there is a monotonicity property for the second component of the diMerent compositions.
Theorem 5.4. If I satis7es (M.1), then dom(S1 ) = dom(S2 ) ∧ S1 ⊆ S2 ⇒ R .bI S1 ⊇ R .bI S2 : If I satis7es (M.2), then S1 ⊆ S2 ⇒ R /bI S1 ⊆ R /bI S2 ; S1 ⊆ S2 ⇒ R /kT
; S; I
S1 ⊆ R /kT
; S; I
S2 :
5.4. On the interaction of the intuitionistic triangular compositions with the union In this subsection we will assume that R; R1 ; R2 ∈ IFR(X × Y ), S; S1 ; S2 ∈ IFR(Y × Z), I is an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. We start with some lemmas. Lemma 5.2. For arbitrary R1 ; R2 ∈ IFR(X × Y ), it holds dom(R1 ∪ R2 ) = dom(R1 ) ∪ dom(R2 ): As a corollary of this lemma it follows that, if dom(R1 ) = dom(R2 ) then dom(R1 ∪ R2 ) = dom(R1 ) = dom(R2 ). Eq. (2) in the following theorem will be a useful tool for proving some of the interactions between the triangular compositions and the union. Lemma 5.3. If I satis7es the following condition: I(sup(a; b); c) = inf (I(a; c); I(b; c)):
(2)
then it satis7es (M.1). If I satis7es (M.1), then I(sup(a; b); c) 6L∗ inf (I(a; c); I(b; c)):
(3)
Note that in fuzzy set theory, Eq. (2) holds for any fuzzy implicator which is decreasing in its 1rst component. For intuitionistic fuzzy implicators, this is not the case. Let, for instance, I be the intuitionistic fuzzy implicator de1ned by, for all x; y ∈ L∗ , (Ns x ∨+;ˆ · y) ∧·;+ˆ y if x1 ¿ x2 ; I(x; y) = Ns x ∨+;ˆ · y else;
G. Deschrijver, Etienne E. Kerre / Fuzzy Sets and Systems 136 (2003) 333 – 361
347
ˆ the probabilistic sum de1ned as, for m; n ∈ [0; 1], m+n ˆ = m + n − m · n. where · is the product, and + Let a = (0:3; 0:2), b = (0:4; 0:5) and c = (0:2; 0:8), then sup(a; b) = (0:4; 0:2) and I(a; c) = ((0:2; 0:3) ∨+;ˆ · (0:2; 0:8)) ∧·;+ˆ (0:2; 0:8) = (0:36; 0:24) ∧·;+ˆ (0:2; 0:8) = (0:072; 0:848); I(b; c) = (0:5; 0:4) ∨+;ˆ · (0:2; 0:8) = (0:6; 0:32); I(sup(a; b); c) = ((0:2; 0:4) ∨+;ˆ · (0:2; 0:8)) ∧·;+ˆ (0:2; 0:8) = (0:36; 0:32) ∧·;+ˆ (0:2; 0:8) = (0:072; 0:864): Hence I(sup(a; b); c)¡L∗ inf {I(a; c); I(b; c)} = I(a; c). Two important classes of intuitionistic fuzzy implicators satisfy the condition (2), the S-implicators and the R-implicators, both de1ned in [9]. Theorem 5.5. Let T be a t-norm, S a t-conorm and N a unary involutive order reversing operator on L∗ . The S-implicator IS; T; N , de7ned as [9] IS;T;N (x; y) = N(x) ∨S;T y;
∀(x; y) ∈ (L∗ )2 ;
satis7es formula (2). Theorem 5.6. Let T be a t-norm and S a t-conorm. The R-implicator IT; S , de7ned as [9] IT;S (x; y) = sup{# ∈ L∗ | x ∧T;S # 6L∗ y} satis7es formula (2). Now we present the diMerent interactions between the triangular compositions and the union. It will be seen that only some semi-distributivity properties hold. Theorem 5.7. If dom R1 = dom R2 and I satis7es (M.1), then (R1 /bI S) ∩ (R2 /bI S) ⊇ (R1 ∪ R2 ) /bI S: If I satis7es formula (2) and not necessarily dom R1 = dom R2 , then (R1 /bI S) ∩ (R2 /bI S) ⊆ (R1 ∪ R2 ) /bI S; (R1 /kT
;S ;I
S) ∩ (R2 /kT
;S ;I
S) ⊆ (R1 ∪ R2 ) /kT
;S ;I
S:
Theorem 5.8. If dom S1 = dom S2 and I satis7es (M.1), then (R .bI S1 ) ∩ (R .bI S2 ) ⊇ R .bI (S1 ∪ S2 ):
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If I satis7es formula (2) and not necessarily dom S1 = dom S2 , then (R .bI S1 ) ∩ (R .bI S2 ) ⊆ R .bI (S1 ∪ S2 ); (R .kT
;S ;I
S1 ) ∩ (R .kT
;S ;I
S2 ) ⊆ R .kT
;S ;I
(S1 ∪ S2 ):
Lemma 5.4. For arbitrary R1 ; R2 ∈ IFR(X × Y ), S ∈ IFR(Y × Z) we have sup(hgt(xR1 ∩T ;S Sz); hgt(xR2 ∩T ;S Sz)) = hgt(x(R1 ∪ R2 ) ∩T ;S Sz): Proof. This property is equivalent to sup T sup T T (R1 sup inf ◦S S) ∪ (R2 inf ◦S S)(x; z) = (R1 ∪ R2 ) inf ◦S S(x; z);
which is proved in [7, Theorem 7]. Theorem 5.9. If I satis7es (M.1) and not necessarily dom R1 = dom R2 , then (R1 ∪ R2 ) /bI S ⊆ (R1 /bI S) ∪ (R2 /bI S); (R1 ∪ R2 ) /kT
;S ;I
S ⊆ (R1 /kT
;S ;I
S) ∪ (R2 /kT
;S ;I
S):
Theorem 5.10. If I satis7es (M.1) and not necessarily dom S1 = dom S2 , then R .bI (S1 ∪ S2 ) ⊆ (R .bI S1 ) ∪ (R .bI S2 ); R .kT
;S ;I
(S1 ∪ S2 ) ⊆ (R .kT
;S ;I
S1 ) ∪ (R .kT
;S ;I
S2 ):
Using the above theorems, we may also conclude the following: • if I satis1es formula (2), then (R1 /bI S) ∩ (R2 /bI S) ⊆ (R1 ∪ R2 ) /bI S ⊆ (R1 /bI S) ∪ (R2 /bI S) and similarly for /kT ; S ; I ; • if I satis1es formula (2) and dom(R1 ) = dom(R2 ), then
(R1 /bI S) ∩ (R2 /bI S) = (R1 ∪ R2 ) /bI S; • if I satis1es formula (2), then (R .bI S1 ) ∩ (R .bI S2 ) ⊆ R .bI (S1 ∪ S2 ) ⊆ (R .bI S1 ) ∪ (R .bI S2 ) and similarly for .kT
; S; I
;
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349
• if I satis1es formula (2) and dom(S1 ) = dom(S2 ), then (R .bI S1 ) ∩ (R .bI S2 ) = R .bI (S1 ∪ S2 ): 5.5. On the interaction of the intuitionistic triangular compositions with the intersection Also in this section it will become clear that the intuitionistic triangular compositions only satisfy some kind of semi-distributivity w.r.t. the intuitionistic fuzzy intersection. Theorem 5.11. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. If I satis7es (M.2) and not necessarily dom R1 = dom R2 , then (R1 ∩ R2 ) .bI S ⊆ (R1 .bI S) ∩ (R2 .bI S); (R1 ∩ R2 ) .kT
;S ;I
S ⊆ (R1 .kT
;S ;I
S) ∩ (R2 .kT
;S ;I
S):
Theorem 5.12. Let R be an IFR from X to Y , S1 and S2 be IFRs from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. If I satis7es (M.2) and not necessarily dom S1 = dom S2 , then R /bI (S1 ∩ S2 ) ⊆ (R /bI S1 ) ∩ (R /bI S2 ); R /kT
;S ;I
(S1 ∩ S2 ) ⊆ (R /kT
;S ;I
S1 ) ∩ (R /kT
;S ;I
S2 ):
In a similar way as Lemma 5.3, the following lemma is proved. Lemma 5.5. If I satis7es the following condition: I(inf (a; b); c) = sup(I(a; c); I(b; c))
(4)
then it satis7es (M.1). If I satis7es (M.1), then I(inf (a; b); c) ¿L∗ sup(I(a; c); I(b; c)):
(5)
In a similar way as above it is proved that S-implicators satisfy condition (4). However R-implicators do in general not satisfy condition (4). Consider Imin; max (x; y) = sup{# ∈ L∗ | (min(x1 ; #1 ); max(x2 ; #2 ))6L∗ y}, and let a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ such that a1 6c1 ¡b1 and a2 ¡c2 6b2 . Then inf (a; b) = (a1 ; b2 ), so we obtain Imin; max (a; c) = (1−c2 ; c2 ), Imin; max (b; c) = (c1 ; 0) and Imin; max ((a1 ; b2 ); c) = (1; 0). Hence Imin; max (inf (a; b); c)¿L∗ sup(Imin; max (a; c); Imin; max (b; c)). Theorem 5.13. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z and I an intuitionistic fuzzy implicator. If I satis7es (M.1) then (R1 /bk S) ∪ (R2 /bk S) ⊆ (R1 ∩ R2 ) /bk S
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Even if dom(R1 ) = dom(R2 ) and if I satis1es condition (4), there exists in general no relation beI I I I tween (R1 /I b S) ∩ (R2 / b S), (R1 / b S) ∪ (R2 / b S) and (R1 ∩ R2 ) / b S. See appendix for more details. Appendix: Proofs and examples A.1. The containment of the intuitionistic compositions Theorem 5.1. Let R ∈ IFR(X × Y ), S ∈ IFR(Y × Z), I be an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. Then R /kT
;S ;I
T S ⊆ R sup inf ◦S S;
R .kT
;S ;I
T S ⊆ R sup inf ◦S S;
R /kT
;S ;I
S ⊆ R /bI S;
R .kT
;S ;I
S ⊆ R .bI S:
Proof. Let x ∈ X , z ∈ Z, then R /kT
;S ;I
S(x; z) = inf {plt I(xR; Sz); hgt(xR ∩T ;S Sz)} 6L∗ hgt(xR ∩T ;S Sz) =
sup T (R (x; y); S (y; z)); inf S ( R (x; y); S (y; z)) y ∈Y
y ∈Y
T = R sup inf ◦S S(x; z):
The second inclusion is proved in an analogous way. To prove the last two inclusions, we observe that xR ∩T ; S Sz ⊆ xR, and thus hgt(xR ∩T ; S Sz)6L∗ hgt(xR). Analogously we obtain hgt(xR ∩T ; S Sz)6L∗ hgt(Sz). Hence hgt(xR ∩T ; S Sz)6L∗ inf {hgt (xR); hgt(Sz)}. Now the inclusions follow easily. sup T To prove that there does not exist a general containment rule between R /I b S and R inf ◦S S, we consider the IFRs R = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:5; 0:5); ((x2 ; y1 ); 0:9; 0:1); ((x2 ; y2 ); 0:9; 0:1)} and S = {((y1 ; z); 0:2; 0:8); ((y2 ; z); 0:1; 0:9)} from X = {x1 ; x2 } to Y = {y1 ; y2 } and from Y to Z = {z} respectively. Let the intuitionistic fuzzy implicator I be such that I(a; b) = Ns (a) ∨ b, ∀a; b ∈ L∗ and let T = min and S = max. Then
R /bI S(x1 ; z) = inf {inf ((0:9; 0:1); (0:5; 0:5)); sup((0:1; 0:9); (0:5; 0:5)); sup((0:2; 0:8); (0:1; 0:9))} = inf {(0:5; 0:5); (0:5; 0:5); (0:2; 0:8)} = (0:2; 0:8);
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351
T R sup inf ◦S S(x1 ; z) = sup((0:1; 0:9); (0:1; 0:9)) = (0:1; 0:9);
R /bI S(x2 ; z) = inf {inf ((0:2; 0:8); (0:1; 0:9)); sup((0:9; 0:1); (0:9; 0:1)); sup((0:2; 0:8); (0:1; 0:9))} = inf {(0:1; 0:9); (0:9; 0:1); (0:2; 0:8)} = (0:1; 0:9); T R sup inf ◦S S(x2 ; z) = sup((0:2; 0:8); (0:1; 0:9)) = (0:2; 0:8):
A.2. On the monotonicity of the intuitionistic triangular compositions Lemma 5.1. If R1 ⊆ R2 , then, for all (x; z) ∈ X × Z, hgt(xR1 ) 6L∗ hgt(xR2 ); hgt(xR1 ∩T ;S Sz) 6L∗ hgt(xR2 ∩T ;S Sz): Proof. We prove the second inequality. Let x ∈ X , z ∈ Z. From R1 ⊆ R2 it follows R1 (x; y)6L∗ R2 (x; y), ∀y ∈ Y , and so R1 (x; y) ∧T ; S S(y; z)6L∗ R2 (x; y) ∧T ; S S(y; z), ∀y ∈ Y . From the monotonicity of sup we obtain sup(R1 (x; y) ∧T ;S S(y; z)) 6L∗ sup(R2 (x; y) ∧T ;S S(y; z)): Theorem 5.3. If I satis7es (M.1), then dom(R1 ) = dom(R2 ) ∧ R1 ⊆ R2 ⇒ R1 /bI S ⊇ R2 /bI S: If I satis7es (M.2), then: R1 ⊆ R2 ⇒ R1 .bI S ⊆ R2 .bI S; R1 ⊆ R2 ⇒ R1 .kT
;S ;I
S ⊆ R2 .kT
;S ;I
S:
Proof. Choose arbitrary x ∈ X and z ∈ Z. From dom(R1 ) = dom(R2 ) it follows hgt(xR1 ) = hgt(xR2 ), so from R1 ⊆ R2 and (M.1) we obtain consequently R1 (x; y) 6L∗ R2 (x; y);
∀y ∈ Y;
I(R1 (x; y); S(y; z)) ¿L∗ I(R2 (x; y); S(y; z));
∀y ∈ Y;
inf I(R1 (x; y); S(y; z)) ¿L∗ inf I(R2 (x; y); S(y; z));
y ∈Y
y ∈Y
R1 /bI S ⊇ R2 /bI S: Analogously, using Lemma 5.1, the second statement is proved.
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In the case of /kT ; S ; I , from dom(R1 ) = dom(R2 ) a monotonicity rule cannot be deduced. Consider, for instance, the relations R1 and R2 from X = {x1 } to Y = {y1 ; y2 } and S from Y to Z = {z1 ; z2 }, and the intuitionistic fuzzy implicator I de1ned as follows:
R1 = {((x1 ; y1 ); 0:1; 0:9); ((x1 ; y2 ); 0:6; 0:4)}; R2 = {((x1 ; y1 ); 0:5; 0:5); ((x1 ; y2 ); 0:6; 0:4)}; S = {((y1 ; z1 ); 0:2; 0:8); ((y2 ; z1 ); 0:1; 0:9); ((y1 ; z2 ); 0:2; 0:8); ((y2 ; z2 ); 0:9; 0:1)}; I(a; b) = Ns (a) ∨ b = (max(a2 ; b1 ); min(a1 ; b2 ));
∀a = (a1 ; a2 ); b = (b1 ; b2 ) ∈ L∗ :
Then R1 /kmin;max;I S(x1 ; z1 ) = inf {inf ((0:9; 0:1); (0:4; 0:6)); sup((0:1; 0:9); (0:1; 0:9))} = (0:1; 0:9); R2 /kmin;max;I S(x1 ; z1 ) = inf {inf ((0:5; 0:5); (0:4; 0:6)); sup((0:2; 0:8); (0:1; 0:9))} = (0:2; 0:8); R1 /kmin;max;I S(x1 ; z2 ) = inf {inf ((0:9; 0:1); (0:9; 0:1)); sup((0:1; 0:9); (0:6; 0:4))} = (0:6; 0:4); R2 /kmin;max;I S(x1 ; z2 ) = inf {inf ((0:5; 0:5); (0:9; 0:1)); sup((0:2; 0:8); (0:6; 0:4))} = (0:5; 0:5): A.3. On the interaction of the intuitionistic triangular compositions with the union Lemma 5.2. For arbitrary R1 ; R2 ∈ IFR(X × Y ), it holds dom(R1 ∪ R2 ) = dom(R1 ) ∪ dom(R2 ): Proof. Let x ∈ X . Then we obtain: dom(R1 ∪ R2 )(x) = hgt x(R1 ∪ R2 ) = sup(R1 ∪ R2 )(x; y) y ∈Y
= sup(sup(R1 (x; y); R2 (x; y))) y ∈Y
= sup(sup(R1 (x; y); R2 (x; y)); inf ( R1 (x; y); R2 (x; y))) y ∈Y
= sup[sup(R1 (x; y); R2 (x; y))]; inf [inf ( R1 (x; y); R2 (x; y))] y ∈Y
y ∈Y
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353
= sup sup R1 (x; y); sup R2 (x; y) ; inf inf R1 (x; y); inf R2 (x; y) y ∈Y
y ∈Y
y ∈Y
y ∈Y
= sup sup R1 (x; y); sup R2 (x; y) y ∈Y
y ∈Y
= sup(hgt xR1 ; hgt xR2 ) = sup(dom(R1 )(x); dom(R2 )(x)): Lemma 5.3. If I satis7es the following condition: I(sup(a; b); c) = inf (I(a; c); I(b; c)):
(2)
then it satis7es (M.1). If I satis7es (M.1), then I(sup(a; b); c) 6L∗ inf (I(a; c); I(b; c)):
(3)
Proof. Let I be an intuitionistic fuzzy implicator which satis1es formula (2). Then, for all x; x ; y ∈ L∗ , from x6L∗ x follows I(x ; y) = I(sup(x; x ); y) = inf (I(x; y); I(x ; y)), i.e. I(x ; y)6L∗ I(x; y). Hence (M.1) is veri1ed. Let now I satisfy (M.1). Then, for all a; b; c ∈ L∗ , from a6L∗ sup(a; b) follows I(a; c)¿L∗ I(sup (a; b); c). Similarly we obtain I(b; c)¿L∗ I(sup(a; b); c). Hence inf (I(a; c); I(b; c))¿L∗ I(sup (a; b); c). Theorem 5.5. Let T be a t-norm, S a t-conorm and N a unary involutive order reversing operator on L∗ . The S-implicator IS; T; N , de7ned as [9] IS;T;N (x; y) = N(x) ∨S;T y;
∀(x; y) ∈ (L∗ )2 ;
satis7es formula (2). Proof. Let a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ . De1ne a = (a1 ; a2 ) = N(a) and b = (b1 ; b2 ) = N(b). Since by de1nition inf (N(a); N(b)) is the largest element in L∗ that is smaller than both N(a) and N(b), we have (∀d ∈ L∗ )
(d 6L∗ N(a) ∧ d 6L∗ N(b) ⇒ d 6L∗ inf (N(a); N(b))):
Since N is order reversing, this is equivalent to (∀d ∈ L∗ )
(N(d) ¿L∗ a ∧ N(d) ¿L∗ b ⇒ N(d) ¿L∗ N(inf (N(a); N(b)))):
Since N is involutive (and thus bijective), this is equivalent to (∀d ∈ L∗ )
(d ¿L∗ a ∧ d ¿L∗ b ⇒ d ¿L∗ N(inf (N(a); N(b)))):
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Hence sup(a; b) = N(inf (N(a); N(b))), or N(sup(a; b)) = inf (N(a); N(b)). Using this equality, we obtain successively: IS;T;N (sup(a; b); c) = N(sup(a; b)) ∨S;T c = inf (N(a); N(b)) ∨S;T c = (S(inf (a1 ; b1 ); c1 ); T (sup(a2 ; b2 ); c2 )) = (S(min(a1 ; b1 ); c1 ); T (max(a2 ; b2 ); c2 )) = (min(S(a1 ; c1 ); S(b1 ; c1 )); max(T (a2 ; c2 ); T (b2 ; c2 ))) = inf (N(a) ∨S;T c; N(b) ∨S;T c) = inf (IS;T;N (a; c); IS;T;N (b; c)): Theorem 5.6. Let T be a t-norm and S a t-conorm. The R-implicator IT; S , de7ned as [9] IT;S (x; y) = sup{# ∈ L∗ | x ∧T;S # 6L∗ y} satis7es formula (2). Proof. Let a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ . {# = (#1 ; #2 ) ∈ L∗ | (T (max(a1 ; b1 ); #1 ); S(min(a2 ; b2 ); #2 )) 6L∗ c} = {# ∈ L∗ | (max(T (a1 ; #1 ); T (b1 ; #1 )); min(S(a2 ; #2 ); S(b2 ; #2 ))) 6L∗ c} = {# ∈ L∗ | T (a1 ; #1 ) 6 c1 ∧ T (b1 ; #1 ) 6 c1 ∧ S(a2 ; #2 ) ¿ c2 ∧ S(b2 ; #2 ) ¿ c2 } = {# ∈ L∗ | T (a1 ; #1 ) 6 c1 ∧ S(a2 ; #2 ) ¿ c2 } ∩ {# ∈ L∗ | T (b1 ; #1 ) 6 c1 ∧ S(b2 ; #2 ) ¿ c2 }: In the last formula, the symbol ∩ represents the intersection of two crisp sets. De1ne A1 = {# ∈ L∗ | T (a1 ; #1 )6c1 ∧ S(a2 ; #2 )¿c2 }, A2 = {# ∈ L∗ | T (b1 ; #1 )6c1 ∧ S(b2 ; #2 )¿c2 }, then IT;S (sup(a; b); c) = sup(A1 ∩ A2 ): From # ∈ A1 and # 6L∗ # follows # ∈ A1 , since ∧T; S is increasing. On the other hand, from # ∈ A1 and # ∈ A1 follows T (a1 ; #1 )6c1 , T (a1 ; #1 )6c1 , S(a2 ; #2 )¿c2 and S(a2 ; #2 )¿c2 , which implies T (a1 ; max(#1 ; #1 )) = max(T (a1 ; #1 ); T (a1 ; #1 ))6c1 and S(a2 ; min(#2 ; #2 )) = min(S(a2 ; #2 ); S(a2 ; #2 ))¿c2 . Hence sup(#; # ) ∈ A1 . It is now easy to see that A1 must have the form as in Fig. 2. Let = (1 ; 2 ) = sup(A1 ) and = (1 ; 2 ) = sup(A2 ). It is easily seen that ∧ is an upper bound for A1 ∩ A2 . Since {# ∈ L∗ | #1 ¡1 ∧ #2 ¿2 } ⊆ A1 and similarly {# ∈ L∗ | #1 ¡1 ∧ #2 ¿2 } ⊆ A2 , we
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Fig. 2. The set A1 = {# ∈ L∗ | T (a1 ; #1 )6c1 ∧ S(a2 ; #2 )¿c2 }.
obtain that A3 = {# ∈ L∗ | #1 ¡ min(1 ; 1 ) ∧ #2 ¿ max(2 ; 2 )} ⊆ A1 ∩ A2 . Hence it is clear that ∧ is the smallest upper bound, i.e. the supremum, of A1 ∩ A2 . It follows that IT;S (sup(a; b); c) = sup(A1 ∩ A2 ) = sup(A1 ) ∧ sup(A2 ) = inf (IT;S (a; c); IT;S (b; c)): Theorem 5.7. If dom R1 = dom R2 and I satis7es (M.1), then (R1 /bI S) ∩ (R2 /bI S) ⊇ (R1 ∪ R2 ) /bI S: If I satis7es formula (2) and not necessarily dom R1 = dom R2 , then (R1 /bI S) ∩ (R2 /bI S) ⊆ (R1 ∪ R2 ) /bI S; (R1 /kT
;S ;I
S) ∩ (R2 /kT
;S ;I
S) ⊆ (R1 ∪ R2 ) /kT
;S ;I
S:
Proof. Since R1 ⊆ R1 ∪ R2 and the same holds for R2 , we obtain by the corollary of Lemma 5.2 and I the monotonicity of the triangular subcomposition that R1 /I b S ⊇ (R1 ∪ R2 ) / b S and similarly for R2 . The 1rst statement follows easily. Let I be an intuitionistic fuzzy implicator satisfying formula (2). Then we obtain successively: inf (plt I(xR1 ; Sz); plt I(xR2 ; Sz)) = inf
inf I(R1 (x; y); S(y; z)); inf I(R2 (x; y); S(y; z))
y ∈Y
y ∈Y
= inf (inf (I(R1 (x; y); S(y; z)); I(R2 (x; y); S(y; z)))) y ∈Y
= inf (I(sup(R1 (x; y); R2 (x; y)); S(y; z))) y ∈Y
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= inf (I(R1 ∪ R2 (x; y); S(y; z))) y ∈Y
= plt I(x(R1 ∪ R2 ); Sz): Using the previous equality, we obtain (R1 /bI S) ∩ (R2 /bI S)(x; z) = inf (inf {plt I(xR1 ; Sz); plt I(xR2 ; Sz)}; inf {hgt(xR1 ); hgt(xR2 )}; hgt(Sz)); 6L∗ inf (plt I(x(R1 ∪ R2 ); Sz); sup{hgt(xR1 ); hgt(xR2 )}; hgt(Sz)) = inf (plt I(x(R1 ∪ R2 ); Sz); hgt(x(R1 ∪ R2 )); hgt(Sz)) = (R1 ∪ R2 ) /bI S(x; z): Since xR1 ⊆ x(R1 ∪ R2 ) and the t-norm T , the t-conorms S and sup are increasing, we obtain inf {hgt(xR1 ∩T ;S Sz); hgt(xR2 ∩T ;S Sz)} 6L∗ hgt(xR1 ∩T ;S Sz) 6L∗ hgt(x(R1 ∪ R2 ) ∩T ;S Sz): In a similar way as above, the last inequality can now be proved. Theorem 5.9. If I satis7es (M.1) and not necessarily dom R1 = dom R2 , then (R1 ∪ R2 ) /bI S ⊆ (R1 /bI S) ∪ (R2 /bI S); (R1 ∪ R2 ) /kT
;S ;I
S ⊆ (R1 /kT
;S ;I
S) ∪ (R2 /kT
;S ;I
S):
Proof. For all a = (a1 ; a2 ); b = (b1 ; b2 ); c = (c1 ; c2 ) ∈ L∗ we have sup(inf (a; b); inf (a; c)) = (max(min(a1 ; b1 ); min(a1 ; c1 )); min(max(a2 ; b2 ); max(a2 ; c2 ))) = (min(a1 ; max(b1 ; c1 )); max(a2 ; min(b2 ; c2 ))) = inf (a; sup(b; c)): Now we can prove the 1rst inequality: (R1 /bI S) ∪ (R2 /bI S)(x; z) = sup(inf {plt I(xR1 ; Sz); hgt(xR1 ); hgt(Sz)}; inf {plt I(xR2 ; Sz); hgt(xR2 ); hgt(Sz)})
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¿L∗ sup(inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR1 ); hgt(Sz)}; inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR2 ); hgt(Sz)}) = sup(inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; hgt(xR1 )}; inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; hgt(xR2 )}) = inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; sup[hgt(xR1 ); hgt(xR2 )]} = inf {inf [plt I(x(R1 ∪ R2 ); Sz); hgt(Sz)]; hgt x(R1 ∪ R2 )} = (R1 ∪ R2 ) /bI S(x; z): The second inequality is proved similarly in the following way: (R1 /kT
; S; I
S) ∪ (R2 /kT
; S; I
S)(x; z)
= sup(inf {plt I(xR1 ; Sz); hgt(xR1 ∩T ;S Sz)}; inf {plt I(xR2 ; Sz); hgt(xR2 ∩T ;S Sz)}) ¿L∗ sup(inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR1 ∩T ;S Sz)}; inf {plt I(x(R1 ∪ R2 ); Sz); hgt(xR2 ∩T ;S Sz)}) = inf (plt I(x(R1 ∪ R2 ); Sz); sup[hgt(xR1 ∩T ;S Sz); hgt(xR2 ∩T ;S Sz)]) = inf {plt I(x(R1 ∪ R2 ); Sz); hgt(x(R1 ∪ R2 ) ∩T ;S Sz)} = (R1 ∪ R2 ) /kT
; S; I
S(x; z):
A.4. On the interaction of the intuitionistic triangular compositions with the intersection Theorem 5.11. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z, I an intuitionistic fuzzy implicator, T a t-norm and S a t-conorm. If I satis7es (M.2) and not necessarily
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dom R1 = dom R2 , then (R1 ∩ R2 ) .bI S ⊆ (R1 .bI S) ∩ (R2 .bI S); (R1 ∩ R2 ) .kT
;S ;I
S ⊆ (R1 .kT
;S ;I
S) ∩ (R2 .kT
;S ;I
S):
Proof. Notice that inf (I(a; b); I(a; c))¿L∗ I(a; inf (b; c)), for all a; b; c ∈ L∗ , if I satis1es (M.2). Let (x; z) ∈ X × Z. Then (R1 .bI S) ∩ (R2 .bI S)(x; z) = inf {inf [plt I(Sz; xR1 ); hgt(xR1 ); hgt(Sz)]; inf [plt I(Sz; xR2 ); hgt(xR2 ); hgt(Sz)]} = inf {inf [plt I(Sz; xR1 ); plt I(Sz; xR2 )]; inf [hgt(xR1 ); hgt(xR2 )]; hgt(Sz)} ¿L∗ inf {plt I(Sz; x(R1 ∩ R2 )); hgt(x(R1 ∩ R2 )); hgt(Sz)} = (R1 ∩ R2 ) .bI S(x; z): In a similar way we can prove the second inequality, since hgt(x(R1 ∩ R2 ) ∩T ; S S)6L∗ inf (hgt(xR1 ∩T ; S S); hgt(xR2 ∩T ; S S)). Theorem 5.13. Let R1 and R2 be IFRs from X to Y , S an IFR from Y to Z and I an intuitionistic fuzzy implicator. If I satis7es (M.1) then (R1 /bk S) ∪ (R2 /bk S) ⊆ (R1 ∩ R2 ) /bk S: Proof. If I satis1es (M.1), then it satis1es condition (5). Hence ((R1 /bk S) ∪ (R2 /bk S))(x; z) = sup{plt I(xR1 ; Sz); plt I(xR2 ; Sz)} = sup inf I(R1 (x; y); S(y; z)); inf I(R2 (x; y); S(y; z)) y ∈Y
y ∈Y
6L∗ inf (sup{I(R1 (x; y); S(y; z)); I(R2 (x; y); S(y; z))}) y ∈Y
6L∗ inf I(inf (R1 (x; y); R2 (x; y)); S(y; z)) y ∈Y
= plt I(x(R1 ∩ R2 ); Sz) = (R1 ∩ R2 ) /bk S(x; z):
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Even if dom(R1 ) = dom(R2 ) and if I satis1es condition (4), there exists in general no relation I I I I between (R1 /I b S) ∩ (R2 / b S), (R1 / b S) ∪ (R2 / b S) and (R1 ∩ R2 ) / b S. Consider the IFRs R1 and R2 from X = {x} to Y = {y1 ; y2 ; y3 } and the IFR S from Y to Z = {z} de1ned as R1 = {((x; y1 ); 0:3; 0:2); ((x; y2 ); 0:3; 0:4); ((x; y3 ); 0:1; 0:9)}; R2 = {((x; y1 ); 0:2; 0:3); ((x; y2 ); 0:3; 0:2); ((x; y3 ); 0:1; 0:9)}; R1 ∩ R2 = {((x; y1 ); 0:2; 0:3); ((x; y2 ); 0:3; 0:4); ((x; y3 ); 0:1; 0:9)}; S = {((y1 ; z); 0:2; 0:3); ((y2 ; z); 0:2; 0:3); ((y3 ; z); 0:9; 0:1)} and the intuitionistic fuzzy implicator I de1ned as I(a; b) = Ns (a) ∨ b, ∀a; b ∈ L∗ . Then R1 /bI S(x; z) = inf {(0:2; 0:3); (0:3; 0:2); (0:9; 0:1)} = (0:2; 0:3); R2 /bI S(x; z) = inf {(0:2; 0:3); (0:3; 0:2); (0:9; 0:1)} = (0:2; 0:3); (R1 ∩ R2 ) /bI S(x; z) = inf {(0:3; 0:3); (0:3; 0:3); (0:9; 0:1)} = (0:3; 0:3): Hence (R1 /bI S) ∪ (R2 /bI S)(x; z) ¡L∗ (R1 ∩ R2 ) /bI S(x; z): Consider now the IFRs R1 and R2 from X to Y and the IFR S from Y to Z de1ned as R1 = {((x; y1 ); 0:2; 0:3); ((x; y2 ); 0:4; 0:5); ((x; y3 ); 0:4; 0:5)}; R2 = {((x; y1 ); 0:4; 0:6); ((x; y2 ); 0:1; 0:3); ((x; y3 ); 0:1; 0:3)}; R1 ∩ R2 = {((x; y1 ); 0:2; 0:6); ((x; y2 ); 0:1; 0:5); ((x; y3 ); 0:1; 0:5)}; S = {((y1 ; z); 0:6; 0:3); ((y2 ; z); 0:6; 0:3); ((y3 ; z); 0:6; 0:3)}: Then R1 /bI S (x; z) = inf {(0:6; 0:3); (0:4; 0:3); (0:6; 0:3)} = (0:4; 0:3); R2 /bI S (x; z) = inf {(0:6; 0:3); (0:4; 0:3); (0:6; 0:3)} = (0:4; 0:3); (R1 ∩ R2 ) /bI S (x; z) = inf {(0:6; 0:2); (0:2; 0:5); (0:6; 0:3)} = (0:2; 0:5):
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Hence (R1 /bI S ) ∩ (R2 /bI S )(x; z) ¿L∗ (R1 ∩ R2 ) /bI S (x; z): as
Consider at last the IFRs R1 and R2 from X to Y = {y1 ; y2 } and the IFR S from Y to Z de1ned R1 = {((x; y1 ); 0:6; 0:2); ((x; y2 ); 0:3; 0:6)}; R2 = {((x; y1 ); 0:2; 0:5); ((x; y2 ); 0:6; 0:2)}; R1 ∩ R2 = {((x; y1 ); 0:2; 0:5); ((x; y2 ); 0:3; 0:6)}; S = {((y1 ; z); 0:2; 0:6); ((y2 ; z); 0:4; 0:5)}:
Then R1 /bI S (x; z) = inf {(0:2; 0:6); (0:6; 0:2); (0:4; 0:5)} = (0:2; 0:6); R2 /bI S (x; z) = inf {(0:4; 0:5); (0:6; 0:2); (0:4; 0:5)} = (0:4; 0:5); (R1 ∩ R2 ) /bI S (x; z) = inf {(0:5; 0:3); (0:3; 0:5); (0:4; 0:5)} = (0:3; 0:5): Hence (R1 /bI S ) ∩ (R2 /bI S )(x; z) ¡L∗ (R1 ∩ R2 ) /bI S (x; z) ¡L∗ (R1 /bI S ) ∪ (R2 /bI S )(x; z): References [1] K.T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, So1a, June 1983 (Deposed in Central Science— Technical Library of Bulg. Academy of Science, 1697=84) (in Bulgarian). [2] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. [3] K.T. Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37–45. [4] K.T. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999. [5] W. Bandler, B. Kohout, Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and arti1cial systems, in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, pp. 341–367. [6] G. BirkhoM, Lattice Theory, AMS Colloquium Publications, Vol. 25, AMS Providence, RI, 1973, pp. 1–19, 111–112. [7] P. Burillo, H. Bustince, Intuitionistic fuzzy relations (Part I), Mathware Soft Comput. 2 (1995) 5–38. [8] H. Bustince, P. Burillo, Structures on intuitionistic fuzzy relations, Fuzzy Sets and Systems 78 (1996) 293–303. [9] C. Cornelis, G. Deschrijver, The compositional rule of inference in an intuitionistic fuzzy logic setting, in: K. Striegnitz (Ed.), Proc. 6th ESSLLI Student Session, Helsinki, August 2001. [10] B. De Baets, E.E. Kerre, Fuzzy relational compositions, Fuzzy Sets and Systems 60 (1993) 109–120.
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