On the composition of intuitionistic fuzzy relations

Fuzzy Sets and Systems 136 (2003) 333–361www.elsevier.com/locate/fss

On the composition of intuitionistic fuzzy relationsGlad 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 objectsare related to each other. However, they cannot model uncertainty: there is no means to attribute reliabilityinformation to the membership degrees. Intuitionistic fuzzy sets, as de1ned by Atanassov (Instuitionistic FuzzySets, Physica-Verlag, Heidelberg, New York, 1999), give us a way to incorporate uncertainty in an additionaldegree. 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 compositionof fuzzy relations to a composition of intuitionistic fuzzy relations. In this paper, we present an intuitionisticfuzzy version of the triangular compositions of Bandler and Kohout (in: P. Wang, S. Chang (Eds.), Theoryand Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, p. 341) and thevariants 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, interactionwith union and intersection. c© 2002 Elsevier Science B.V. All rights reserved.

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 originatedfrom the observation that real-life objects can be related to each other to a certain degree. How-ever, in real-life situations, a person may assume that a certain object A is in relation R with anotherobject 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).

0165-0114/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0165-0114(02)00269-5

334 G. Deschrijver, Etienne E. Kerre / Fuzzy Sets and Systems 136 (2003) 333–361

be a hesitation or uncertainty about the degree that is assigned to the relationship between A andB. 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]. Intuitionisticfuzzy 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 allx∈X , the number �A(x) = 1−�A(x)− A(x) is called the hesitation degree or the intuitionistic indexof x to A.

In fuzzy set theory, the non-membership degree of an element x of the universe is de1ned as oneminus the membership degree (using the standard negation) and thus it is 1xed. In intuitionistic fuzzyset theory, the non-membership degree is a more-or-less independent degree: the only condition isthat it is smaller than one minus the membership degree. Note that both �A and A can be seen asfuzzy sets on X (that are not completely independent from each other, because of the condition thatthe sum of the two degrees should be less than or equal to 1). In this way, the negation of the non-membership degree w.r.t. the standard fuzzy negation can be seen as a degree of membership. Sofor each element x∈X there exist two degrees that model the membership of x in the intuitionisticfuzzy set A, namely �A(x) and 1−A(x). The length of the interval [�A(x); 1−A(x)], which is givenby �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 intuitionisticfuzzy set in X ×Y [3,4]. In a similar way as above, if R is a relation between X and Y , x∈X andy∈Y , then �R(x; y) denotes the degree to which x is in relation R with y and �R(x; y) denotes theuncertainty degree to which x and y are in relation R with each other. So the “real” degree to whichx 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 considerthe 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 compositionR ◦ 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 zw.r.t S is de1ned as Sz = {y∈Y | (y; z)∈ S}. The relation R can be identi1ed with its characteristicmapping, namely

R : X ×Y → {0; 1}(x; y) �→ 1 if (x; y) ∈ R;

(x; y) �→ 0 else:

G. Deschrijver, Etienne E. Kerre / Fuzzy Sets and Systems 136 (2003) 333–361 335

Then the composition R ◦ S of R and S can also be written as

R ◦ S : X × Z → [0; 1]

(x; z) �→ supy∈Y

R(x; y) ∧B S(y; z);

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 compo-

sition R ◦ S is a relation from X to Y , consisting of those couples (x; z) for which there exists atleast 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 symptomsand 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 symptomy is a symptom of illness z, and S(y; z) = 0 else. Then R ◦ S(x; z) = 1 if patient x shows at least onesymptom of illness z. However, therapists may also want to know whether the symptoms shownby patient x are all symptoms of illness z, or whether all the symptoms of illness z are shown bypatient x.

Bandler and Kohout have de1ned the triangular compositions, in order to be able to model situa-tions 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) = infy∈Y

R(x; y) ⇒B S(y; z);

R .bk S(x; z) = infy∈Y

S(y; z) ⇒B R(x; 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 bypatient x are symptoms of illness z, and it belongs to R .bk S if patient x shows all symptoms ofillness z.

The domain of a (crisp) relation R is de1ned as dom(R) = {x∈X | xR = ∅}, and the range of Ris 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:


The 1rst expression means that if x is not in the domain of R, then x is in relation R /bk S with allelements of Z , even if there is no element of Y that is in relation Rt with x. A similar remark holdsfor the second expression. De Baets and Kerre [10,11,12] have therefore introduced the followingmodi1ed 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 equivalentways:

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 investigatewhether the results of De Baets and Kerre still hold. In Section 2 we 1rst recall the de1nitions ofthe 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 someoperators which we will need later on. In Section 4 we give the de1nition of intuitionistic fuzzyrelation and introduce the extension to intuitionistic fuzzy relations of the diMerent types of relationalcomposition. 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 de1nedby 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 foran arbitrary fuzzy set A in a universe X the height of A is de1ned as hgt(A) = supx∈X A(x). Therange 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 Tbe a triangular norm, then the sup−T composition R ◦T S of a fuzzy relation R from X to Y and a


fuzzy relation S from Y to Z is the fuzzy relation from X to Z de1ned by

R ◦T S(x; z) = supy∈Y

T (R(x; y); S(y; z)) = hgt T (xR; Sz):

Bandler and Kohout have extended their triangular compositions to fuzzy relations in the followingway:

R /Ibk S(x; z) = infy∈Y

I(R(x; y); S(y; z)) = plt I(xR; Sz);

R .Ibk S(x; z) = infy∈Y

I(S(y; z); R(x; y)) = plt I(Sz; xR);

where I is a fuzzy implicator, i.e. a [0; 1]2 → [0; 1] mapping that satis1es the boundary conditionsI(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 universeX is de1ned as plt(A) = infx∈X A(x).

De Baets and Kerre have introduced two classes of improved de1nitions based on the improvedversions in the crisp case [11]:

R /Ib S(x; z) = min(plt I(xR; Sz); hgt(xR); hgt(Sz));

R .Ib S(x; z) = min(plt I(Sz; xR); hgt(xR); hgt(Sz));

and

R /T;Ik S(x; z) = min(plt I(xR; Sz); hgt T (xR; Sz));

R .T;Ik 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 thelattice 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 anobject 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 ).


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 de-1ned by replacing min by an arbitrary t-norm T and max by an arbitrary t-conorm S. The generalizedintersection and union yield IFSs if (∀(x; y)∈ [0; 1]2)(T (x; y)6S∗(x; y)) (shortly T6S∗) 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] | x1A = ∅}; inf{x2 ∈ [0; 1] |Ax2 = ∅});

inf A = (inf{x1 ∈ [0; 1] | x1A = ∅}; 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)):


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 T6S∗. It has been shown that the condition T6S∗is necessary and suOcient for these operators to be well de1ned and that they are increasing in bothcomponents [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 } corre-

sponds 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, forthe IFS A we will also use the notation A(x) = (�A(x); A(x)). From now on, we will assume that ifa∈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 cal-culating with membership and non-membership degrees, since the pair formed by the two degreesis 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 theintuitionistic 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 borderconditions

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, andincreasing 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.


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

(∀(x; y) ∈ X × Y ) (�R(x; y) + R(x; y) 6 1):

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 }. Theheight of an IFS A in X is de1ned as hgt(A) = supx∈X A(x) = (supx∈X �A(x); infx∈X A(x)) (notice thathgt(A)∈L∗). The plinth of an IFS A in X is de1ned as plt(A) = infx∈X A(x) = (infx∈X �A(x); supx∈XA(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); infx∈X R(x; y)). The converse relation Rt of R is the IFR from Y to Xde1ned 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 casethat 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) = �

y∈Y(�(�R(x; y); �S(y; z)));

��◦�

�(x; z) = �

y∈Y(�(R(x; y); S(y; z)));

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 compositionful1lls 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 fuzzyimplicator, T ′ a t-norm and S ′ a t-conorm, then the triangular sub- and supercomposition and their


improved versions are de1ned as IFRs from X to Z de1ned by

R /Ibk S(x; z) = infy∈Y

I(R(x; y); S(y; z)) = plt I(xR; Sz);

R .Ibk S(x; z) = infy∈Y

I(S(y; z); R(x; y)) = plt I(Sz; xR);

R /Ib S(x; z) = inf

{infy∈Y

I(R(x; y); S(y; z)); supy∈Y

R(x; y); supy∈Y

S(y; z)

}

= inf{plt I(xR; Sz); hgt(xR); hgt(Sz)};

R .Ib S(x; z) = inf{plt I(Sz; xR); hgt(xR); hgt(Sz)};

R /T ′ ;S′ ;Ik S(x; z) = inf

{infy∈Y


(R(x; y) ∧T ′ ;S′ S(y; z))

}

= inf{plt I(xR; Sz); hgt(xR∩T ′ ;S′ Sz)};

R .T ′ ;S′ ;Ik 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 /Ibk S(x; z)is an element of L∗ given by(

infy∈Y

pr1 I((�R(x; y); R(x; y)); (�S(y; z); S(y; z)));

supy∈Y

pr2 I((�R(x; y); R(x; y)); (�S(y; z); S(y; z)))

);

where pri :L∗→ [0; 1] : (a1; a2) �→ ai, i = 1; 2. Then the IFR R /Ibk S is given by

R /Ibk S = {((x; z); pr1R /Ibk S(x; z); pr2R /Ibk S(x; z)) | x ∈ X; z ∈ Z}:

Example 4.1. Consider a set of patients X , a set of symptoms Y and a set of illnesses Z . Let R bethe 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


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 furthermorethat

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 /Ibk S(x; z) = (1; 0). This means that the degree to which all symptoms shown by patientx are symptoms of illness z is equal to 1, although patient x is only showing symptom y4 todegree 0:1, and with only an uncertainty given by 0:05. Such surprising results are the conse-quence of the fact that the Bandler and Kohout compositions do not take into account the degreeof emptiness or non-emptyness of the foresets and the aftersets involved. For the improved ver-sions we get R /Ib 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 /∩W ;+b;I

k S(x; z) = inf{(1; 0); (0; 0:95)}= (0; 0:95). Theseresults 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 to1nd out to which degree a patient x suMers from illness z. The therapist determines to which degreethe patient shows the symptoms y1; : : : ; y5 and for each symptom he gives a degree of how sure heis about that degree. Assume he gets the following table (in the 1rst row is written to which degreepatient x shows each symptom, in the second row is written the degree of how sure the therapist iswhen 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 wetake the fuzzy negation of each element in the second row, then we get the degree of uncertainty towhich (x; yi)∈R, for each i∈{1; : : : ; 5}. For instance, for y2, the membership degree of (x; y2) to R isgiven by �R(x; y2) = 0:5 and the degree of uncertainty is given by �R(x; y2) = 1−0:8 = 0:2. We obtainimmediately the degree of non-membership as R(x; y2) = 1 − �R(x; y2) − �R(x; y2) = 0:3. Applying


the same method for the other symptoms y1; y3; : : : ; y5, we obtain the following IFS modelling thedegree 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 arecharacteristic 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)}:

Let T ′ = min, S ′ = max, I(a; b) = sup(Ns(a); b). Then we obtain for instance, R supinf ◦min

max S(x; z) =(0:9; 0:1), R /Ib S(x; z) = (0:3; 0:5), and R .Ib S(x; z) = (0:4; 0:5). This means that the degree to whichpatient x shows at least one symptom of illness z is equal to 0:9 with no hesitation, the degree towhich 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 onlyconsider the sup

inf◦minmax 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- andsupercomposition, 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 composi-tions: 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 /T ′ ;S′ ;Ik S ⊆ R sup

inf ◦T ′S′ S;

R .T ′ ;S′ ;Ik S ⊆ R sup

inf ◦T ′S′ S;

R /T ′ ;S′ ;Ik S ⊆ R /Ib S;

R .T ′ ;S′ ;Ik S ⊆ R .Ib S:

A general containment rule between R /Ib S and R supinf ◦T

′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. Letthe intuitionistic fuzzy implicator I be such that I(a; b) =Ns(a)∨ b, ∀a; b∈L∗ and let T ′ = min


and S ′ = max. Then we obtain

R /Ib S(x1; z) = (0:2; 0:8) ¿L∗ R supinf ◦T ′

S′ S(x1; z) = (0:1; 0:9)

and

R /Ib S(x2; z) = (0:1; 0:9) ¡L∗ R supinf ◦T ′

S′ S(x2; z) = (0:2; 0:8):

Similarly there is no containment rule between R/Ibk S and R supinf ◦T

′S′ S(x1; z), since R/Ibk S(x1; z) = (0:5;

0:5)¿L∗R supinf ◦T

′S′ S(x1; z) and R .Ibk S(x2; z) = (0:1; 0:9)¡L∗R sup

inf ◦T′

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 thenumber of pairs (x; z) for which there exists at least one y that is in relation Rt with x and in relationS with z. The theorem shows that this intuitively natural property still holds for intuitionistic fuzzyrelations, but only for /T ′ ; S′ ;I

k . We then have that for any (x; z)∈X ×Z , the degree to which all they∈Y that are in relation Rt with x are also in relation with z, is smaller than the degree to whichthere 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 compositionsin 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:

(R /Ibk S)t = St .Ibk Rt;

(R .Ibk S)t = St /Ibk Rt;

(R /Ib S)t = St .Ib Rt;

(R .Ib S)t = St /Ib Rt;

(R /T ′ ; S′ ; Ik S)t = St .T ′ ; S′ ; I

k Rt;

(R .T ′ ; S′ ; Ik S)t = St /T ′ ; S′ ; I

k Rt:

Proof. Notice that (R /T ′ ; S′ ;Ik S)t ∈IFR(Z ×X ). Hence we calculate, for x∈X and z ∈Z :

(R /T ′ ; S′ ; Ik S)t(z; x) = R /T ′ ; S′ ; I

k S(x; z)

= inf

{infy∈Y


(R(x; y) ∧T ′ ;S′ S(y; z))

}


= inf

{infy∈Y

I(Rt(y; x); St(z; y)); supy∈Y

(St(z; y) ∧T ′ ;S′ Rt(y; x))

}

= St .T ′ ;S′ ;Ik 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 anintuitionistic 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 /Ib S ⊇ R2 /Ib S:

If I satis7es (M.2), then:

R1 ⊆ R2 ⇒ R1 .Ib S ⊆ R2 .Ib S;

R1 ⊆ R2 ⇒ R1 .T ′ ;S′ ;Ik S ⊆ R2 .T ′ ;S′ ;I

k S:

In the case of /T ′ ; S′ ;Ik , 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/min;max;Ik S(x1; z1) = (0:1; 0:9)¡L∗R2/

min;max;Ik S(x1; z1) = (0:2; 0:8) and R1/

min;max;Ik S(x1; z2) =

(0:6; 0:4)¿L∗R2 /min;max;Ik S(x1; z2) = (0:5; 0:5).


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(x1R1

∩ Sz1) = (0:1; 0:9)¡L∗(0:2; 0:8) = hgt(x1R2 ∩ 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 .Ib S1 ⊇ R .Ib S2:

If I satis7es (M.2), then

S1 ⊆ S2 ⇒ R /Ib S1 ⊆ R /Ib S2;

S1 ⊆ S2 ⇒ R /T ′ ; S′ ; Ik S1 ⊆ R /T ′ ; S′ ; I

k 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 isan 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 betweenthe 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 its1rst component. For intuitionistic fuzzy implicators, this is not the case. Let, for instance, I be theintuitionistic fuzzy implicator de1ned by, for all x; y∈L∗,

I(x; y) =

{(Nsx ∨+̂;· y) ∧·;+̂ y if x1 ¿ x2;

Nsx ∨+̂;· y else;


where · is the product, and +̂ the probabilistic sum de1ned as, for m; n∈ [0; 1], m+̂n=m+n−m ·n.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) ∨+̂;· (0:2; 0:8)) ∧·;+̂ (0:2; 0:8) = (0:36; 0:24) ∧·;+̂ (0:2; 0:8) = (0:072; 0:848);

I(b; c) = (0:5; 0:4) ∨+̂;· (0:2; 0:8) = (0:6; 0:32);

I(sup(a; b); c) = ((0:2; 0:4) ∨+̂;· (0:2; 0:8)) ∧·;+̂ (0:2; 0:8) = (0:36; 0:32) ∧·;+̂ (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 operatoron 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. Itwill be seen that only some semi-distributivity properties hold.

Theorem 5.7. If dom R1 = dom R2 and I satis7es (M.1), then

(R1 /Ib S) ∩ (R2 /Ib S) ⊇ (R1 ∪ R2) /Ib S:

If I satis7es formula (2) and not necessarily dom R1 = dom R2, then

(R1 /Ib S) ∩ (R2 /Ib S) ⊆ (R1 ∪ R2) /Ib S;

(R1 /T ′ ;S′ ;Ik S) ∩ (R2 /T ′ ;S′ ;I

k S) ⊆ (R1 ∪ R2) /T ′ ;S′ ;Ik S:

Theorem 5.8. If dom S1 = dom S2 and I satis7es (M.1), then

(R .Ib S1) ∩ (R .Ib S2) ⊇ R .Ib (S1 ∪ S2):


If I satis7es formula (2) and not necessarily dom S1 = dom S2, then

(R .Ib S1) ∩ (R .Ib S2) ⊆ R .Ib (S1 ∪ S2);

(R .T ′ ;S′ ;Ik S1) ∩ (R .T ′ ;S′ ;I

k S2) ⊆ R .T ′ ;S′ ;Ik (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

(R1supinf ◦T ′

S′ S) ∪ (R2supinf ◦T ′

S′ S)(x; z) = (R1 ∪ R2) supinf ◦T ′

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) /Ib S ⊆ (R1 /Ib S) ∪ (R2 /Ib S);

(R1 ∪ R2) /T ′ ;S′ ;Ik S ⊆ (R1 /T ′ ;S′ ;I

k S) ∪ (R2 /T ′ ;S′ ;Ik S):

Theorem 5.10. If I satis7es (M.1) and not necessarily dom S1 = dom S2, then

R .Ib (S1 ∪ S2) ⊆ (R .Ib S1) ∪ (R .Ib S2);

R .T ′ ;S′ ;Ik (S1 ∪ S2) ⊆ (R .T ′ ;S′ ;I

k S1) ∪ (R .T ′ ;S′ ;Ik S2):

Using the above theorems, we may also conclude the following:

• if I satis1es formula (2), then

(R1 /Ib S) ∩ (R2 /Ib S) ⊆ (R1 ∪ R2) /Ib S ⊆ (R1 /Ib S) ∪ (R2 /Ib S)

and similarly for /T ′ ; S′ ;Ik ;

• if I satis1es formula (2) and dom(R1) = dom(R2), then

(R1 /Ib S) ∩ (R2 /Ib S) = (R1 ∪ R2) /Ib S;

• if I satis1es formula (2), then

(R .Ib S1) ∩ (R .Ib S2) ⊆ R .Ib (S1 ∪ S2) ⊆ (R .Ib S1) ∪ (R .Ib S2)

and similarly for .T ′ ; S′ ;Ik ;


• if I satis1es formula (2) and dom(S1) = dom(S2), then

(R .Ib S1) ∩ (R .Ib S2) = R .Ib (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 satisfysome 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 intuition-istic fuzzy implicator, T ′ a t-norm and S ′ a t-conorm. If I satis7es (M.2) and not necessarilydom R1 = dom R2, then

(R1 ∩ R2) .Ib S ⊆ (R1 .Ib S) ∩ (R2 .Ib S);

(R1 ∩ R2) .T ′ ;S′ ;Ik S ⊆ (R1 .T ′ ;S′ ;I

k S) ∩ (R2 .T ′ ;S′ ;Ik S):

Theorem 5.12. Let R be an IFR from X to Y , S1 and S2 be IFRs from Y to Z , I an intuition-istic fuzzy implicator, T ′ a t-norm and S ′ a t-conorm. If I satis7es (M.2) and not necessarilydom S1 = dom S2, then

R /Ib (S1 ∩ S2) ⊆ (R /Ib S1) ∩ (R /Ib S2);

R /T ′ ;S′ ;Ik (S1 ∩ S2) ⊆ (R /T ′ ;S′ ;I

k S1) ∩ (R /T ′ ;S′ ;Ik S2):

In a similar way as Lemma 5.3, the following lemma is proved.


I(inf (a; b); c) = sup(I(a; c);I(b; c)) (4)


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). HoweverR-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 a16c1¡b1 anda2¡c26b2. 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 intuitionisticfuzzy implicator. If I satis7es (M.1) then

(R1 /bk S) ∪ (R2 /bk S) ⊆ (R1 ∩ R2) /bk S


Even if dom(R1) = dom(R2) and if I satis1es condition (4), there exists in general no relation be-tween (R1 /Ib S)∩ (R2 /Ib S), (R1 /Ib S)∪ (R2 /Ib S) and (R1 ∩R2) /Ib 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 /T ′ ;S′ ;Ik S ⊆ R sup

inf ◦T ′S′ S;

R .T ′ ;S′ ;Ik S ⊆ R sup

inf ◦T ′S′ S;

R /T ′ ;S′ ;Ik S ⊆ R /Ib S;

R .T ′ ;S′ ;Ik S ⊆ R .Ib S:

Proof. Let x∈X , z ∈Z , then

R /T ′ ;S′ ;Ik S(x; z) = inf{plt I(xR; Sz); hgt(xR∩T ′ ;S′ Sz)}

6L∗ hgt(xR∩T ′ ;S′ Sz)

=

(supy∈Y

T ′(�R(x; y); �S(y; z)); infy∈Y

S ′(R(x; y); S(y; z))

)

= R supinf ◦T ′

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.

To prove that there does not exist a general containment rule between R /Ib S and R supinf ◦T

′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)} andS = {((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 /Ib 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);


R supinf ◦T ′

S′ S(x1; z) = sup((0:1; 0:9); (0:1; 0:9)) = (0:1; 0:9);

R /Ib 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);

R supinf ◦T ′

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 mono-tonicity 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 /Ib S ⊇ R2 /Ib S:

If I satis7es (M.2), then:

R1 ⊆ R2 ⇒ R1 .Ib S ⊆ R2 .Ib S;

R1 ⊆ R2 ⇒ R1 .T ′ ;S′ ;Ik S ⊆ R2 .T ′ ;S′ ;I

k 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;

infy∈Y

I(R1(x; y); S(y; z)) ¿L∗ infy∈Y

I(R2(x; y); S(y; z));

R1 /Ib S ⊇ R2 /Ib S:

Analogously, using Lemma 5.1, the second statement is proved.


In the case of /T ′ ; S′ ;Ik , 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 /min;max;Ik 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 /min;max;Ik 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) = supy∈Y

(R1 ∪ R2)(x; y)

= supy∈Y

(sup(R1(x; y); R2(x; y)))

= supy∈Y

(sup(�R1(x; y); �R2(x; y)); inf (R1(x; y); R2(x; y)))

=

(supy∈Y

[sup(�R1(x; y); �R2(x; y))]; infy∈Y

[inf (R1(x; y); R2(x; y))]

)


=

(sup

[supy∈Y

�R1(x; y); supy∈Y

�R2(x; y)

]; inf

[infy∈Y

R1(x; y); infy∈Y

R2(x; y)])

= sup

(supy∈Y

R1(x; y); supy∈Y

R2(x; y)

)

= sup(hgt xR1; hgt xR2)

= sup(dom(R1)(x); dom(R2)(x)):


I(sup(a; b); c) = inf (I(a; c);I(b; c)): (2)


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 operatoron 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′ = (a′1; a′2) =N(a) and b′ = (b′1; b′2) =N(b). Since by de1nition inf (N(a);N(b)) is the largest element in L∗ that is smaller than bothN(a) and N(b), we have

(∀d ∈ L∗) (d6L∗ N(a) ∧ d6L∗ N(b) ⇒ d6L∗ 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)))):


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 (a′1; b′1); c1); T (sup(a′2; b

′2); c2))

= (S(min(a′1; b′1); c1); T (max(a′2; b

′2); c2))

= (min(S(a′1; c1); S(b′1; c1));max(T (a′2; c2); T (b′2; 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 boundfor A1 ∩A2. Since {#∈L∗ | #1¡�1 ∧ #2¿�2}⊆A1 and similarly {#∈L∗ | #1¡�1 ∧ #2¿�2}⊆A2, we


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 /Ib S) ∩ (R2 /Ib S) ⊇ (R1 ∪ R2) /Ib S:

If I satis7es formula (2) and not necessarily dom R1 = dom R2, then

(R1 /Ib S) ∩ (R2 /Ib S) ⊆ (R1 ∪ R2) /Ib S;

(R1 /T ′ ;S′ ;Ik S) ∩ (R2 /T ′ ;S′ ;I

k S) ⊆ (R1 ∪ R2) /T ′ ;S′ ;Ik S:

Proof. Since R1 ⊆R1 ∪R2 and the same holds for R2, we obtain by the corollary of Lemma 5.2 andthe monotonicity of the triangular subcomposition that R1 /Ib S ⊇ (R1 ∪R2) /Ib 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(

infy∈Y

I(R1(x; y); S(y; z)); infy∈Y

I(R2(x; y); S(y; z)))

= infy∈Y

(inf (I(R1(x; y); S(y; z));I(R2(x; y); S(y; z))))

= infy∈Y

(I(sup(R1(x; y); R2(x; y)); S(y; z)))


= infy∈Y

(I(R1 ∪ R2(x; y); S(y; z)))

= plt I(x(R1 ∪ R2); Sz):

Using the previous equality, we obtain

(R1 /Ib S) ∩ (R2 /Ib 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) /Ib 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) /Ib S ⊆ (R1 /Ib S) ∪ (R2 /Ib S);

(R1 ∪ R2) /T ′ ;S′ ;Ik S ⊆ (R1 /T ′ ;S′ ;I

k S) ∪ (R2 /T ′ ;S′ ;Ik 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 /Ib S) ∪ (R2 /Ib S)(x; z)

= sup(inf{plt I(xR1; Sz); hgt(xR1); hgt(Sz)};

inf{plt I(xR2; Sz); hgt(xR2); hgt(Sz)})


¿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) /Ib S(x; z):

The second inequality is proved similarly in the following way:

(R1 /T ′ ; S′ ; Ik S) ∪ (R2 /T ′ ; S′ ; I

k 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) /T ′ ; S′ ; Ik 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 intuition-istic 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) .Ib S ⊆ (R1 .Ib S) ∩ (R2 .Ib S);

(R1 ∩ R2) .T ′ ;S′ ;Ik S ⊆ (R1 .T ′ ;S′ ;I

k S) ∩ (R2 .T ′ ;S′ ;Ik 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 .Ib S) ∩ (R2 .Ib 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) .Ib 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 intuitionisticfuzzy 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{

infy∈Y

I(R1(x; y); S(y; z)); infy∈Y

I(R2(x; y); S(y; z))}

6L∗ infy∈Y

(sup{I(R1(x; y); S(y; z));I(R2(x; y); S(y; z))})

6L∗ infy∈Y

I(inf (R1(x; y); R2(x; y)); S(y; z))

= plt I(x(R1 ∩ R2); Sz)

= (R1 ∩ R2) /bk S(x; z):


Even if dom(R1) = dom(R2) and if I satis1es condition (4), there exists in general no relationbetween (R1 /Ib S)∩ (R2 /Ib S), (R1 /Ib S)∪ (R2 /Ib S) and (R1 ∩R2) /Ib S. Consider the IFRs R1 andR2 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 /Ib S(x; z) = inf{(0:2; 0:3); (0:3; 0:2); (0:9; 0:1)} = (0:2; 0:3);

R2 /Ib S(x; z) = inf{(0:2; 0:3); (0:3; 0:2); (0:9; 0:1)} = (0:2; 0:3);

(R1 ∩ R2) /Ib S(x; z) = inf{(0:3; 0:3); (0:3; 0:3); (0:9; 0:1)} = (0:3; 0:3):

Hence

(R1 /Ib S) ∪ (R2 /Ib S)(x; z) ¡L∗ (R1 ∩ R2) /Ib S(x; z):

Consider now the IFRs R′1 and R′

2 from X to Y and the IFR S ′ from Y to Z de1ned as

R′1 = {((x; y1); 0:2; 0:3); ((x; y2); 0:4; 0:5); ((x; y3); 0:4; 0:5)};

R′2 = {((x; y1); 0:4; 0:6); ((x; y2); 0:1; 0:3); ((x; y3); 0:1; 0:3)};

R′1 ∩ R′

2 = {((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

R′1 /Ib S ′(x; z) = inf{(0:6; 0:3); (0:4; 0:3); (0:6; 0:3)} = (0:4; 0:3);

R′2 /Ib S ′(x; z) = inf{(0:6; 0:3); (0:4; 0:3); (0:6; 0:3)} = (0:4; 0:3);

(R′1 ∩ R′

2) /Ib S ′(x; z) = inf{(0:6; 0:2); (0:2; 0:5); (0:6; 0:3)} = (0:2; 0:5):


Hence

(R′1 /Ib S ′) ∩ (R′

2 /Ib S ′)(x; z) ¿L∗ (R′1 ∩ R′

2) /Ib S ′(x; z):

Consider at last the IFRs R′′1 and R′′

2 from X to Y = {y1; y2} and the IFR S ′′ from Y to Z de1nedas

R′′1 = {((x; y1); 0:6; 0:2); ((x; y2); 0:3; 0:6)};

R′′2 = {((x; y1); 0:2; 0:5); ((x; y2); 0:6; 0:2)};

R′′1 ∩ R′′

2 = {((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

R′′1 /Ib S ′′(x; z) = inf{(0:2; 0:6); (0:6; 0:2); (0:4; 0:5)} = (0:2; 0:6);

R′′2 /Ib S ′′(x; z) = inf{(0:4; 0:5); (0:6; 0:2); (0:4; 0:5)} = (0:4; 0:5);

(R′′1 ∩ R′′

2 ) /Ib S ′′(x; z) = inf{(0:5; 0:3); (0:3; 0:5); (0:4; 0:5)} = (0:3; 0:5):

Hence

(R′′1 /Ib S ′′) ∩ (R′′

2 /Ib S ′′)(x; z) ¡L∗ (R′′1 ∩ R′′

2 ) /Ib S ′′(x; z)

¡L∗ (R′′1 /Ib S ′′) ∪ (R′

2 /Ib S ′′)(x; z):

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On the composition of intuitionistic fuzzy relations

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