#### Abstract

We establish, by means of a large class of continuous t-representable intuitionistic fuzzy t-conorms, a factorization of an intuitionistic fuzzy relation (IFR) into a unique indifference component and a family of regular strict components. This result generalizes a previous factorization obtained by Dimitrov (2002) with the intuitionistic fuzzy t-conorm. We provide, for a continuous t-representable intuitionistic fuzzy t-norm , a characterization of the -transitivity of an IFR. This enables us to determine necessary and sufficient conditions on a -transitive IFR under which a strict component of satisfies pos-transitivity and negative transitivity.

#### 1. Introduction

In the real life, individual or collective preferences are not always crisp; they can be also ambiguous. Since 1965 when Zadeh [1] introduced fuzzy set theory, researchers [2β11] modelled such preferences by (binary) fuzzy relation (simply denoted by FR) on that is, a function where is a set of alternatives with In this case, for is interpreted as the degree to which is βat least as good as" If then is crisp, and we denote by and by Literature on the theory of fuzzy relations and on applications of fuzzy relations in other fields such as economics and in particular social choice theory is growing.

Since 1983 when Atanassov [12, 13] introduced intuitionistic fuzzy sets (IFSs), some scholars [14β19] modelled ambiguous preferences by a (binary) intuitionistic fuzzy relation (IFR) on , that is, a function where In this case, is the degree to which is βat least as good as" , and is the degree to which is not βat least as good as" The positive real number (since usually called fuzzy index, indicates the degree of incomparability between and . In this paper, we simply write . Clearly, we have two particular cases: (i) if that is, then becomes an FR on , and (ii) if and , then becomes the well-known (binary) crisp relation. In the first case, we simply write and in the second case, we have (i.e., ).

A factorization of a binary relation is an important question in preference modelling. In that view, Dimitrov [18] established a factorization of an IFR into an indifference and a strict component in the particular case where the union is defined by means of the () -representable intuitionistic fuzzy -conorm. Recently, Cornelis et al. [20] established some results on -representable intuitionistic fuzzy -norms (i.e., where is a fuzzy -conorm, and is a fuzzy -norm satisfying on -representable intuitionistic fuzzy -conorms (i.e., ) and on intuitionistic fuzzy implications. Thereby, our goal is to generalize Dimitrov's framework [18] and to establish some results on IFRs by means of continuous -representable intuitionistic fuzzy -norms and -conorms.

The aim of this paper is (i) to study the standard completeness of an IFR, (ii) to establish a characterization of the -transitivity of an IFR, (iii) to generalize the factorization of an IFR established by Dimitrov [18], and (iv) to determine necessary and sufficient conditions on a -transitive IFR under which a given strict component of (obtained in our factorization) satisfies respectively pos-transitivity and negative transitivity.

First we establish some useful results on -representable intuitionistic fuzzy -norms, -representable intuitionistic fuzzy -conorms, and intuitionistic fuzzy implications.

The paper is organized as follows. In Section 2, we recall some basic notions and properties on fuzzy operators and intuitionistic fuzzy operators which we need throughout the paper. We also establish some useful results on fuzzy implications and intuitionistic fuzzy implications. Section 3 has three subsections. In Section 3.1, we recall some basic and useful definitions on IFRs. In Section 3.2, we introduce the standard completeness, namely, a -completeness of an IFR. We make clear that the notion of completeness introduced by Dimitrov [18] is not standard, but it is weaker than a standard one. In Section 3.3, we establish, for a given a characterization of the -transitivity of an IFR. Section 4 is devoted to a new factorization of an IFR, and it has two subsections. In Section 4.1, we recall the factorization of an IFR established by Dimitrov [18] with the intuitionistic fuzzy -conorm. We point out some intuitive difficulties of the strict component obtained in [18]. In Section 4.2, we introduce definitions of an indifference and a strict component of an IFR, and we establish a general factorization of an IFR for a large class of continuous -representable intuitionistic fuzzy -conorms. Section 5 contains two subsections. In Section 5.1, we introduce intuitionistic fuzzy counterparts of pos-transitivity and negative transitivity of a crisp relation. We justify that there exists some IFRs (noncrisp and non FRs) which violate each of these two properties. This forces us in Section 5.2 to establish necessary and sufficient conditions on a -transitive IFR such that a strict component of satisfies, respectively, pos-transitivity and negative transitivity. Section 6 contains some concluding remarks. The proofs of our results are in the Appendix. (This was suggested by an anonymous referee.)

#### 2. Preliminaries on Operators

Let be an order in defined by and . is a complete lattice. and are the units of

In the following section, we recall some definitions, examples, and well-known results on fuzzy -norms, fuzzy -conorms, fuzzy implications, and fuzzy coimplicators.

##### 2.1. Review on Fuzzy Operators

We firstly recall notions on fuzzy -norms and fuzzy -conorms (see [21, 22]).

A fuzzy -norm (resp. a fuzzy -conorm) is an increasing, commutative, and associative binary operation on with a neutral 1 (resp. 0). The dual of a fuzzy -norm is a fuzzy -conorm that is, , .

Let us recall two usual families of fuzzy -norms and fuzzy -conorms. The Frank -norms that is, , where , and are the minimum fuzzy -norm, the product fuzzy -norm, and the ukasiewicz fuzzy -norm, respectively. The Frank -conorms that is, , where , and are the maximum fuzzy -conorm, the product fuzzy -conorm, and the ukasiewicz fuzzy -conorm, respectively.

A fuzzy -norm (fuzzy -conorm ) is strict if , implies (resp. , , implies ). The product fuzzy -norm (resp. the product fuzzy -conorm) is an example of a strict fuzzy -norm (resp. fuzzy -conorm).

We have the following properties:

Throughout the paper, is a continuous fuzzy -norm, and is a continuous fuzzy -conorm.

In the following, we recall some definitions and examples on fuzzy implications and fuzzy coimplicators based on fuzzy -norms and fuzzy -conorms, respectively (see [21β23]).

The fuzzy -implication associated to is a binary operation on defined by The fuzzy coimplicator associated to is a binary operation on defined by

Let us recall some usual examples of these fuzzy operators.

The fuzzy -implication associated to is defined by The fuzzy coimplicator associated to is defined by The fuzzy -implication associated to is defined by The fuzzy coimplicator associated to is defined by The fuzzy -implication associated to is defined by The fuzzy coimplicator associated to is defined by

We complete the previous examples by giving expressions of fuzzy R-implications of the other Frank fuzzy -norms and fuzzy coimplicators of the other Frank fuzzy -conorms:

We recall some useful properties on fuzzy implications and fuzzy coimplicators.

Proposition 2.1 (See [4, 5, 9, 21, 23]). *For all ,*

(1)*, and ;*(2)*, and ;*(3)*(4)**(5)*

In the following, we recall some useful definitions and results on intuitionistic fuzzy operators.

##### 2.2. Review on Intuitionistic Fuzzy Operators

*Definition 2.2 (See [20]). * An intuitionistic fuzzy -norm is an increasing, commutative, and associative binary operation on satisfying

() An intuitionistic fuzzy -conorm is an increasing, commutative, associative binary operation on satisfying

Cornelis et al. [20] introduced an important class of intuitionistic fuzzy -norms (resp. -conorms) based on fuzzy -norms (resp. fuzzy -conorms).

*Definition 2.3. *An intuitionistic fuzzy -norm (resp. -conorm ) is called -representable if there exists a fuzzy -norm and a fuzzy -conorm (resp. a fuzzy -conorm and fuzzy -norm ) such that (resp. .

and (resp. and ) are called the representants of (resp.

The theorem below states conditions under which a pair of connectives on gives rise to a -representable intuitionistic fuzzy -norm (-conorm).

Theorem 2.4 (see Cornelis et al. [20, Theorem , pages 60β61]). *Given a fuzzy -norm and a fuzzy -conorm satisfying **The mappings and defined by, for and in and , are, respectively, a -representable intuitionistic fuzzy -norm and -representable intuitionistic fuzzy -conorm.*

Throughout the paper, we consider only continuous -representable intuitionistic fuzzy -conorms (shortly if--conorm) and continuous -representable intuitionistic fuzzy -norms (shortly if--norm). They are denoted by and , respectively, where

From the previous result, we deduce some examples of if--norms and if--conorms.

*Example 2.5. * and are, respectively, if--norm and if--conorm associated to and since

and are, respectively, if--norm and if--conorm associated to and since and are, respectively, if--norm and if--conorm associated to and since

*Definition 2.6 (see Cornelis et al. [20, Definition , page 64]). * The intuitionistic fuzzy R-implication (shortly if-R-implication) associated with an if--norm is a binary operation on defined by: , .

The intuitionistic fuzzy coimplicator (shortly if-coimplicator) associated with an if--conorm is a binary operation on defined by: , .

We establish in the sequel some new and basic results on the previous implications. These results will be useful later.

##### 2.3. Some Basic Results on Fuzzy Implications and If-Implications

The following result establishes two links between the fuzzy R-implication and the fuzzy coimplicator

Proposition 2.7. *Let and such that Then** for all ** if and are dual, then *

The following result gives expressions of an if-R-implication and an if-coimplicator by means of and

Lemma 2.8. *For all ,*(1)*;*(2)*.*

We now introduce a new condition which can be satisfied by a if--conorm

*Definition 2.9. * satisfies condition if

Let us end this section by giving some examples of if--conorms satisfying condition This justifies that the class of continuous -representable if--conorms satisfying condition is not empty.

Proposition 2.10. * For all satisfies condition ** If and are dual, then the restriction of on satisfies condition *

In the next section, we recall some basic notions on IFRs and study its standard completeness (see Atanassov [12], Bustince and Burillo [15], and Dimitrov [17, 18]). We establish, for a given a characterization of the -transitivity of an IFR

#### 3. Preliminaries on IFRs

##### 3.1. Review on IFRs

An IFS in is an expression given by , where and are functions satisfying the condition The numbers and denote, respectively, the degree of membership and the degree of nonmembership of the element in The number is an index of the element in Obviously, when that is, the IFS is a fuzzy set (simply denoted by FS) in . In this case,

Let and be two IFSs, and let . The intuitionistic fuzzy union associated to is an IFS defined by (we recall that if and are FSs, and and are dual, then becomes the well-known fuzzy union defined by ). And if and are crisp, becomes the crisp union). As defined in the Introduction, an IFR in is an IFS in .

We complete some basic definitions on IFRs.

*Definition 3.1. *Let be an IFR.

is reflexive if . is symmetric if and . is -symmetric if , is perfect antisymmetric if The converse of is the IFR denoted and defined by , and .

In the following, we recall the well-known notion of completeness of a crisp relation in We then present definition of the standard completeness of a FR and its two usual and particular cases (weak completeness and strong completeness). Following that line, we introduce the definition of the standard completeness of an IFR. We establish a link between that standard definition and the one introduced by Dimitrov (see [17, 18]). And we write the two particular cases of that standard definition.

##### 3.2. Intuitionistic Fuzzy Standard Completeness (-Completeness)

Let be a reflexive IFR and .

When is a crisp relation, is complete if that is,

When is a FR, for the fuzzy -conorm is -complete if that is, In particular, if , we simply say that is strongly complete, that is, If , we simply say that is weakly complete, that is, (see Fono and Andjiga [7, Definition , page 375]).

In the general case where is an IFR and , we have the following generic version of the standard completeness of

*Definition 3.2. * is -complete if that is,

*Remark 3.3. *If an IFR becomes a FR, and and are dual, then -completeness becomes -completeness. Furthermore, if becomes crisp, then -completeness and -completeness become crisp completeness.

Dimitrov (see [17, Definition , page 151]) introduced the following version of completeness of an IFR: is -complete if ,

It is important to notice that -completeness is not a version of the standard completeness. However, the following result shows that it is weaker than each version of the standard completeness.

Proposition 3.4. *If is -complete, then is -complete.*

As for FRs, we deduce the two following interesting particular cases of -completeness when .

*Example 3.5. *Let be a reflexive IFR and .

(1)If , then is -complete if ,
In this case, we simply say that is strongly complete.(2)If , then is -complete if ,
In this case, we simply say that is weakly complete.

We notice that, if becomes a FR, then intuitionistic strong completeness of and the intuitionistic weak completeness of become, respectively, fuzzy strong completeness of and fuzzy weak completeness of Furthermore, as for FRs, intuitionistic strong completeness implies intuitionistic weak completeness.

Throughout the paper, is a reflexive, weakly complete, and -symmetric IFR.

In the sequel, we define -transitivity of an IFR We introduce and analyze four elements of They enable us to obtain a characterization of the -transitivity of which generalizes the one obtained earlier by Fono and Andjiga [7] for FRs.

##### 3.3. -transitivity of an IFR: Definition and Characterization

Let and be the if-R-implication associated to

*Definition 3.6. * is -transitive if ,

If becomes a FR and and are dual, then the -transitivity becomes the usual -transitivity, that is, If becomes a crisp relation, then the -transitivity and the -transitivity become the crisp transitivity, that is,

We write the particular case of the -transitivity where

*Example 3.7. *If then R is -transitive if ,
We simply say that is transitive.

To establish a characterization of the -transitivity of , we need the following four elements of associated to

Let us introduce and analyze these elements of .

*Definition 3.8. *For all ,

(1)(2)(3) = [, [(), , (4) = , [(), , .

The next result shows that are elements of and deduces expressions of and

Proposition 3.9. (1)*For all .*(2)*(i) is the minimum of and (ii)(iii) is the minimum of and (iv)*

The following remark gives some comparisons of those elements of .

*Remark 3.10. *For all ,(1)(2)

The following result shows that in the particular case where is strongly complete, the four reals and become simple.

Corollary 3.11. *Let be an IFR, and let *

(1)*If is strongly complete and
then
*(2)*If is strongly complete and
then
*

In the particular case where and are dual and becomes a FR, we have some links between and for Furthermore, we obtain expressions of introduced earlier by Fono and Andjiga (see [7, page 375]).

Corollary 3.12. *If and are dual and is a FR, then *

(1)*(2)*

We end this section by establishing by means of those four elements of a characterization of the -transitivity of an IFR . Before that, let us recall a characterization of the -transitivity of a FR: if and are dual, and the IFR becomes a FR, then Fono and Andjiga (see [7, Lemma , page 375]) used the four reals , and defined in Corollary 3.12, to obtain the following characterization of the -transitivity of

R is -transitive on if and only if

We generalize that result for an IFR. Therefore, we obtain our first key result.

Lemma 3.13. *Let **The two following statements are equivalent:*

(i)* is -transitive on ;*(ii)

In the following section, we study a factorization of For that, we proceed as follows: (i) we recall the factorization of an IFR established by Dimitrov [18]; (ii) we notice some similarities between that factorization and those established earlier on FRs by Dutta [3], Richardson [10], and Fono and Andjiga [7]; (iii) using the vocabulary used by these authors for the factorization of FRs, we write Dimitrov's [18] factorization in a simple and elegant way for the if--conorm (see Lemma 4.2); (iv) we point out some intuitive difficulties of the strict component obtained in [18]; (v) we introduce definitions of an indifference and a strict component of an IFR, and we complete Lemma 4.2 to obtain a general factorization of an IFR when the union is defined by means of a given continuous -representable if--conorm satisfying condition

#### 4. Factorization of an IFR

##### 4.1. Review on Dimitrov's Results and Some Comments

Dimitrov proposed a factorization of an IFR and obtained the following result.

Proposition 4.1 (see Dimitrov [18, Proposition ], or Dimitrov [17, Proposition , page 152]). *Let be the if-t-conorm, and let be an IFR which is reflexive, -complete and -symmetric; and are two IFRs such that*

(i)*,*(ii)* is symmetric,*(iii)* is perfect antisymmetric,*(iv)*Then, for all ,*

(1)* where
*(2)* where
*

After a careful check, we notice that, in the particular case where becomes a FR, conditions (4.1) and (4.3) become some known notions introduced earlier by Dutta [3] and, used by Richardson [10] and, Fono and Andjiga [7].

Let be an IFR.

(1)If becomes a FR, the strict component of becomes the fuzzy strict component of and thus, condition (4.1) of the previous result becomes condition β is simple," that is, For convenience and as in fuzzy case, we also call condition (4.1): β is simple."(2)The strict component of obtained in the previous result satisfies the following condition:If becomes a FR, condition (4.5) becomes the condition β is regular," that is, , For convenience and as in fuzzy case, we also call condition (4.5): β is regular."

With these remarks on the intuitionistic fuzzy strict component obtained in Dimitrov [18], we rewrite Proposition 4.1 as follows:

β is regular and is defined by (4.2) if is perfect antisymmetric and simple, is symmetric, and for βAn interesting question is to check if this version of Dimitrov's result remains true for

The following result shows that this is true. More precisely, it establishes a generalization of the previous version of Dimitrov's result. And we obtain our second key result.

Lemma 4.2. *Let be a reflexive, weakly complete and -symmetric IFR; and are two IFRs such that: (i) , (ii) is symmetric, (iii) is perfect antisymmetric, (iv) is simple. Then,*

(1)* is defined by (4.2);*(2)* is regular.*

Otherwise, let us also point out some intuitive difficulties of the strict component obtained by Dimitrov in the factorization of Proposition 4.1.

(1)The component defined by (4.3) is obtained for the particular t-representable if-t-conorm .(2)The discontinuity of that is,(i)for all , the degree is insensitive for the variability of and . For illustration, if or then But if then (ii)for all , the degree is insensitive for the variability of and . For illustration, if or , then But if thenThe previous observations force us to complete and generalize the factorization of Dimitrov for a if-t-conorm satisfying condition .

##### 4.2. A New and General Factorization of an IFR

First at all, we introduce formally a definition of βindifference of an IFR" and βstrict component of an IFR."

*Definition 4.3. *Let satisfying condition be an IFR; and are two IFRs. and are βindifference of " and βstrict component of " associated to , respectively, if the following conditions are satisfied:

With the results of Lemma 4.2, the equality of (4.7) becomes the following equation: which is equivalent to the following system:

To establish a new and general factorization, we need the following lemma which is our third key result.

Lemma 4.4. *Let be an IFR, and let such that
**
Then,*

(i)*(4.9)() and (4.9)() have at least one solution;*(ii)*each solution of (4.9)() is strictly positive, and each solution of (4.9)() is strictly least than 1;*(iii)*if satisfies condition then (4.8) or (4.9) has at least one solution;*(iv)*if satisfies condition then the element is the optimal solution of (4.8) or (4.9), that is, is the upper solution of (4.9), and is the lowest solution of (4.9)*(v)*furthermore, if or is a strict if-t-conorm (i.e., and are strict) satisfying condition then is the unique solution of (4.8) or (4.9).*

We now establish the result of factorization which is the first main result of our paper.

Theorem 4.5. *Let satisfying condition be an IFR; and are two IFRs.**The two following statements are equivalent:*

(1)* and are βindifference" and βstrict component of " associated to , respectively;*(2)*(i)(ii) (4.9), (4.9), and
*

The previous factorization gives a unique indifference of However, as in fuzzy case and contrary to the crisp case, for an IFR and for satisfying the previous result generates a family of strict components of More interesting is that family has an optimal element called the optimal strict component of associated to

Let us give expressions of optimal intuitionistic fuzzy strict components of associated to in the general case and for the three particular cases where

*Example 4.6. * If satisfying then Theorem 4.5 implies that has an optimal strict component defined by, ,

If then the optimal strict component of is defined by, ,
This version is the one obtained by Dimitrov (see [18] or Proposition 4.1). If then the optimal strict component of is defined by, ,
If then the optimal strict component of is defined by, ,