A Binary Intuitionistic Fuzzy Relation: Some New Results, a General Factorization, and Two Properties of Strict Components

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 (max,min) intuitionistic fuzzy t-conorm. We provide, for a continuous t-representable intuitionistic fuzzy t-norm 𝒯, a characterization of the 𝒯-transitivity of an IFR. This enables us to determine necessary and sufficient conditions on a 𝒯-transitive IFR 𝑅 under which a strict component of 𝑅 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 [2–11] modelled such preferences by (binary) fuzzy relation (simply denoted by FR) on 𝑋, that is, a function 𝑅∶𝑋×𝑋→[0,1] where 𝑋 is a set of alternatives with Card(𝑋)=|𝑋|≥3. In this case, for (𝑥,𝑦)∈𝑋²,𝑅(𝑥,𝑦) is interpreted as the degree to which 𝑥 is “at least as good as" 𝑦. If ∀𝑥,𝑦∈𝑋,𝑅(𝑥,𝑦)∈{0,1}, then 𝑅 is crisp, and we denote 𝑅(𝑥,𝑦)=1 by 𝑥𝑅𝑦 and 𝑅(𝑥,𝑦)=0 by not(𝑥𝑅𝑦). 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 [14–19] modelled ambiguous preferences by a (binary) intuitionistic fuzzy relation (IFR) on 𝑋, that is, a function 𝑅∶𝑋×𝑋→𝐿^∗={(𝑎₁,𝑎₂)∈[0,1]²,𝑎₁+𝑎₂≤1} where ∀𝑥,𝑦∈𝑋,𝑅(𝑥,𝑦)=(𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦)). In this case, 𝜇_𝑅(𝑥,𝑦) is the degree to which 𝑥 is “at least as good as" 𝑦, and 𝜈_𝑅(𝑥,𝑦) is the degree to which 𝑥 is not “at least as good as" 𝑦. The positive real number 𝜋_𝑅(𝑥,𝑦)=1−𝜇_𝑅(𝑥,𝑦)−𝜈_𝑅(𝑥,𝑦) (since 𝜇_𝑅(𝑥,𝑦)+𝜈_𝑅(𝑥,𝑦)≤1), usually called fuzzy index, indicates the degree of incomparability between 𝑥 and 𝑦. In this paper, we simply write ∀𝑥,𝑦∈𝑋,𝑅(𝑥,𝑦)=(𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦)). Clearly, we have two particular cases: (i) if ∀(𝑥,𝑦)∈𝑋×𝑋,𝜋_𝑅(𝑥,𝑦)=0, that is, 𝜈_𝑅(𝑥,𝑦)=1−𝜇_𝑅(𝑥,𝑦), then 𝑅 becomes an FR on 𝑋, and (ii) if ∀(𝑥,𝑦)∈𝑋×𝑋,𝜋_𝑅(𝑥,𝑦)=0 and 𝜈_𝑅(𝑥,𝑦)∈{0,1}, then 𝑅 becomes the well-known (binary) crisp relation. In the first case, we simply write 𝑅(𝑥,𝑦)=𝜇_𝑅(𝑥,𝑦), and in the second case, we have 𝑥𝑅𝑦⇔𝜇_𝑅(𝑥,𝑦)=1 (i.e., 𝜈_𝑅(𝑥,𝑦)=0).

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 (max,min) t-representable intuitionistic fuzzy t-conorm. Recently, Cornelis et al. [20] established some results on t-representable intuitionistic fuzzy t-norms (i.e., 𝒯=(𝑇,𝑆) where 𝑆 is a fuzzy t-conorm, and 𝑇 is a fuzzy t-norm satisfying ∀𝑎,𝑏∈[0,1],𝑇(𝑎,𝑏)≤1−𝑆(1−𝑎,1−𝑏)), on t-representable intuitionistic fuzzy t-conorms (i.e., 𝒥=(𝑆,𝑇)) 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 t-representable intuitionistic fuzzy t-norms and t-conorms.

The aim of this paper is (i) to study the standard completeness of an IFR, (ii) to establish a characterization of the 𝒯-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 𝒯-transitive IFR 𝑅 under which a given strict component of 𝑅 (obtained in our factorization) satisfies respectively pos-transitivity and negative transitivity.

First we establish some useful results on t-representable intuitionistic fuzzy t-norms, t-representable intuitionistic fuzzy t-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 (𝑆,𝑇)-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 𝒯, a characterization of the 𝒯-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 (max,min) intuitionistic fuzzy t-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 t-representable intuitionistic fuzzy t-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 𝒯-transitive IFR 𝑅, such that a strict component of 𝑅 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 ≤_𝐿^∗ be an order in 𝐿^∗ defined by ∀(𝑎₁,𝑎₂),(𝑏₁,𝑏₂)∈𝐿^∗,(𝑎₁,𝑎₂)≤_𝐿^∗(𝑏₁,𝑏₂)⇔(𝑎₁≤𝑏₁ and 𝑎₂≥𝑏₂). (𝐿^∗,≤_𝐿^∗) is a complete lattice. 0_𝐿^∗=(0,1) and 1_𝐿^∗=(1,0) are the units of 𝐿^∗.

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

2.1. Review on Fuzzy Operators

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

A fuzzy t-norm (resp. a fuzzy t-conorm) is an increasing, commutative, and associative binary operation on [0,1] with a neutral 1 (resp. 0). The dual of a fuzzy t-norm 𝑇 is a fuzzy t-conorm 𝑆, that is, ∀𝑎,𝑏∈[0,1], 𝑇(𝑎,𝑏)=1−𝑆(1−𝑎,1−𝑏).

Let us recall two usual families of fuzzy t-norms and fuzzy t-conorms. The Frank t-norms (𝑇^𝑙_𝐹)_{𝑙∈[0,∞]}, that is, ∀𝑙∈[0,∞],∀𝑎,𝑏∈[0,1], 𝑇^𝑙_𝐹𝑇(𝑎,𝑏)=^𝐌⋀𝑇(𝑎,𝑏)=min(𝑎,𝑏)=𝑎𝑏,if𝑙=0,_𝐏𝑇(𝑎,𝑏)=𝑎×𝑏,if𝑙=1,_𝐋(𝑎,𝑏)=max(𝑎+𝑏−1,0),if𝑙=+∞,log_𝑙(1+(𝑙^𝑎−1)(𝑙^𝑏−1)𝑙−1),otherwise, (2.1) where 𝑇_𝐌,𝑇_𝐏, and 𝑇_𝐋 are the minimum fuzzy t-norm, the product fuzzy t-norm, and the Łukasiewicz fuzzy t-norm, respectively. The Frank t-conorms (𝑆^𝑙_𝐹)_{𝑙∈[0,∞]}, that is, ∀𝑙∈[0,∞],∀𝑎,𝑏∈[0,1], 𝑆^𝑙_𝐹𝑆(𝑎,𝑏)=^𝐌𝑆(𝑎,𝑏)=max(𝑎,𝑏)=𝑎∨𝑏,if𝑙=0,_𝐏𝑆(𝑎,𝑏)=𝑎+𝑏−𝑎×𝑏,if𝑙=1,_𝐋(𝑎,𝑏)=min(𝑎+𝑏,1),if𝑙=+∞,1−log_𝑙(1+(𝑙^1−𝑎−1)(𝑙^1−𝑏−1)𝑙−1),otherwise, (2.2) where 𝑆_𝐌,𝑆_𝐏, and 𝑆_𝐋 are the maximum fuzzy t-conorm, the product fuzzy t-conorm, and the Łukasiewicz fuzzy t-conorm, respectively.

A fuzzy t-norm 𝑇 (fuzzy t-conorm 𝑆) is strict if ∀𝑎,𝑏∈[0,1], ∀𝑐∈]0,1],𝑎<𝑏 implies 𝑇(𝑎,𝑐)<𝑇(𝑏,𝑐) (resp. ∀𝑎,𝑏∈[0,1], ∀𝑐∈[0,1[, 𝑎<𝑏 implies 𝑆(𝑎,𝑐)<𝑆(𝑏,𝑐)). The product fuzzy t-norm (resp. the product fuzzy t-conorm) is an example of a strict fuzzy t-norm (resp. fuzzy t-conorm).

We have the following properties: ∀𝑎,𝑏∈[0,1],(i)𝑇(𝑎,𝑏)≤min(𝑎,𝑏)(ii)max(𝑎,𝑏)≤𝑆(𝑎,𝑏). (2.3)

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

In the following, we recall some definitions and examples on fuzzy implications and fuzzy coimplicators based on fuzzy t-norms and fuzzy t-conorms, respectively (see [21–23]).

The fuzzy R-implication 𝐼_𝑇 associated to 𝑇 is a binary operation on [0,1] defined by ∀𝑎,𝑏∈[0,1],𝐼_𝑇(𝑎,𝑏)=max{𝑡∈[0,1],𝑇(𝑎,𝑡)≤𝑏}. The fuzzy coimplicator 𝐽_𝑆 associated to 𝑆 is a binary operation on [0,1] defined by ∀𝑎,𝑏∈[0,1],𝐽_𝑆(𝑎,𝑏)=min{𝑡∈[0,1],𝑏≤𝑆(𝑎,𝑡)}.

Let us recall some usual examples of these fuzzy operators.

The fuzzy R-implication associated to 𝑇_𝐌 is defined by ∀𝑎,𝑏∈[0,1],𝐼_𝑇𝐌=1,if𝑎≤𝑏,𝑏,if𝑎>𝑏. (2.4) The fuzzy coimplicator associated to 𝑆_𝐌 is defined by ∀𝑎,𝑏∈[0,1],𝐽_𝑆𝐌(𝑎,𝑏)=𝑏,if𝑎<𝑏,0,if𝑎≥𝑏. (2.5) The fuzzy R-implication associated to 𝑇_𝐋 is defined by ∀𝑎,𝑏∈[0,1],𝐼_𝑇𝐋(𝑎,𝑏)=1,if𝑎≤𝑏,1−𝑎+𝑏,if𝑎>𝑏. (2.6) The fuzzy coimplicator associated to 𝑆_𝐋 is defined by ∀𝑎,𝑏∈[0,1],𝐽_𝑆𝐋(𝑎,𝑏)=𝑏−𝑎,if𝑎<𝑏,0,if𝑎≥𝑏. (2.7) The fuzzy R-implication associated to 𝑇_𝑃 is defined by ∀𝑎,𝑏∈[0,1],𝐼_𝑇𝐏(𝑎,𝑏)=𝑏1,if𝑎≤𝑏,𝑎,if𝑎>𝑏. (2.8) The fuzzy coimplicator associated to 𝑆_𝑃 is defined by ∀𝑎,𝑏∈[0,1],𝐽_𝑆𝐏(𝑎,𝑏)=𝑏−𝑎1−𝑎,if𝑎<𝑏,0,if𝑎≥𝑏. (2.9)

We complete the previous examples by giving expressions of fuzzy R-implications of the other Frank fuzzy t-norms and fuzzy coimplicators of the other Frank fuzzy t-conorms: ∀𝑙∈]0,1[∪]1,+∞[,∀𝑎,𝑏∈[0,1],𝐼_𝑇^𝑙_𝐹(𝑎,𝑏)=1,if𝑎≤𝑏,log_𝑙(1+(𝑙−1)(𝑙^𝑏−1)𝑙^𝑎𝐽−1),if𝑎>𝑏,_𝑆𝑙𝐹(𝑎,𝑏)=0,if𝑎≥𝑏,1−log_𝑙(1+(𝑙−1)(𝑙^1−𝑏−1)𝑙^(1−𝑎)−1),if𝑎<𝑏. (2.10)

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

Proposition 2.1 (See [4, 5, 9, 21, 23]). For all 𝑎,𝑏,𝑐∈[0,1],
(1)𝐼_𝑇(𝑎,𝑎)=1;𝐽_𝑆(𝑎,𝑎)=0, and 𝐽_𝑆(𝑎,𝑏)≤𝑏≤𝐼_𝑇(𝑎,𝑏);(2)𝑇(𝑎,𝐼_𝑇(𝑎,𝑏))=min(𝑎,𝑏), and 𝑆(𝑎,𝐽_𝑆(𝑎,𝑏))=max(𝑎,𝑏);(3)𝑏<𝑎⟺𝐼_𝑇(𝑎,𝑏)<1,𝑎<𝑏⟺𝐽_𝑆(𝑎,𝑏)>0; (2.11)(4)𝐼𝑎≤𝑏⇒^𝑇(𝑏,𝑐)≤𝐼_𝑇𝐼(𝑎,𝑐),_𝑇(𝑐,𝑎)≤𝐼_𝑇(𝑐,𝑏); (2.12)(5)𝐽𝑎≤𝑏⇒^𝑆(𝑏,𝑐)≤𝐽_𝑆𝐽(𝑎,𝑐),_𝑆(𝑐,𝑎)≤𝐽_𝑆(𝑐,𝑏); (2.13)

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]). (1) An intuitionistic fuzzy t-norm is an increasing, commutative, and associative binary operation 𝒯 on 𝐿^∗ satisfying ∀(𝑎,𝑏)∈𝐿^∗,𝒯((𝑎,𝑏),(1,0))=(𝑎,𝑏).
(2) An intuitionistic fuzzy t-conorm is an increasing, commutative, associative binary operation 𝒥 on 𝐿^∗ satisfying ∀(𝑎,𝑏)∈𝐿^∗,𝒥((𝑎,𝑏),(0,1))=(𝑎,𝑏).

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

Definition 2.3. An intuitionistic fuzzy t-norm 𝒯 (resp. t-conorm 𝒥) is called t-representable if there exists a fuzzy t-norm 𝑇 and a fuzzy t-conorm 𝑆 (resp. a fuzzy t-conorm 𝑆 and fuzzy t-norm 𝑇) such that ∀𝑎=(𝑎₁,𝑎₂),𝑏=(𝑏₁,𝑏₂)∈𝐿^∗,𝒯(𝑎,𝑏)=(𝑇(𝑎₁,𝑏₁),𝑆(𝑎₂,𝑏₂)) (resp. 𝒥(𝑎,𝑏))=(𝑆(𝑎₁,𝑏₁),𝑇(𝑎₂,𝑏₂)).
𝑇 and 𝑆 (resp. 𝑆 and 𝑇) are called the representants of 𝒯 (resp. 𝒥).

The theorem below states conditions under which a pair of connectives on [0,1] gives rise to a t-representable intuitionistic fuzzy t-norm (t-conorm).

Theorem 2.4 (see Cornelis et al. [20, Theorem 2, pages 60–61]). Given a fuzzy t-norm 𝑇 and a fuzzy t-conorm 𝑆 satisfying ∀𝑎₁,𝑎₂∈[0,1],𝑇(𝑎₁,𝑎₂)≤1−𝑆(1−𝑎₁,1−𝑎₂).
The mappings 𝒯 and 𝒥 defined by, for 𝑥=(𝑥₁,𝑥₂) and 𝑦=(𝑦₁,𝑦₂) in 𝐿^∗∶𝒯(𝑥,𝑦)=(𝑇(𝑥₁,𝑦₁),𝑆(𝑥₂,𝑦₂)) and 𝒥(𝑥,𝑦)=(𝑆(𝑥₁,𝑦₁),𝑇(𝑥₂,𝑦₂)), are, respectively, a t-representable intuitionistic fuzzy t-norm and t-representable intuitionistic fuzzy t-conorm.

Throughout the paper, we consider only continuous t-representable intuitionistic fuzzy t-conorms (shortly if-t-conorm) and continuous t-representable intuitionistic fuzzy t-norms (shortly if-t-norm). They are denoted by 𝒥=(𝑆,𝑇) and 𝒯=(𝑇,𝑆), respectively, where ∀𝑎,𝑏∈[0,1],𝑆(𝑎,𝑏)≤1−𝑇(1−𝑎,𝑎−𝑏).

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

Example 2.5. (1)𝒯_𝐌=(𝑇_𝐌,𝑆_𝐌) and 𝒥_𝐌=(𝑆_𝐌,𝑇_𝐌) are, respectively, if-t-norm and if-t-conorm associated to 𝑇_𝐌 and 𝑆_𝐌 since ∀𝑎,𝑏∈[0,1],𝑇_𝐌(𝑎,𝑏)≤1−𝑆_𝐌(1−𝑎,1−𝑏).
(2)𝒯_𝐋=(𝑇_𝐋,𝑆_𝐋) and 𝒥_𝐋=(𝑆_𝐋,𝑇_𝐋) are, respectively, if-t-norm and if-t-conorm associated to 𝑇_𝐋 and 𝑆_𝐋 since ∀𝑎,𝑏∈[0,1],𝑇_𝐋(𝑎,𝑏)≤1−𝑆_𝐋(1−𝑎,1−𝑏).(3)𝒯_𝐏=(𝑇_𝐏,𝑆_𝐏) and 𝒥_𝐏=(𝑆_𝐏,𝑇_𝐏) are, respectively, if-t-norm and if-t-conorm associated to 𝑇_𝐏 and 𝑆_𝐏 since ∀𝑎,𝑏∈[0,1],𝑇_𝐏(𝑎,𝑏)≤1−𝑆_𝐏(1−𝑎,1−𝑏).

Definition 2.6 (see Cornelis et al. [20, Definition 8, page 64]). (1) The intuitionistic fuzzy R-implication (shortly if-R-implication) associated with an if-t-norm 𝒯=(𝑇,𝑆) is a binary operation on 𝐿^∗ defined by: ∀𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈𝐿^∗, 𝐼_𝒯(𝑥,𝑦)=sup{𝑧∈𝐿^∗,𝒯(𝑥,𝑧)≤_𝐿^∗𝑦}=sup{𝑧=(𝑧₁,𝑧₂)∈𝐿^∗,𝑇(𝑥₁,𝑧₁)≤𝑦₁and𝑆(𝑥₂,𝑧₂)≥𝑦₂}.
(2) The intuitionistic fuzzy coimplicator (shortly if-coimplicator) associated with an if-t-conorm 𝒥=(𝑆,𝑇) is a binary operation on 𝐿^∗ defined by: ∀𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈𝐿^∗, 𝐽_𝒥(𝑥,𝑦)=inf{𝑧∈𝐿^∗,𝑦≤_𝐿^∗𝒥(𝑥,𝑧)}=inf{𝑧=(𝑧₁,𝑧₂)∈𝐿^∗/𝑦₁≤𝑆(𝑥₁,𝑧₁)and𝑦₂≥𝑇(𝑥₂,𝑧₂)}.

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 𝐼_𝑇 and the fuzzy coimplicator 𝐽_𝑆.

Proposition 2.7. Let 𝑆 and 𝑇 such that ∀𝑎,𝑏∈[0,1],𝑇(𝑎,𝑏)≤1−𝑆(1−𝑎,1−𝑏). Then
(1) for all 𝑎,𝑏∈[0,1],𝐼_𝑇(𝑎,𝑏)≥1−𝐽_𝑆(1−𝑎,1−𝑏);(2) if 𝑇 and 𝑆 are dual, then ∀𝑎,𝑏∈[0,1],𝐼_𝑇(𝑎,𝑏)=1−𝐽_𝑆(1−𝑎,1−𝑏).

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

Lemma 2.8. For all 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈𝐿^∗,(1)𝐽_𝒥(𝑥,𝑦)=(𝐽_𝑆(𝑥₁,𝑦₁),min(𝐼_𝑇(𝑥₂,𝑦₂),1−𝐽_𝑆(𝑥₁,𝑦₁)));(2)𝐼_𝒯(𝑥,𝑦)=(min(𝐼_𝑇(𝑥₁,𝑦₁),1−𝐽_𝑆(𝑥₂,𝑦₂)),𝐽_𝑆(𝑥₂,𝑦₂)).

We now introduce a new condition which can be satisfied by a if-t-conorm 𝒥=(𝑆,𝑇).

Definition 2.9. 𝒥=(𝑆,𝑇) satisfies condition 𝐺 if ∀(𝑎₁,𝑎₂),(𝑏₁,𝑏₂)∈𝐿^∗,𝑎₁>𝑏₁𝑎₂<𝑏₂𝑎₁+𝑎₂=𝑏₁+𝑏₂⇒𝐼_𝑇(𝑏₂,𝑎₂)+𝐽_𝑆(𝑏₁,𝑎₁)≤1. (2.14)

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

Proposition 2.10. (1) For all 𝑙∈[0,+∞],𝒥^𝐹_𝑙=(𝑆^𝐹_𝑙,𝑇^𝐹_𝑙) satisfies condition 𝐺.(2) If 𝑆 and 𝑇 are dual, then the restriction of 𝒥=(𝑆,𝑇) on 𝐿^∗₁={(𝑥₁,𝑥₂)∈𝐿^∗,𝑥₁+𝑥₂=1} satisfies condition 𝐺.

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 𝒯=(𝑇,𝑆), a characterization of the 𝒯-transitivity of an IFR 𝑅.

3. Preliminaries on IFRs

3.1. Review on IFRs

An IFS in 𝑋 is an expression 𝐴 given by 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩,𝑥∈𝑋}, where 𝜇_𝐴∶𝑋→[0,1] and 𝜈_𝐴∶𝑋→[0,1] are functions satisfying the condition ∀𝑥∈𝑋,𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1. The numbers 𝜇_𝐴(𝑥) and 𝜈_𝐴(𝑥) denote, respectively, the degree of membership and the degree of nonmembership of the element 𝑥 in 𝐴. The number 𝜋_𝐴(𝑥)=1−𝜇_𝐴(𝑥)−𝜈_𝐴(𝑥) is an index of the element 𝑥 in 𝑋. Obviously, when ∀𝑥∈𝑋,𝜈_𝐴(𝑥)=1−𝜇_𝐴(𝑥), that is, 𝜋_𝐴(𝑥)=0, the IFS 𝐴 is a fuzzy set (simply denoted by FS) in 𝑋. In this case, ∀𝑥∈𝑋,𝐴(𝑥)=𝜇_𝐴(𝑥).

Let 𝐴 and 𝐵 be two IFSs, and let 𝒥=(𝑆,𝑇). The intuitionistic fuzzy union 𝐴∪_𝒥𝐵 associated to 𝒥 is an IFS defined by 𝜇∀𝑥∈𝑋,^{𝐴∪𝒥𝐵}(𝑥)=𝑆(𝜇_𝐴(𝑥),𝜇_𝐵𝜈(𝑥)),_{𝐴∪𝒥𝐵}(𝑥)=𝑇(𝜈_𝐴(𝑥),𝜈_𝐵(𝑥)) (3.1) (we recall that if 𝐴 and 𝐵 are FSs, and 𝑇 and 𝑆 are dual, then 𝐴∪_𝒥𝐵 becomes the well-known fuzzy union 𝐴∪_𝑆𝐵 defined by ∀𝑥∈𝑋,𝐴∪_𝑆𝐵(𝑥)=𝑆(𝐴(𝑥),𝐵(𝑥)). And if 𝐴 and 𝐵 are crisp, 𝐴∪_𝒥𝐵=𝐴∪_𝑆𝐵 becomes the crisp union). As defined in the Introduction, an IFR in 𝑋 is an IFS in 𝑋×𝑋.

We complete some basic definitions on IFRs.

Definition 3.1. Let 𝑅 be an IFR.
(1)𝑅 is reflexive if ∀𝑥∈𝑋,𝜇_𝑅(𝑥,𝑥)=1.(2)𝑅 is symmetric if ∀𝑥,𝑦∈𝑋,𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥).(3)𝑅 is 𝜋-symmetric if ∀𝑥,𝑦∈𝑋, 𝜋_𝑅(𝑥,𝑦)=𝜋_𝑅(𝑦,𝑥).(4)𝑅 is perfect antisymmetric if ∀(𝑥,𝑦)∈𝑋×𝑋,𝑥≠𝑦,𝜇_𝑅(𝑥,𝑦)>0or(𝜇_𝑅(𝑥,𝑦)=0,𝜈_𝑅⇒𝜇(𝑥,𝑦)<1)^𝑅𝜈(𝑦,𝑥)=0,_𝑅(𝑦,𝑥)=1. (3.2)(5) The converse of 𝑅 is the IFR denoted 𝑅⁻¹ and defined by ∀𝑥,𝑦∈𝑋, 𝜇_𝑅−1(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅⁻¹(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥).

In the following, we recall the well-known notion of completeness of a crisp relation in 𝑋. 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 (𝒥-Completeness)

Let 𝑅 be a reflexive IFR and 𝒥=(𝑆,𝑇).

When 𝑅 is a crisp relation, 𝑅 is complete if 𝑅∪𝑅⁻¹=𝑋², that is, ∀𝑥,𝑦∈𝑋,𝑥𝑅𝑦or𝑦𝑅𝑥.

When 𝑅 is a FR, for the fuzzy t-conorm 𝑆,𝑅 is 𝑆-complete if 𝑅∪_𝑆𝑅⁻¹=𝑋², that is, ∀𝑥,𝑦∈𝑋,𝑆(𝑅(𝑥,𝑦),𝑅(𝑦,𝑥))=1. In particular, if 𝑆=𝑆_𝑀, we simply say that 𝑅 is strongly complete, that is, ∀𝑥,𝑦∈𝑋,max(𝑅(𝑥,𝑦),𝑅(𝑦,𝑥))=1. If 𝑆=𝑆_𝐿, we simply say that 𝑅 is weakly complete, that is, ∀𝑥,𝑦∈𝑋,𝑅(𝑥,𝑦)+𝑅(𝑦,𝑥)≥1 (see Fono and Andjiga [7, Definition 2, page 375]).

In the general case where 𝑅 is an IFR and 𝒥=(𝑆,𝑇), we have the following generic version of the standard completeness of 𝑅.

Definition 3.2. 𝑅 is 𝒥-complete if 𝑅∪_𝒥𝑅⁻¹=𝑋², that is, ∀𝑥,𝑦∈𝑋,𝑆(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥))=1,𝑇(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥))=0. (3.3)

Remark 3.3. If an IFR 𝑅 becomes a FR, and 𝑆 and 𝑇 are dual, then (𝑆,𝑇)-completeness becomes 𝑆-completeness. Furthermore, if 𝑅 becomes crisp, then 𝒥-completeness and 𝑆-completeness become crisp completeness.

Dimitrov (see [17, Definition 2, page 151]) introduced the following version of completeness of an IFR: 𝑅 is 𝐷-complete if ∀𝑥,𝑦∈𝑋, [(𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦))=(0,1)]⇒[𝜇_𝑅(𝑦,𝑥)>0,𝜈_𝑅(𝑦,𝑥)<1]. (3.4)

It is important to notice that 𝐷-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 𝑅 is (𝑆,𝑇)-complete, then 𝑅 is 𝐷-complete.

As for FRs, we deduce the two following interesting particular cases of 𝒥-completeness when 𝒥∈{𝒥_𝐌,𝒥_𝐋}.

Example 3.5. Let 𝑅 be a reflexive IFR and 𝒥=(𝑆,𝑇).
(1)If 𝒥=𝒥_𝐌=(max,min), then 𝑅 is 𝒥-complete if ∀𝑥,𝑦∈𝑋, max(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥))=1,min(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥))=0. (3.5) In this case, we simply say that 𝑅 is strongly complete.(2)If 𝒥=𝒥_𝐋=(𝑆_𝐋,𝑇_𝐋), then 𝑅 is 𝒥-complete if ∀𝑥,𝑦∈𝑋, min(1,𝜇_𝑅(𝑥,𝑦)+𝜇_𝑅(𝑦,𝑥))=1,max(0,𝜈_𝑅(𝑥,𝑦)+𝜈_𝑅𝜇(𝑦,𝑥)−1)=0,i.e.,^𝑅(𝑥,𝑦)+𝜇_𝑅𝜈(𝑦,𝑥)≥1,_𝑅(𝑥,𝑦)+𝜈_𝑅(𝑦,𝑥)≤1. (3.6) In this case, we simply say that 𝑅 is weakly complete.

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

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

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

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

Let 𝒯=(𝑇,𝑆) and 𝐼_𝒯 be the if-R-implication associated to 𝒯.

Definition 3.6. 𝑅 is 𝒯-transitive if ∀𝑥,𝑦,𝑧∈𝑋, 𝒯(𝑅(𝑥,𝑦),𝑅(𝑦,𝑧))≤_𝐿^∗𝜇𝑅(𝑥,𝑧),i.e.,^𝑅(𝑥,𝑧)≥𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅𝜈(𝑦,𝑧)),_𝑅(𝑥,𝑧)≤𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑧)). (3.7)

If 𝑅 becomes a FR and 𝑇 and 𝑆 are dual, then the 𝒯-transitivity becomes the usual 𝑇-transitivity, that is, ∀𝑥,𝑦,𝑧∈𝑋,𝑅(𝑥,𝑧)≥𝑇(𝑅(𝑥,𝑦),𝑅(𝑦,𝑧)). If 𝑅 becomes a crisp relation, then the 𝒯-transitivity and the 𝑇-transitivity become the crisp transitivity, that is, ∀𝑥,𝑦,𝑧∈𝑋,(𝑥𝑅𝑦and𝑦𝑅𝑧)⇒𝑥𝑅𝑧.

We write the particular case of the 𝒯-transitivity where 𝒯=𝒯_𝐌.

Example 3.7. If 𝒯=𝒯_𝐌, then R is 𝒯_𝐌-transitive if ∀𝑥,𝑦,𝑧∈𝑋, 𝜇_𝑅(𝑥,𝑧)≥min(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅𝜈(𝑦,𝑧)),_𝑅(𝑥,𝑧)≤max(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑧)). (3.8) We simply say that 𝑅 is transitive.

To establish a characterization of the 𝒯-transitivity of 𝑅, we need the following four elements of [0,1]² associated to 𝑅.

Let us introduce and analyze these elements of [0,1]².

Definition 3.8. For all 𝑥,𝑦,𝑧∈𝑋,
(1)(𝛼₁(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧))=(𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑦,𝑥)),𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥)));(2)(𝛼₂(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧))=(𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑧)),𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑧)));(3)(𝛼₃(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧)) = min(𝐼_𝒯[(𝜇_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑧)),(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥))], 𝐼_𝒯[(𝜇_𝑅(𝑥,𝑦), 𝜈_𝑅(𝑥,𝑦)), (𝜇_𝑅(𝑧,𝑦),𝜈_𝑅(𝑧,𝑦))]);(4)(𝛼₄(𝑥,𝑦,𝑧),𝛽₄(𝑥,𝑦,𝑧)) = min(𝐼_𝒯[(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥)),(𝜇_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑧))], 𝐼_𝒯[(𝜇_𝑅(𝑧,𝑦), 𝜈_𝑅(𝑧,𝑦)), (𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦))]).

The next result shows that (𝛼_𝑖(𝑥,𝑦,𝑧),𝛽_𝑖(𝑥,𝑦,𝑧))_{𝑖∈{1,2,3,4}} are elements of 𝐿^∗ and deduces expressions of 𝛼₃(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧),𝛼₄(𝑥,𝑦,𝑧) and 𝛽₄(𝑥,𝑦,𝑧).

Proposition 3.9. (1)For all 𝑖∈{1,2,3,4},(𝛼_𝑖(𝑥,𝑦,𝑧),𝛽_𝑖(𝑥,𝑦,𝑧))∈𝐿^∗.(2)(i)𝛼₃(𝑥,𝑦,𝑧) is the minimum of min[𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦)),1−𝐽_𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑧,𝑦))] and min[𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)),1−𝐽_𝑆(𝜈_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑥))].(ii)𝛽₃(𝑥,𝑦,𝑧)=max(𝐽_𝑆(𝜈_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑥)),𝐽_𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑧,𝑦))).(iii)𝛼₄(𝑥,𝑦,𝑧) is the minimum of min[𝐼_𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑥,𝑦)),1−𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑥,𝑦))] and min[𝐼_𝑇(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑦,𝑧)),1−𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑧)))].(iv)𝛽₄(𝑥,𝑦,𝑧)=max(𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑧)),𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑥,𝑦))).

The following remark gives some comparisons of those elements of 𝐿^∗.

Remark 3.10. For all 𝑥,𝑦,𝑧∈𝑋,(1)(𝛼₁(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧))≤_𝐿^∗(𝛼₃(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧));(2)(𝛼₂(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧))≤_𝐿^∗(𝛼₄(𝑥,𝑦,𝑧),𝛽₄(𝑥,𝑦,𝑧)).

The following result shows that in the particular case where 𝑅 is strongly complete, the four reals 𝛼₃(𝑥,𝑦,𝑧),𝛼₄(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧) and 𝛽₄(𝑥,𝑦,𝑧) become simple.

Corollary 3.11. Let 𝑅 be an IFR, and let 𝑥,𝑦,𝑧∈𝑋.
(1)If 𝑅 is strongly complete and (𝜇_𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧)or^𝑅(𝑦,𝑥)=𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧)or^𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜇_𝑅(𝑦,𝑧)), (3.9) then 𝛼₁(𝑥,𝑦,𝑧)=𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅𝛼(𝑦,𝑥))<1,₃(𝑥,𝑦,𝑧)=min(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅𝛼(𝑧,𝑦)),₂(𝑥,𝑦,𝑧)=𝛼₄(𝑥,𝑦,𝑧)=1. (3.10)(2)If 𝑅 is strongly complete and (𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧)or^𝑅(𝑦,𝑥)=𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧)or^𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜈_𝑅(𝑦,𝑧)), (3.11) then 𝛽₁(𝑥,𝑦,𝑧)=𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅𝛽(𝑦,𝑥))>0,₃(𝑥,𝑦,𝑧)=max(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅𝛽(𝑧,𝑦)),₂(𝑥,𝑦,𝑧)=𝛽₄(𝑥,𝑦,𝑧)=0. (3.12)

In the particular case where 𝑇 and 𝑆 are dual and 𝑅 becomes a FR, we have some links between 𝛼_𝑖 and 𝛽_𝑖 for 𝑖∈{1,2,3,4}. Furthermore, we obtain expressions of (𝛼_𝑖(𝑥,𝑦,𝑧))_{𝑖∈{1,2,3,4}} introduced earlier by Fono and Andjiga (see [7, page 375]).

Corollary 3.12. If 𝑇 and 𝑆 are dual and 𝑅 is a FR, then ∀𝑥,𝑦,𝑧∈𝑋,
(1)𝛽₁(𝑥,𝑦,𝑧)=1−𝛼₁𝛽(𝑥,𝑦,𝑧),₂(𝑥,𝑦,𝑧)=1−𝛼₂𝛽(𝑥,𝑦,𝑧),₃(𝑥,𝑦,𝑧)=1−𝛼₃𝛽(𝑥,𝑦,𝑧),₄(𝑥,𝑦,𝑧)=1−𝛼₄(𝑥,𝑦,𝑧). (3.13)(2)𝛼₁(𝑥,𝑦,𝑧)=𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅𝛼(𝑦,𝑥)),₂(𝑥,𝑦,𝑧)=𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅𝛼(𝑦,𝑧)),₃(𝑥,𝑦,𝑧)=min(𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦)),𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅𝛼(𝑦,𝑥))),₄(𝑥,𝑦,𝑧)=min(𝐼_𝑇(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑦,𝑧)),𝐼_𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑥,𝑦))). (3.14)

We end this section by establishing by means of those four elements of 𝐿^∗ a characterization of the 𝒯-transitivity of an IFR 𝑅. Before that, let us recall a characterization of the 𝑇-transitivity of a FR: ∀𝑥,𝑦,𝑧∈𝑋, if 𝑇 and 𝑆 are dual, and the IFR 𝑅 becomes a FR, then Fono and Andjiga (see [7, Lemma 1, page 375]) used the four reals 𝛼₁(𝑥,𝑦,𝑧),𝛼₂(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧), and 𝛼₄(𝑥,𝑦,𝑧) defined in Corollary 3.12, to obtain the following characterization of the 𝑇-transitivity of 𝑅.

R is 𝑇-transitive on {𝑥,𝑦,𝑧} if and only if 𝑅(𝑥,𝑧)=𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₄(𝑥,𝑦,𝑧)],𝑅(𝑧,𝑥)=𝜇_𝑅(𝑧,𝑥)∈[𝛼₁(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)]. (3.15)

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

Lemma 3.13. Let {𝑥,𝑦,𝑧}⊆𝑋.
The two following statements are equivalent:
(i)𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧};(ii)𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₄(𝑥,𝑦,𝑧)],𝜈_𝑅(𝑥,𝑧)∈[𝛽₄(𝑥,𝑦,𝑧),𝛽₂𝜇(𝑥,𝑦,𝑧)],_𝑅(𝑧,𝑥)∈[𝛼₁(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)],𝜈_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧)]. (3.16)

In the following section, we study a factorization of 𝑅. 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 (max,min) if-t-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 t-representable if-t-conorm satisfying condition 𝐺.

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 1], or Dimitrov [17, Proposition 3, page 152]). Let 𝒥=𝒥_𝑀=(max,min) be the if-t-conorm, and let 𝑅 be an IFR which is reflexive, 𝐷-complete and 𝜋-symmetric; 𝐼 and 𝑃 are two IFRs such that
(i)𝑅=𝐼∪_𝒥𝑀𝑃,(ii)𝐼 is symmetric,(iii)𝑃 is perfect antisymmetric,(iv)𝜇∀(𝑥,𝑦)∈𝑋×𝑋,^𝑅(𝑥,𝑦)=𝜇_𝑅𝜈(𝑦,𝑥)_𝑅(𝑥,𝑦)=𝜈_𝑅⇒𝜇(𝑦,𝑥)^𝑃(𝑥,𝑦)=𝜇_𝑃𝜈(𝑦,𝑥)_𝑃(𝑥,𝑦)=𝜈_𝑃(𝑦,𝑥). (4.1)
Then, for all 𝑥,𝑦∈𝑋,
(1)𝐼(𝑥,𝑦)=(𝜇_𝐼(𝑥,𝑦),𝜈_𝐼(𝑥,𝑦)), where 𝜇_𝐼(𝑥,𝑦)=min(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅𝜈(𝑦,𝑥)),_𝐼(𝑥,𝑦)=max(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥)); (4.2)(2)𝑃(𝑥,𝑦)=(𝜇_𝑃(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦)), where 𝜇_𝑃𝜇(𝑥,𝑦)=^𝑅(𝑥,𝑦),if𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅𝜈(𝑦,𝑥),0,otherwise,_𝑃𝜈(𝑥,𝑦)=^𝑅(𝑥,𝑦),if𝜈_𝑅(𝑥,𝑦)>𝜈_𝑅(𝑦,𝑥),1,otherwise. (4.3)

After a careful check, we notice that, in the particular case where 𝑅 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 𝑅 be an IFR.

(1)If 𝑅 becomes a FR, the strict component 𝑃 of 𝑅 becomes the fuzzy strict component of 𝑅 and thus, condition (4.1) of the previous result becomes condition “𝑃 is simple," that is, ∀(𝑥,𝑦)∈𝑋×𝑋,𝑅(𝑥,𝑦)=𝜇_𝑅(𝑥,𝑦)=𝑅(𝑦,𝑥)=𝜇_𝑅(𝑦,𝑥)⇒𝑃(𝑥,𝑦)=𝜇_𝑃(𝑥,𝑦)=𝑃(𝑦,𝑥)=𝜇_𝑃(𝑦,𝑥). (4.4) For convenience and as in fuzzy case, we also call condition (4.1): “𝑃 is simple."(2)The strict component 𝑃 of 𝑅 obtained in the previous result satisfies the following condition: 𝜇∀𝑥,𝑦∈𝑋,^𝑅(𝑥,𝑦)≤𝜇_𝑅𝜈(𝑦,𝑥)_𝑅(𝑥,𝑦)≥𝜈_𝑅⟺𝜇(𝑦,𝑥)𝜈^𝑃(𝑥,𝑦)=0_𝑃(𝑥,𝑦)=1. (4.5)

If 𝑅 becomes a FR, condition (4.5) becomes the condition “𝑃 is regular," that is, ∀(𝑥,𝑦)∈𝑋×𝑋, 𝑅(𝑥,𝑦)=𝜇_𝑅(𝑥,𝑦)≤𝑅(𝑦,𝑥)=𝜇_𝑅(𝑦,𝑥)⇒𝑃(𝑥,𝑦)=𝜇_𝑃(𝑥,𝑦)=0. (4.6) For convenience and as in fuzzy case, we also call condition (4.5): “𝑃 is regular."

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

“𝑃 is regular and 𝐼 is defined by (4.2) if 𝑃 is perfect antisymmetric and simple, 𝐼 is symmetric, and 𝑅=𝐼∪_𝒥𝑃 for 𝒥=𝒥_𝑀=(max,min).”

An interesting question is to check if this version of Dimitrov's result remains true for 𝒥=(𝑆,𝑇).

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 𝒥=(𝑆,𝑇),𝑅 be a reflexive, weakly complete and 𝜋-symmetric IFR; 𝐼 and 𝑃 are two IFRs such that: (i) 𝑅=𝐼∪_𝒥𝑃, (ii) 𝐼 is symmetric, (iii) 𝑃 is perfect antisymmetric, (iv) 𝑃 is simple. Then,
(1)𝐼 is defined by (4.2);(2)𝑃 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 𝑃 defined by (4.3) is obtained for the particular t-representable if-t-conorm 𝒥=𝒥_𝐌=(max,min).(2)The discontinuity of 𝑃, that is,(i)for all 𝑥,𝑦∈𝑋, the degree 𝜇_𝑃(𝑥,𝑦) is insensitive for the variability of 𝜇_𝑅(𝑥,𝑦) and 𝜇_𝑅(𝑦,𝑥). For illustration, if (𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥))=(1,0.999) or (𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥))=(1,0), then 𝜇_𝑃(𝑥,𝑦)=1. But if (𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥))=(1,1), then 𝜇_𝑃(𝑥,𝑦)=0;(ii)for all 𝑥,𝑦∈𝑋, the degree 𝜈_𝑃(𝑥,𝑦) is insensitive for the variability of 𝜈_𝑅(𝑥,𝑦) and 𝜈_𝑅(𝑦,𝑥). For illustration, if (𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥))=(0,0.001) or (𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥))=(0,1), then 𝜈_𝑃(𝑥,𝑦)=0. But if (𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥))=(0,0), then 𝜈_𝑃(𝑥,𝑦)=1.

The previous observations force us to complete and generalize the factorization of Dimitrov for a if-t-conorm 𝒥=(𝑆,𝑇) satisfying condition 𝐺.

4.2. A New and General Factorization of an IFR

First at all, we introduce formally a definition of “indifference of an IFR" and “strict component of an IFR."

Definition 4.3. Let 𝒥=(𝑆,𝑇) satisfying condition 𝐺,𝑅 be an IFR; 𝐼 and 𝑃 are two IFRs. 𝐼 and 𝑃 are “indifference of 𝑅" and “strict component of 𝑅" associated to 𝒥, respectively, if the following conditions are satisfied: 𝑅=𝐼∪_(𝑆,𝑇)𝑃,𝑃issimpleandperfectantisymmetric,𝐼issymmetric. (4.7)

With the results of Lemma 4.2, the equality of (4.7) becomes the following equation: ∀𝑥,𝑦∈𝑋,(𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦))=𝒥[(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥)),(𝑎,𝑏)] (4.8) which is equivalent to the following system: 𝑎+𝑏≤1,𝑆(𝜇_𝑅(𝑦,𝑥),𝑎)=𝜇_𝑅(𝑥,𝑦)(𝐸₁),𝑇(𝜈_𝑅(𝑦,𝑥),𝑏)=𝜈_𝑅(𝑥,𝑦)(𝐸₂). (4.9)

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

Lemma 4.4. Let 𝒥=(𝑆,𝑇),𝑅 be an IFR, and let 𝑥,𝑦∈𝑋 such that 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅𝜈(𝑦,𝑥),_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥). (4.10) Then,
(i)(4.9)(𝐸₁) and (4.9)(𝐸₂) have at least one solution;(ii)each solution of (4.9)(𝐸₁) is strictly positive, and each solution of (4.9)(𝐸₂) is strictly least than 1;(iii)if 𝒥 satisfies condition 𝐺, then (4.8) or (4.9) has at least one solution;(iv)if 𝒥 satisfies condition 𝐺, then the element (𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)),𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦))) is the optimal solution of (4.8) or (4.9), that is, 𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦)) is the upper solution of (4.9)(𝐸₂), and 𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)) is the lowest solution of (4.9)(𝐸₁).(v)furthermore, if 𝒥=𝒥_𝐌=(max,min) or 𝒥 is a strict if-t-conorm (i.e., 𝑆 and 𝑇 are strict) satisfying condition 𝐺, then (𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)),𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦))) 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 𝒥=(𝑆,𝑇) satisfying condition 𝐺,𝑅 be an IFR; 𝐼 and 𝑃 are two IFRs.
The two following statements are equivalent:
(1)𝐼 and 𝑃 are “indifference" and “strict component of 𝑅" associated to 𝒥, respectively;(2)(i)𝜇∀𝑥,𝑦∈𝑋,^𝐼(𝑥,𝑦)=𝜇_𝐼(𝑦,𝑥)=min(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅𝜈(𝑦,𝑥)),_𝐼(𝑥,𝑦)=𝜈_𝐼(𝑦,𝑥)=max(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥)), (4.11)(ii)∀𝑥,𝑦∈𝑋,∃(𝑐_𝑥𝑦,𝑔_𝑥𝑦)∈𝐿^∗suchthat𝑐_𝑥𝑦>0,𝑐_𝑥𝑦isasolutionof (4.9)(𝐸₁), 𝑔_𝑥𝑦<1,𝑔_𝑥𝑦isasolutionof (4.9)(𝐸₂), and 𝜇_𝑃(𝑥,𝑦)=0,if𝜇_𝑅(𝑥,𝑦)≤𝜇_𝑅𝑐(𝑦,𝑥),_𝑥𝑦𝜈,otherwise,_𝑃(𝑥,𝑦)=1,if𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅𝑔(𝑦,𝑥),_𝑥𝑦,otherwise. (4.12)

The previous factorization gives a unique indifference of 𝑅. However, as in fuzzy case and contrary to the crisp case, for an IFR 𝑅 and for 𝒥=(𝑆,𝑇) satisfying 𝐺, the previous result generates a family of strict components of 𝑅. More interesting is that family has an optimal element called the optimal strict component 𝑃 of 𝑅 associated to 𝒥.

Let us give expressions of optimal intuitionistic fuzzy strict components 𝑃 of 𝑅 associated to 𝒥 in the general case and for the three particular cases where 𝒥∈{𝒥_𝐌,𝒥_𝐋,𝒥_𝐏}.

Example 4.6. (1) If 𝒥=(𝑆,𝑇) satisfying 𝐺, then Theorem 4.5 implies that 𝑅 has an optimal strict component 𝑃 defined by, ∀𝑎,𝑏∈𝑋, 𝜇_𝑃(𝑎,𝑏)=𝐽_𝑆(𝜇_𝑅(𝑏,𝑎),𝜇_𝑅𝜈(𝑎,𝑏)),_𝑃(𝑎,𝑏)=𝐼_𝑇(𝜈_𝑅(𝑏,𝑎),𝜈_𝑅(𝑎,𝑏)). (4.13)
(2) If 𝒥=𝒥_𝐌=(𝑆_𝐌,𝑇_𝐌), then the optimal strict component 𝑃 of 𝑅 is defined by, ∀𝑎,𝑏∈𝑋, 𝜇_𝑃(𝑎,𝑏)=𝐽_𝑆𝐌(𝜇_𝑅(𝑏,𝑎),𝜇_𝑅=(𝑎,𝑏))0,if𝜇_𝑅(𝑎,𝑏)≤𝜇_𝑅𝜇(𝑏,𝑎)_𝑅𝜈(𝑎,𝑏),otherwise,_𝑃(𝑎,𝑏)=𝐼_𝑇𝐌(𝜈_𝑅(𝑏,𝑎),𝜈_𝑅=(𝑎,𝑏))1,if𝜈_𝑅(𝑎,𝑏)≥𝜈_𝑅𝜈(𝑏,𝑎),_𝑅(𝑎,𝑏),otherwise. (4.14) This version is the one obtained by Dimitrov (see [18] or Proposition 4.1).(3) If 𝒥=𝒥_𝐋=(𝑆_𝐋,𝑇_𝐋), then the optimal strict component 𝑃 of 𝑅 is defined by, ∀𝑎,𝑏∈𝑋, 𝜇_𝑃(𝑎,𝑏)=𝐽_𝑆𝐋(𝜇_𝑅(𝑏,𝑎),𝜇_𝑅=(𝑎,𝑏))0,if𝜇_𝑅(𝑎,𝑏)≤𝜇_𝑅𝜇(𝑏,𝑎)_𝑅(𝑎,𝑏)−𝜇_𝑅𝜈(𝑏,𝑎),otherwise,_𝑃(𝑎,𝑏)=𝐼_𝑇𝐋(𝜈_𝑅(𝑏,𝑎),𝜈_𝑅=(𝑎,𝑏))1,if𝜈_𝑅(𝑎,𝑏)≥𝜈_𝑅(𝑏,𝑎),1+𝜈_𝑅(𝑎,𝑏)−𝜈_𝑅(𝑏,𝑎),otherwise. (4.15)(4) If 𝒥=𝒥_𝐏=(𝑆_𝐏,𝑇_𝐏), then the optimal strict component 𝑃 of 𝑅 is defined by, ∀𝑎,𝑏∈𝑋, 𝜇_𝑃(𝑎,𝑏)=𝐽_𝑆𝐏(𝜇_𝑅(𝑏,𝑎),𝜇_𝑅=(𝑎,𝑏))0,if𝜇_𝑅(𝑎,𝑏)≤𝜇_𝑅𝜇(𝑏,𝑎),_𝑅(𝑎,𝑏)−𝜇_𝑅(𝑏,𝑎)1−𝜇_𝑅𝜈(𝑏,𝑎),otherwise,_𝑃(𝑎,𝑏)=𝐼_𝑇𝐏(𝜈_𝑅(𝑏,𝑎),𝜈_𝑅=(𝑎,𝑏))1,if𝜈_𝑅(𝑎,𝑏)≥𝜈_𝑅𝜈(𝑏,𝑎),_𝑅(𝑎,𝑏)𝜈_𝑅(𝑏,𝑎),otherwise. (4.16)

It is interesting to give some cases of 𝒥 where the family of intuitionistic fuzzy strict components of 𝑅 has a unique element (i.e., becomes the optimal intuitionistic fuzzy strict component).

The following result specifies that we have a unique intuitionistic fuzzy strict component of an IFR 𝑅 if 𝒥 is a strict t-conorm satisfying 𝐺 or 𝒥=𝒥_𝐌.

Corollary 4.7. Let 𝒥=(𝑆,𝑇),𝑅 be an IFR; 𝐼 and 𝑃 are two IFRs.
If 𝒥 is a strict t-conorm satisfying 𝐺, or 𝒥=𝒥_𝐌, thus the two following statements are equivalent:(1)𝐼 and 𝑃 are “indifference of R" and “strict component of 𝑅" associated to 𝒥 respectively.(2)𝐼 and 𝑃 are, respectively, defined by, ∀𝑥,𝑦∈𝑋, 𝜇_𝐼(𝑥,𝑦)=𝜇_𝐼(𝑦,𝑥)=min(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅𝜈(𝑦,𝑥)),_𝐼(𝑥,𝑦)=𝜈_𝐼(𝑦,𝑥)=max(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅𝜇(𝑦,𝑥)),_𝑃(𝑥,𝑦)=𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅𝜈(𝑥,𝑦)),_𝑃(𝑥,𝑦)=𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦)). (4.17)

The following result shows that when the IFR 𝑅 becomes a FR, the previous theorem becomes the factorization established by Fono and Andjiga (see [7, Proposition 3, page 378]).

Corollary 4.8. Let 𝒥=(𝑆,𝑇), 𝑅 be an IFR; 𝐼 and 𝑃 are two IFRs.
If 𝑇 and 𝑆 are dual and 𝑅 becomes a FR, then the two following statements are equivalent:(1)𝐼 and 𝑃 are “indifference of 𝑅" and “strict component of 𝑅" associated to 𝒥, respectively.(2)𝐼 and 𝑃 are, respectively, defined by, ∀𝑥,𝑦∈𝑋;(i)𝜇_𝐼(𝑥,𝑦)=𝜇_𝐼(𝑦,𝑥)=min(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥));(ii)𝜇_𝑅(𝑥,𝑦)≤𝜇_𝑅(𝑦,𝑥)⇔𝜇_𝑃(𝑥,𝑦)=0;(iii)𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥)⇔(𝜇_𝑃(𝑥,𝑦)>0, and 𝜇_𝑃(𝑥,𝑦) is a solution of (4.9)(𝐸₁)).

In this case, “indifference of an IFR" and “strict component of an IFR" become “indifference of a FR" and “strict component of a FR" associated to 𝑆, respectively.

In the rest of the paper, we study two properties of a given strict component of an IFR.

In literature of (binary) crisp relations, it is well-known that the unique strict component 𝑃 of a given reflexive, complete, and transitive crisp relation 𝑅 satisfies those two interesting and usual properties, namely, pos-transitivity, that is, ∀𝑥,𝑦,𝑧∈𝑋,(𝑥𝑃𝑦and𝑦𝑃𝑧)⇒𝑥𝑃𝑧 and negative transitivity, that is, ∀𝑥,𝑦,𝑧∈𝑋,𝑥𝑃𝑧⇒(𝑥𝑃𝑦or𝑦𝑃𝑧).

Fono and Andjiga [7] showed that this result is no true in the fuzzy case. More precisely, they introduced fuzzy versions of these properties (see [7, Definition 5, page 379]), showed that some strict components violate these fuzzy versions (see [7, Example 2, page 383]). They determined necessary and sufficient conditions on a reflexive, weakly complete and 𝑇-transitive FR 𝑅 such that a regular fuzzy strict component of 𝑅 satisfies each of these properties (see [7, Propositions 6 and 7, page 381]).

Following this line, the aim of the sequel is to (i) introduce a version of pos-transitivity for IFRs and a version of negative transitivity for IFRs and (ii) determine necessary and sufficient conditions on a given 𝒯-transitive IFR 𝑅 under which a strict component of 𝑅 satisfies the introduced properties.

5. Properties of a Strict Component of an IFR

5.1. Definitions and Examples of Properties, and New Conditions on an IFR

Definition 5.1. Let 𝑅 be an IFR, and let 𝑃 be a strict component of 𝑅.
(1)𝑃 is pos-transitive if ∀𝑥,𝑦,𝑧∈𝑋,𝜇_𝑃𝜈(𝑥,𝑦)>0_𝑃𝜇(𝑥,𝑦)<1,𝜈^𝑃(𝑦,𝑧)>0_𝑃𝜇(𝑦,𝑧)<1imply^𝑃𝜈(𝑥,𝑧)>0_𝑃(𝑥,𝑧)<1. (5.1)(2)𝑃 is negative transitive if ∀𝑥,𝑦,𝑧∈𝑋, 𝜇_𝑃𝜈(𝑥,𝑦)=0_𝑃𝜇(𝑥,𝑦)=1,𝜈^𝑃(𝑦,𝑧)=0_𝑃𝜇(𝑦,𝑧)=1imply^𝑃𝜈(𝑥,𝑧)=0_𝑃(𝑥,𝑧)=1. (5.2)

Let us give the following remark on these definitions.

Remark 5.2. (1) As 𝑃 is regular, we can rewrite the pos-transitivity as follows: ∀𝑥,𝑦,𝑧∈𝑋, 𝜇(i)^𝑅(𝑥,𝑦)>𝜇_𝑅𝜇(𝑦,𝑥)_𝑅(𝑦,𝑧)>𝜇_𝑅(𝑧,𝑦)imply𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅𝜈(𝑧,𝑥),(ii)^𝑅(𝑥,𝑦)<𝜈_𝑅𝜈(𝑦,𝑥)_𝑅(𝑦,𝑧)<𝜈_𝑅(𝑧,𝑦)imply𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥). (5.3)
(2) As 𝑃 is regular, the negative transitivity is equivalent to the following conjunction: ∀𝑥,𝑦,𝑧∈𝑋, 𝜇_𝑅(𝑦,𝑥)>𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜇_𝑅𝜇(𝑦,𝑧)or_𝑅(𝑦,𝑥)=𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜇_𝑅𝜇(𝑦,𝑧)or_𝑅(𝑦,𝑥)>𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜇_𝑅𝜇(𝑦,𝑧)or_𝑅(𝑥,𝑦)=𝜇_𝑅𝜇(𝑦,𝑥)_𝑅(𝑦,𝑧)=𝜇_𝑅(𝑧,𝑦)implies𝜇_𝑅(𝑥,𝑧)≤𝜇_𝑅𝜈(𝑧,𝑥),_𝑅(𝑦,𝑥)<𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜈_𝑅𝜈(𝑦,𝑧)or_𝑅(𝑦,𝑥)=𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜈_𝑅𝜈(𝑦,𝑧)or_𝑅(𝑦,𝑥)<𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜈_𝑅𝜈(𝑦,𝑧)or_𝑅(𝑥,𝑦)=𝜈_𝑅𝜈(𝑦,𝑥)_𝑅(𝑦,𝑧)=𝜈_𝑅(𝑧,𝑦)implies𝜈_𝑅(𝑥,𝑧)≥𝜈_𝑅(𝑧,𝑥). (5.4)(5.5)(3) Since 𝑅 is 𝜋-symmetric, then 𝜇_𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧),^𝑅(𝑦,𝑥)=𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧),^𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜇_𝑅(𝑦,𝑧), (5.6) are equivalent to 𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧),^𝑅(𝑦,𝑥)=𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧),^𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜈_𝑅(𝑦,𝑧), (5.7) respectively.

One of the main questions is to wonder if a strict component of a given IFR, which is not a FR, satisfies each of these two properties.

In the following, we justify that there exists an IFR 𝑅 (distinct to FRs) such that some strict components of 𝑅 violate each of these properties.

Example 5.3. Let 𝑋={𝑥,𝑦,𝑧} and 𝒥_𝐋 (thus by Proposition 2.10, 𝒥_𝐋 satisfies condition 𝐺).
(1) We determine an IFR 𝑅 on 𝑋 such that there exists a strict component 𝑃 of 𝑅 which violates pos-transitivity, (i.e., 𝑃 satisfies: there exists 𝑢,𝑣,𝑤∈𝑋 such that 𝜇_𝑃𝜈(𝑢,𝑣)>0_𝑃𝜇(𝑢,𝑣)<1,𝜈^𝑃(𝑣,𝑤)>0_𝑃(𝑣,𝑤)<1,(𝜇_𝑃(𝑢,𝑤)=0or𝜈_𝑃(𝑢,𝑤)=1). (5.8)
Let 𝑅 be defined by, ∀𝑑∈𝑋,𝑅(𝑑,𝑑)=(1,0);𝑅(𝑥,𝑦)=(0.7,0.2);𝑅(𝑦,𝑥)=(0.5,0.4);𝑅(𝑦,𝑧)=(0.6,0.2);𝑅(𝑧,𝑦)=(0.5,0.3); 𝑅(𝑥,𝑧)=(0.5,0.4); 𝑅(𝑧,𝑥)=(0.8,0.1).
Clearly, 𝑅 is reflexive, weakly complete, and 𝜋-symmetric. By Theorem 4.5, 𝑅 has an optimal strict component 𝑃_𝐋 defined by (4.15).
We have: 𝜇_𝑃𝐋(𝑥,𝑦)=𝜇_𝑅(𝑥,𝑦)−𝜇_𝑅(𝑦,𝑥)=0.2>0; 𝜇_𝑃𝐋(𝑦,𝑧)=𝜇_𝑅(𝑦,𝑧)−𝜇_𝑅(𝑧,𝑦)=0.1>0; 𝜈_𝑃𝐋(𝑥,𝑦)=1+𝜈_𝑅(𝑥,𝑦)−𝜈_𝑅(𝑦,𝑥)=0.8<1 and 𝜈_𝑃𝐋(𝑦,𝑧)=1+𝜈_𝑅(𝑦,𝑧)−𝜈_𝑅(𝑧,𝑦)=0.9<1, whereas 𝜇_𝑃𝐋(𝑥,𝑧)=0 and 𝜈_𝑃𝐋(𝑥,𝑧)=1.
In other words, 𝑥 is strictly preferred to y (since 𝜇_𝑃𝐋(𝑥,𝑦)>0 and 𝜈_𝑃𝐋(𝑥,𝑦)<1), and 𝑦 is strictly preferred to 𝑧 (since 𝜇_𝑃𝐋(𝑦,𝑧)>0 and 𝜈_𝑃𝐋(𝑦,𝑧)<1), but 𝑥 is not strictly preferred to 𝑧 (since 𝜇_𝑃𝐋(𝑥,𝑧)=0 and 𝜈_𝑃𝐋(𝑥,𝑧)=1).
Hence 𝑃_𝐋 violates pos-transitivity.
(2) We determine an IFR 𝑅 on 𝑋 such that there exists a strict component 𝑃 of 𝑅 which violates negative transitivity, (i.e., 𝑃 satisfies: there exists 𝑢,𝑣,𝑤∈𝑋 such that 𝜇_𝑃𝜈(𝑢,𝑣)=0_𝑃𝜇(𝑢,𝑣)=1,𝜈^𝑃(𝑣,𝑤)=0_𝑃(𝑣,𝑤)=1,(𝜇_𝑃(𝑢,𝑤)>0or𝜈_𝑃(𝑢,𝑤)<1). (5.9)
Let 𝑅 be defined by, ∀𝑑∈𝑋,𝑅(𝑑,𝑑)=(1,0);𝑅(𝑥,𝑦)=(0.5,0.4);𝑅(𝑦,𝑥)=(0.7,0.2);𝑅(𝑦,𝑧)=(0.4,0.5);𝑅(𝑧,𝑦)=(0.8,0.1); 𝑅(𝑥,𝑧)=(0.6,0.1); 𝑅(𝑧,𝑥)=(0.4,0.3).
Clearly, 𝑅 is reflexive, weakly complete, and 𝜋-symmetric. By Theorem 4.5, 𝑅 has an optimal strict component 𝑃_𝐋 defined by (4.15).
We have 𝜇_𝑃𝐋(𝑥,𝑦)=𝜇_𝑃𝐋(𝑦,𝑧)=0 and 𝜈_𝑃𝐋(𝑥,𝑦)=𝜈_𝑃𝐋(𝑦,𝑧)=1, whereas 𝜇_𝑃𝐋(𝑥,𝑧)=𝜇_𝑅(𝑥,𝑧)−𝜇_𝑅(𝑧,𝑥)=0.2>0, and 𝜈_𝑃𝐋(𝑥,𝑧)=1+𝜈_𝑅(𝑥,𝑧)−𝜈_𝑅(𝑧,𝑥)=0.8<1.
In other words, 𝑥 is strictly preferred to 𝑧 (since 𝜇_𝑃𝐋(𝑥,𝑧)>0 and 𝜈_𝑃𝐋(𝑥,𝑧)<1), but 𝑥 is not strictly preferred to 𝑦 (since 𝜇_𝑃𝐋(𝑥,𝑦)=0 and 𝜈_𝑃𝐋(𝑥,𝑦)=1), and 𝑦 is not strictly preferred to 𝑧 (since 𝜇_𝑃𝐋(𝑦,𝑧)=0 and 𝜈_𝑃𝐋(𝑦,𝑧)=1).
Hence 𝑃_𝐋 violates negative transitivity.

This enables us to determine necessary and sufficient conditions on an IFR 𝑅 such that a given strict component 𝑃_𝑅 of 𝑅 is pos-transitive and negative transitive. For that we introduce the following conditions.

Definition 5.4. Let 𝑅 be an IFR.
(1)(i)𝑅 satisfies condition 𝐶^𝜇𝑅₁ if ∀𝑥,𝑦,𝑧∈𝑋, 𝜇_𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧)⇒(^𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃𝜇(𝑥,𝑦,𝑧)]_𝑅(𝑧,𝑥)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)]⇒𝜇_𝑅(𝑧,𝑥)<𝜇_𝑅(𝑥,𝑧)). (5.10)(ii)𝑅 satisfies condition 𝐶^𝜈𝑅₁ if ∀𝑥,𝑦,𝑧∈𝑋, 𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧)⇒(^𝑅(𝑥,𝑧)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂𝜈(𝑥,𝑦,𝑧)]_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧)]⇒𝜈_𝑅(𝑧,𝑥)>𝜈_𝑅(𝑥,𝑧)). (5.11)(2)(i)𝑅 satisfies condition 𝐶^𝜇𝑅₂ if ∀𝑥,𝑦,𝑧∈𝑋, (𝜇_𝑅(𝑦,𝑥)=𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧)or^𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜇_𝑅)⇒𝜇(𝑦,𝑧)_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃𝜇(𝑥,𝑦,𝑧)]_𝑅(𝑧,𝑥)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃⇓𝜇(𝑥,𝑦,𝑧)]_𝑅(𝑧,𝑥)<𝜇_𝑅.(𝑥,𝑧) (5.12)(ii)𝑅 satisfies condition 𝐶^𝜈𝑅₂ if ∀𝑥,𝑦,𝑧∈𝑋, (𝜈_𝑅(𝑦,𝑥)=𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧)or^𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜈_𝑅)⇒𝜈(𝑦,𝑧)_𝑅(𝑥,𝑧)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂𝜈(𝑥,𝑦,𝑧)]_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂⇓𝜈(𝑥,𝑦,𝑧)]_𝑅(𝑧,𝑥)>𝜈_𝑅.(𝑥,𝑧) (5.13)

The next result shows that a strongly complete IFR 𝑅 satisfies the previous conditions.

Proposition 5.5. Let 𝑅 be an IFR.
If 𝑅 is strongly complete, then 𝑅 satisfies conditions 𝐶^𝜇𝑅₁,𝐶^𝜈𝑅₁,𝐶^𝜇𝑅₂, and 𝐶^𝜈𝑅₂.

5.2. Characterization of Some Properties of a Strict Component of an IFR

We now establish, using 𝐶^𝜇𝑅₁ and 𝐶^𝜈𝑅₁, our second main result which determines all 𝒯-transitive IFRs whose strict components are pos-transitive.

Theorem 5.6. Let 𝑅 be a 𝒯-transitive IFR, and let 𝑃 be a strict component of 𝑅. Then 𝑅satisfiescondition𝐶^𝜇𝑅₁or𝑅satisfiescondition𝐶^𝜈𝑅₁⟺(𝑃ispos-transitive). (5.14)

In the particular case where 𝑅 is strongly complete, the previous result becomes as follows:

Corollary 5.7. Let 𝑅 be a 𝒯-transitive IFR, and let 𝑃 be a strict component of 𝑅.
If 𝑅 is strongly complete, then 𝑃 is pos-transitive.

Given a 𝒯-transitive IFR 𝑅, the next result establishes an equivalence between conditions 𝐶^𝜇𝑅₂ and 𝐶^𝜈𝑅₂, and two properties of 𝑃.

Lemma 5.8. Let 𝑅 be a 𝒯-transitive IFR, and let 𝑃 be a strict component of 𝑅.
The two following statements are equivalent:
(1)𝑅 satisfies condition 𝐶^𝜇𝑅₂ or condition 𝐶^𝜈𝑅₂;(2)𝜇∀𝑥,𝑦,𝑧∈𝑋(i)(^𝑃𝜈(𝑥,𝑦)>0_𝑃,𝜇(𝑥,𝑦)<1^𝑃(𝑦,𝑧)=𝜇_𝑃𝜈(𝑧,𝑦)=0_𝑃(𝑦,𝑧)=𝜈_𝑃)𝜇(𝑧,𝑦)=1or(ii)(^𝑃(𝑥,𝑦)=𝜇_𝑃𝜈(𝑦,𝑥)=0_𝑃(𝑥,𝑦)=𝜈_𝑃,𝜇(𝑦,𝑥)=1𝜈^𝑃(𝑦,𝑧)>0_𝑃)⇒𝜇(𝑦,𝑧)<1^𝑃𝜈(𝑥,𝑧)>0_𝑃(𝑥,𝑧)<1. (5.15)

It is important to notice that as 𝑃 is regular and 𝑅 is 𝜋-symmetric, we can rewrite (5.15) as follows: 𝜇∀𝑥,𝑦,𝑧∈𝑋,(i)(^𝑅(𝑦,𝑥)=𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧)or^𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜇_𝑅(𝑦,𝑧))⇒𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅𝜈(𝑧,𝑥)or(ii)(^𝑅(𝑦,𝑥)=𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧)or^𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜈_𝑅(𝑦,𝑧))⇒𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅.(𝑧,𝑥) (5.16)

The third and last main result of our paper determines all 𝒯-transitive IFRs whose strict components are negative transitive.

Theorem 5.9. Let 𝑅 be a 𝒯-transitive IFR, and let 𝑃 be a strict component of 𝑅. Then 𝑅satisfiesconditions𝐶^𝜇𝑅₁and𝐶^𝜇𝑅₂or𝑅satisfiesconditions𝐶^𝜈𝑅₁and𝐶^𝜈𝑅₂⟺(𝑃isnegativetransitive). (5.17)

In the particular case where 𝑅 is strongly complete, the previous result becomes as follows:

Corollary 5.10. Let 𝑅 be a 𝒯-transitive IFR, and let 𝑃 be a strict component of 𝑅.
If 𝑅 is strongly complete, then 𝑃 is negative transitive.

6. Concluding Remarks

In this paper, we establish a characterization of the 𝒯-transitivity of 𝑅. We also establish a general factorization of an intuitionistic fuzzy binary relation when the union is defined by a continuous t-representable intuitionistic fuzzy t-conorm satisfying condition 𝐺. This factorization gives a family of regular strict components. Furthermore, given an IFR 𝑅, we introduce two conditions 𝐶^𝜇𝑅₁ and 𝐶^𝜇𝑅₂ (or equivalently 𝐶^𝜈𝑅₁ and 𝐶^𝜈𝑅₂). And we show that, when 𝑅 is 𝒯-transitive, these conditions are necessary and sufficient to obtain pos-transitivity and negative transitivity of a given strict component of 𝑅.

An open problem is to apply these results especially in social choice theory when individual and social preferences are modelled by reflexive, weakly complete, and 𝒯-transitive IFRs. Another open problem is to study the properties of the class of continuous t-representable intuitionistic fuzzy t-conorms satisfying condition 𝐺.

Appendix

The Proofs of our Results

Proof of Proposition 2.7. (1) Let 𝑎,𝑏∈[0,1]. Since 𝑇(𝑎,𝑏)≤1−𝑆(1−𝑎,1−𝑏), then {𝑡∈[0,1],1−𝑆(1−𝑎,1−𝑡)≤𝑏}⊆{𝑡∈[0,1],𝑇(𝑎,𝑡)≤𝑏}. Set 𝑡^′=1−𝑡, we have 𝐼_𝑇(𝑎,𝑏)≥max{𝑡∈[0,1],1−𝑆(1−𝑎,1−𝑡)≤𝑏} =max{𝑡∈[0,1],𝑆(1−𝑎,1−𝑡)≥1−𝑏} =max{1−𝑡^′∈[0,1],𝑆(1−𝑎,𝑡^′)≥1−𝑏}=1 −min{𝑡^′∈[0,1],𝑆(1−𝑎,𝑡^′)≥1−𝑏}=1−𝐽_𝑆(1−𝑎,1−𝑏).
(2) The proof is obvious.

Proof of Lemma 2.8. (1) Let 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈𝐿^∗. Set 𝐴={𝑧=(𝑧₁,𝑧₂)∈𝐿^∗,𝑦₁≤𝑆(𝑥₁,𝑧₁) and 𝑦₂≥𝑇(𝑥₂,𝑧₂)}≠∅. That is to show that inf𝐴=(min{𝑧₁∈[0,1],𝑦₁≤𝑆(𝑥₁,𝑧₁)},min(max{𝑧₂∈[0,1],𝑇(𝑥₂,𝑧₂)≤𝑦₂},1−min{𝑧₁∈[0,1],𝑦₁≤𝑆(𝑥₁,𝑧₁)})).
Since 𝐴⊆𝐿^∗ and (𝐿^∗,≤_𝐿^∗) is a complete lattice, and 𝑇 and 𝑆 are continuous functions on the compact [0,1]×[0,1], then the definition of lower limit in 𝐿^∗ gives inf𝐴=(inf{𝑧₁∈[0,1],∃𝑡₁∈[0,1],(𝑧₁,𝑡₁)∈𝐴},sup{𝑧₂∈[0,1],∃𝑡₂∈[0,1],(𝑡₂,𝑧₂)∈𝐴})=(inf{𝑧₁∈[0,1],∃𝑡₁𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁≤1},sup{𝑧₂∈[0,1],∃𝑡₂𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂≤1})=(min{𝑧₁∈[0,1],∃𝑡₁𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁≤1},max{𝑧₂∈[0,1],∃𝑡₂𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂≤1}). (A.1)
Otherwise, with the second result of Proposition 2.1, we have 𝑦₁≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁⟺𝑧≤1¹∈[𝐽_𝑆(𝑥₁,𝑦₁𝑡),1]₁∈[0,𝐼_𝑇(𝑥₂,𝑦₂𝑧)]₁+𝑡₁𝑦≤1,₁≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂⟺𝑡≤1²∈[𝐽_𝑆(𝑥₁,𝑦₁𝑧),1]₂∈[0,𝐼_𝑇(𝑥₂,𝑦₂𝑧)]₂+𝑡₂≤1. (A.2) We distinguish three cases.
(i)If 𝑥₁<𝑦₁ and 𝑥₂≤𝑦₂, then with the third result of Proposition 2.1, we have 𝐽_𝑆(𝑥₁,𝑦₁)>0 and 𝐼_𝑇(𝑥₂,𝑦₂)=1. This implies 𝐼_𝑇(𝑥₂,𝑦₂)>1−𝐽_𝑆(𝑥₁,𝑦₁). Thus, min{𝑧₁∈[0,1],∃𝑡₁𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁≤1}=𝐽_𝑆(𝑥₁,𝑦₁),max{𝑧₂∈[0,1],∃𝑡₂𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂≤1}=1−𝐽_𝑆(𝑥₁,𝑦₁). (A.3) Hence 𝐽_𝒥(𝑥,𝑦)=(𝐽_𝑆(𝑥₁,𝑦₁),1−𝐽_𝑆(𝑥₁,𝑦₁)).(ii)If (𝑥₁≥𝑦₁ and 𝑥₂≤𝑦₂) or (𝑥₁≥𝑦₁ and 𝑥₂>𝑦₂), then with the third result of Proposition 2.1, it is easy to show that min{𝑧₁∈[0,1],∃𝑡₁𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁≤1}=𝐽_𝑆(𝑥₁,𝑦₁),max{𝑧₂∈[0,1],∃𝑡₂𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂≤1}=𝐼_𝑇(𝑥₂,𝑦₂)≤1−𝐽_𝑆(𝑥₁,𝑦₁). (A.4) Hence 𝐽_𝒥(𝑥,𝑦)=(𝐽_𝑆(𝑥₁,𝑦₁),𝐼_𝑇(𝑥₂,𝑦₂)).(iii)If 𝑥₁<𝑦₁ and 𝑥₂>𝑦₂, then with the third result of Proposition 2.1, 𝐽_𝑆(𝑥₁,𝑦₁)>0, and 𝐼_𝑇(𝑥₂,𝑦₂)<1. We distinguish two cases.(a)If 𝐼_𝑇(𝑥₂,𝑦₂)≤1−𝐽_𝑆(𝑥₁,𝑦₁), thus min{𝑧₁∈[0,1],∃𝑡₁𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁≤1}=𝐽_𝑆(𝑥₁,𝑦₁),max{𝑧₂∈[0,1],∃𝑡₂𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂≤1}=𝐼_𝑇(𝑥₂,𝑦₂). (A.5) Hence 𝐽_𝒥(𝑥,𝑦)=(𝐽_𝑆(𝑥₁,𝑦₁),𝐼_𝑇(𝑥₂,𝑦₂)).(b)If 𝐼_𝑇(𝑥₂,𝑦₂)>1−𝐽_𝑆(𝑥₁,𝑦₁), thus min{𝑧₁∈[0,1],∃𝑡₁𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑧₁)𝑦₂≥𝑇(𝑥₂,𝑡₁)𝑧₁+𝑡₁≤1}=𝐽_𝑆(𝑥₁,𝑦₁),max{𝑧₂∈[0,1],∃𝑡₂𝑦∈[0,1],¹≤𝑆(𝑥₁,𝑡₂)𝑦₂≥𝑇(𝑥₂,𝑧₂)𝑧₂+𝑡₂≤1}=1−𝐽_𝑆(𝑥₁,𝑦₁). (A.6) Hence 𝐽_𝒥(𝑥,𝑦)=(𝐽_𝑆(𝑥₁,𝑦₁),1−𝐽_𝑆(𝑥₁,𝑦₁)).
Finally, we obtain 𝐽_𝒥(𝑥,𝑦)=(𝐽_𝑆(𝑥₁,𝑦₁),min(𝐼_𝑇(𝑥₂,𝑦₂),1−𝐽_𝑆(𝑥₁,𝑦₁))).
(2) The proof of the second result is analogous to the previous one.

Proof of Proposition 2.10. (1) Let (𝑎₁,𝑎₂),(𝑏₁,𝑏₂)∈𝐿^∗ such that 𝑎₁>𝑏₁,𝑎₂<𝑏₂,𝑎₁+𝑎₂=𝑏₁+𝑏₂. (A.7)
(i)Since 𝒥^𝐹₀=𝒥_𝐌,𝒥^𝐹₁=𝒥_𝐏 and 𝒥^𝐹_∞=𝒥_𝐋, with Example 2.5, it is obvious to show that ∀𝑙∈{0,1,+∞},𝒥^𝐹_𝑙 satisfies condition 𝐺.(ii)Assume that 𝑙∈]0,1[, and let us show that 𝒥^𝐹_𝑙=(𝑆^𝑙_𝐹,𝑇^𝑙_𝐹) satisfies condition 𝐺, that is, 𝐼_𝑇(𝑏₂,𝑎₂)+𝐽_𝑆(𝑏₁,𝑎₁)≤1. It suffices to show that ∀𝑙∈]0,1[,log_𝑙(1+((𝑙−1)(𝑙^𝑎2−1))/(𝑙^𝑏2−1))≤log_𝑙(1+(𝑙−1)(𝑙^1−𝑎1−1)/(𝑙^1−𝑏1−1)). Set 𝑐=𝑎₁+𝑎₂=𝑏₁+𝑏₂.(a) Since 𝑎₁>𝑏₁ and the mapping 𝑙^𝑡 is decreasing, then 𝑙^𝑎1≤𝑙^𝑏1, that is, 𝑙^−𝑏1≤𝑙^−𝑎1. (b)Since 𝑙^𝑐−𝑙≥0, we obtain 𝑙^−𝑏1(𝑙^𝑐−𝑙)≤𝑙^−𝑎1(𝑙^𝑐−𝑙), that is, 𝑙^𝑏2+𝑙^1−𝑎1≤𝑙^𝑎2+𝑙^1−𝑏1.(c)Since 𝑎₂−𝑏₁=𝑏₂−𝑎₁, the previous inequality becomes 𝑙^{1+𝑎2−𝑏1}−𝑙^𝑎2−𝑙^1−𝑏1+1≤𝑙^{1+𝑏2−𝑎1}−𝑙^𝑏2−𝑙^1−𝑎1+1, that is, (𝑙^𝑎2−1)(𝑙^1−𝑏1−1)≤(𝑙^𝑏2−1)(𝑙^1−𝑎1−1), that is, (𝑙^𝑎2−1)/(𝑙^𝑏2−1)≤(𝑙^1−𝑎1−1)/(𝑙^1−𝑏1−1), that is, 1+(𝑙−1)(𝑙^𝑎2−1)/(𝑙^𝑏2−1)≥1+(𝑙−1)(𝑙^1−𝑎1−1)/(𝑙^1−𝑏1−1). And we have log_𝑙(1+(𝑙−1)(𝑙^𝑎2−1)/(𝑙^𝑏2−1))≤log_𝑙(1+(𝑙−1)(𝑙^1−𝑎1−1)/(𝑙^1−𝑏1−1)).(iii)The proof of the case 𝑙∈]1,+∞[ is similar to the previous one.
(2) Let 𝒥^′ be a restriction of 𝒥=(𝑆,𝑇) on 𝐿^∗₁={(𝑥₁,𝑥₂)∈𝐿^∗,𝑥₁+𝑥₂=1}. Assume that 𝑆 and 𝑇 are dual and show that 𝒥^′ satisfies 𝐺. Let (𝑎₁,𝑎₂),(𝑏₁,𝑏₂)∈𝐿^∗₁ such that 𝑎₁>𝑏₁,𝑎₂<𝑏₂,𝑎₁+𝑎₂=𝑏₁+𝑏₂=1. (A.8) Let us show that 𝐼_𝑇(𝑏₂,𝑎₂)+𝐽_𝑆(𝑏₁,𝑎₁)≤1.
Since 𝑎₁+𝑎₂=𝑏₁+𝑏₂=1 and 𝑆 are 𝑇 are dual, then 𝐼_𝑇(𝑏₂,𝑎₂)=𝐼_𝑇(1−𝑏₁,1−𝑎₁). Thus, 𝐼_𝑇(𝑏₂,𝑎₂)+𝐽_𝑆(𝑏₁,𝑎₁)=𝐼_𝑇(1−𝑏₁,1−𝑎₁)+𝐽_𝑆(𝑏₁,𝑎₁)=1. Hence the result.

Proof of Proposition 3.4. Suppose that 𝑅 is (𝑆,𝑇)-complete and let us show that 𝑅 is 𝐷-complete.
Let (𝑥,𝑦)∈𝑋×𝑋 such that (𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦))=(0,1). Thus 𝑆(0,𝜇_𝑅(𝑦,𝑥))=1 and 𝑇(1,𝜈_𝑅(𝑦,𝑥))=0. Since 𝑆(0,𝜇_𝑅(𝑦,𝑥))=𝜇_𝑅(𝑦,𝑥) and 𝑇(1,𝜈_𝑅(𝑦,𝑥))=𝜈_𝑅(𝑦,𝑥), we have 𝜇_𝑅(𝑦,𝑥)=1 and 𝜈_𝑅(𝑦,𝑥)=0. Hence 𝜇_𝑅(𝑦,𝑥)>0 and 𝜈_𝑅(𝑦,𝑥)<1.

Proof of Proposition 3.9. For all 𝑥,𝑦,𝑧∈𝑋, we have the following.
(1)Let us show that (𝛼₁(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧))∈𝐿^∗. In fact, (𝛼₁(𝑥,𝑦,𝑧), 𝛽₁(𝑥,𝑦,𝑧)) = (𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑦,𝑥)), 𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥))) = 𝒯[(𝜇_𝑅(𝑧,𝑦),𝜈_𝑅(𝑧,𝑦)),(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥))]∈𝐿^∗ since 𝒯 is a t-norm on 𝐿^∗.(a)Let us show that (𝛼₃(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧))∈𝐿^∗. In fact, (𝛼₃(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧)) = min(𝐼_𝒯[(𝜇_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑧)),(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥))], 𝐼_𝒯[(𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦)), (𝜇_𝑅(𝑧,𝑦),𝜈_𝑅(𝑧,𝑦))]). Since, 𝐼_𝒯[(𝜇_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑧)),(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥))]∈𝐿^∗,𝐼_𝒯[(𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦)),(𝜇_𝑅(𝑧,𝑦),𝜈_𝑅(𝑧,𝑦))]∈𝐿^∗ and Min=(min,max) is a t-norm in 𝐿^∗, then (𝛼₃(𝑥,𝑦,𝑧),𝛽₃(𝑥,𝑦,𝑧))∈𝐿^∗.(b)The proofs of the assertions (𝛼₂(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧))∈𝐿^∗ and (𝛼₄(𝑥,𝑦,𝑧),𝛽₄(𝑥,𝑦,𝑧))∈𝐿^∗ are analogous to the previous ones.(2)The proof of the last result is deduced from the second result of Lemma 2.8.

Proof of Corollary 3.11. Assume that 𝑅 is strongly complete.
(1)Assume also that 𝑅 satisfies (3.9). And in the three cases, we have 𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑧)=1, and 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑧)=0. This implies 𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑥,𝑦))=𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑧))=0, 𝐽_𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑧,𝑦))=𝜈_𝑅(𝑧,𝑦), 𝐽_𝑆(𝜈_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑥))=𝜈_𝑅(𝑦,𝑥) and 𝐼_𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑥,𝑦))=𝐼_𝑇(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑦,𝑧))=1, 𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦))=𝜇_𝑅(𝑧,𝑦), 𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥))=𝜇_𝑅(𝑦,𝑥). And we obtain 𝛼₂(𝑥,𝑦,𝑧)=𝛼₄(𝑥,𝑦,𝑧)=1, and 𝛼₃(𝑥,𝑦,𝑧)=min(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑧,𝑦)). Thus 𝛼₁(𝑥,𝑦,𝑧)<1 and 𝛼₃(𝑥,𝑦,𝑧)<1.(2)Assume that 𝑅 satisfies (3.11). Thus 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑧)=0 and (𝜈_𝑅(𝑦,𝑥)>0 or 𝜈_𝑅(𝑧,𝑦)>0). Thus, 𝛽₂(𝑥,𝑦,𝑧)=𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑧))=𝑆(0,0)=0.
Assume to the contrary that 𝛽₁(𝑥,𝑦,𝑧)=0. We have max(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥))≤𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥))=𝛽₁(𝑥,𝑦,𝑧)=0. Hence 𝜈_𝑅(𝑦,𝑥)=0 and 𝜈_𝑅(𝑧,𝑦)=0. This contradicts the assertion (𝜈_𝑅(𝑦,𝑥)>0 or 𝜈_𝑅(𝑧,𝑦)>0).
Let us show that 𝛽₃(𝑥,𝑦,𝑧)=max(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑧,𝑦)) and 𝛽₄(𝑥,𝑦,𝑧)=max(0,0)=0.
Proposition 3.9 implies that 𝛽₃(𝑥,𝑦,𝑧)=max(𝐽_𝑆(0,𝜈_𝑅(𝑦,𝑥)),𝐽_𝑆(0,𝜈_𝑅𝛽(𝑧,𝑦))),₄(𝑥,𝑦,𝑧)=max(𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),0),𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),0)). (A.9)
By definition, 𝐽_𝑆(0,𝜈_𝑅(𝑦,𝑥))=min{𝑡∈[0,1]/𝑆(0,𝑡)≥𝜈_𝑅(𝑦,𝑥)}=min{𝑡∈[0,1]/𝑡≥𝜈_𝑅(𝑦,𝑥)}=min[𝜈_𝑅𝜈(𝑦,𝑥),1]=_𝑅(𝑦,𝑥) and 𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),0)=min{𝑡∈[0,1]/𝑆(𝜈_𝑅(𝑦,𝑥),𝑡)≥0}=min[0,1]=0.
Analogously, 𝐽_𝑆(0,𝜈_𝑅(𝑧,𝑦))=𝜈_𝑅(𝑧,𝑦) and 𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),0))=0. Thus 𝛽₃(𝑥,𝑦,𝑧)=max(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑧,𝑦)), and 𝛽₄(𝑥,𝑦,𝑧)=max(0,0)=0.

Proof of Corollary 3.12. Assume that 𝑇 and 𝑆 are dual and 𝑅 is a FR. Let 𝑥,𝑦,𝑧∈𝑋.(1)(i) Let us show that 𝛽₁(𝑥,𝑦,𝑧)=1−𝛼₁(𝑥,𝑦,𝑧).
By definition, 𝛽₁(𝑥,𝑦,𝑧)=𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥)) and 𝛼₁(𝑥,𝑦,𝑧)=𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑦,𝑥)). Since 𝑅 is a FR, we have 𝜈_𝑅(𝑧,𝑦)=1−𝜇_𝑅(𝑧,𝑦) and 𝜈_𝑅(𝑦,𝑥)=1−𝜇_𝑅(𝑦,𝑥). Otherwise, since 𝑇 and 𝑆 are dual, we have 𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥))=1−𝑇(1−𝜈_𝑅(𝑧,𝑦),1−𝜈_𝑅(𝑦,𝑥)). Thus 𝛽₁(𝑥,𝑦,𝑧)=1−𝑇(1−(1−𝜇_𝑅(𝑧,𝑦)),1−(1−𝜇_𝑅(𝑦,𝑥)))=1−𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑦,𝑥))=1−𝛼₁(𝑥,𝑦,𝑧). Hence the result.
(ii) The proof of equality 𝛽₂(𝑥,𝑦,𝑧)=1−𝛼₂(𝑥,𝑦,𝑧) is analogous to the previous one.(iii) Let us show that 𝛽₃(𝑥,𝑦,𝑧)=1−𝛼₃(𝑥,𝑦,𝑧). With the previous proposition, 𝛽₃(𝑥,𝑦,𝑧)=max(𝐽_𝑆(𝜈_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑥)),𝐽_𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑧,𝑦))). Since 𝑅 is a FR, we have 𝜈_𝑅(𝑦,𝑧)=1−𝜇_𝑅(𝑦,𝑧), 𝜈_𝑅(𝑧,𝑦)=1−𝜇_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥)=1−𝜇_𝑅(𝑦,𝑥), and 𝜈_𝑅(𝑥,𝑦)=1−𝜇_𝑅(𝑥,𝑦). Otherwise, since 𝑇 and 𝑆 are dual, we have 𝐽_𝑆(𝜈_𝑅(𝑦,𝑧),𝜈_𝑅(𝑦,𝑥))=1−𝐼_𝑇(1−𝜈_𝑅(𝑦,𝑧),1−𝜈_𝑅(𝑦,𝑥)) and 𝐽_𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑧,𝑦))=1−𝐼_𝑇(1−𝜈_𝑅(𝑥,𝑦),1−𝜈_𝑅(𝑧,𝑦)). Thus 𝛽₃(𝑥,𝑦,𝑧)=max(1−𝐼_𝑇(1−(1−𝜇_𝑅(𝑦,𝑧)),1−(1−𝜇_𝑅(𝑦,𝑥))),1−𝐼_𝑇(1−(1−𝜇_𝑅(𝑥,𝑦)),1−(1−𝜇_𝑅(𝑧,𝑦))))=max(1−𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)),1−𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦)))=1−min(𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)),𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦))).
Otherwise, 𝛼₃(𝑥,𝑦,𝑧)=min[min(𝐼_𝑇(𝜇_𝑅(𝑦,𝑧), 𝜇_𝑅(𝑦,𝑥)), 1−𝐽_𝑆(𝜈_𝑅(𝑦,𝑧), 𝜈_𝑅(𝑦,𝑥))), min(𝐼_𝑇(𝜇_𝑅(𝑥,𝑦), 𝜇_𝑅(𝑧,𝑦)), 1−𝐽_𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑧,𝑦)))] = min[min(𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)), 𝐼¹_𝑇(1−𝜈_𝑅(𝑦,𝑧), 1−𝜈_𝑅(𝑦,𝑥))),min(𝐼_𝑇(𝜇_𝑅(𝑥,𝑦), 𝜇_𝑅(𝑧,𝑦)), 𝐼_𝑇(1−𝜈_𝑅(𝑥,𝑦),1−𝜈_𝑅(𝑧,𝑦)))] = min[min(𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)), 𝐼_𝑇(𝜇_𝑅(𝑦,𝑧), 𝜇_𝑅(𝑦,𝑥))), min(𝐼_𝑇(𝜇_𝑅(𝑥,𝑦), 𝜇_𝑅(𝑧,𝑦)), 𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦)))] = min[𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)), 𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦))] = 1−𝛽₃(𝑥,𝑦,𝑧). Hence the result.
(iv) The proof of the equality 𝛽₄(𝑥,𝑦,𝑧)=1−𝛼₄(𝑥,𝑦,𝑧) is analogous to the previous one.
(2) The proof of the last result is obvious.

Proof of Lemma 3.13. (i)⇒(ii): Assume that 𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧} and show that (3.16).
Since 𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧}, then we have the following twelve inequalities: (i)𝜈_𝑅(𝑥,𝑧)≤𝑆(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑧));𝜈_𝑅(𝑥,𝑦)≤𝑆(𝜈_𝑅(𝑥,𝑧),𝜈_𝑅𝜈(𝑧,𝑦));_𝑅(𝑦,𝑧)≤𝑆(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑧));(ii)𝜈_𝑅(𝑧,𝑥)≤𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑦,𝑥));𝜈_𝑅(𝑧,𝑦)≤𝑆(𝜈_𝑅(𝑧,𝑥),𝜈_𝑅𝜈(𝑥,𝑦));_𝑅(𝑦,𝑥)≤𝑆(𝜈_𝑅(𝑦,𝑧),𝜈_𝑅(𝑧,𝑥));(iii)𝜇_𝑅(𝑥,𝑧)≥𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑧));𝜇_𝑅(𝑥,𝑦)≥𝑇(𝜇_𝑅(𝑥,𝑧),𝜇_𝑅𝜇(𝑧,𝑦));_𝑅(𝑦,𝑧)≥𝑇(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑧));(iv)𝜇_𝑅(𝑧,𝑥)≥𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑦,𝑥));𝜇_𝑅(𝑧,𝑦)≥𝑇(𝜇_𝑅(𝑧,𝑥),𝜇_𝑅𝜇(𝑥,𝑦));_𝑅(𝑦,𝑥)≥𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑧,𝑥)). (A.10) Let us show that 𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₄(𝑥,𝑦,𝑧)].
By definition of 𝐼_𝑇, the two first inequalities of (iii) of (A.10) imply 𝛼₂(𝑥,𝑦,𝑧)=𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑧))≤𝜇_𝑅(𝑥,𝑧)≤𝐼_𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑥,𝑦)). And the second inequalities of (i) give 𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑥,𝑦))≤𝜈_𝑅(𝑥,𝑧)≤1−𝜇_𝑅(𝑥,𝑧), that is, 𝜇_𝑅(𝑥,𝑧)≤1−𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑥,𝑦)). Thus, 𝛼₂(𝑥,𝑦,𝑧)≤𝜇_𝑅(𝑥,𝑧)≤min(𝐼_𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑥,𝑦)),1−𝐽_𝑆(𝜈_𝑅(𝑧,𝑦),𝜈_𝑅(𝑥,𝑦))).
Otherwise, the third inequality of (iii) gives 𝜇_𝑅(𝑥,𝑧)≤𝐼_𝑇(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑦,𝑧)). The third inequality of (i) gives 𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑧))≤𝜈_𝑅(𝑥,𝑧)≤1−𝜇_𝑅(𝑥,𝑧). Thus, 𝜇_𝑅(𝑥,𝑧)≤min(𝐼_𝑇(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑦,𝑧)),1−𝐽_𝑆(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑧))). Hence the result.
With the two first inequalities of (iv) of (A.10), the second inequality of (ii) of (A.10), the third inequality of (iv) of (A.10), and the third inequality of (ii) of (A.10), we show analogously that 𝜇_𝑅(𝑧,𝑥)∈[𝛼₁(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)].
With the two first inequalities of (i) of (A.10) and the third inequality of (i) of (A.10), we show analogously that 𝜈_𝑅(𝑥,𝑧)∈[𝛽₄(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧)].
With the two first inequalities of (ii) of (A.10) and the third inequality of (ii) of (A.10), we show analogously that 𝜈_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧)]. Hence the result.
(ii)⇒(i): Assume (3.16), and let us show that 𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧}. That is to show the twelve inequalities of (A.10).
The assertion 𝜇_𝑅(𝑧,𝑥)∈[𝛼₁(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)] implies 𝛼₁(𝑥,𝑦,𝑧)=𝑇(𝜇_𝑅(𝑧,𝑦),𝜇_𝑅(𝑦,𝑥))≤𝜇_𝑅(𝑧,𝑥), 𝜇_𝑅(𝑧,𝑥)≤𝐼_𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑦,𝑥)) and 𝜇_𝑅(𝑧,𝑥)≤𝐼_𝑇(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑧,𝑦)). Thus, the second result of Proposition 2.1 and the last inequalities imply 𝜇_𝑅(𝑧,𝑦)≥𝑇(𝜇_𝑅(𝑧,𝑥),𝜇_𝑅(𝑥,𝑦)) and 𝜇_𝑅(𝑦,𝑥)≥𝑇(𝜇_𝑅(𝑦,𝑧),𝜇_𝑅(𝑧,𝑥)). Hence (iv) of (A.10).
Analogously, the assertion 𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₄(𝑥,𝑦,𝑧)] and the second result of Proposition 2.1 imply (iii) of (A.10); the assertion 𝜈_𝑅(𝑥,𝑧)∈[𝛽₄(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧)] and the second result of Proposition 2.1 imply (i) of (A.10); the assertion 𝜈_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧)], and the second result of Proposition 2.1 imply (ii) of (A.10).

Proof of Lemma 4.2. Let 𝑥,𝑦∈𝑋.
(1) Let us show that 𝜈_𝐼(𝑥,𝑦)=max(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥)).
Since 𝑅=𝐼∪_𝒥𝑃 and 𝐼 is symmetric, thus 𝜈_𝑅(𝑥,𝑦)=𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦)) and 𝜈_𝑅(𝑦,𝑥)=𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑦,𝑥)). We distinguish two cases.
(i) Suppose that 𝜇_𝑃(𝑥,𝑦)>0 or (𝜇_𝑃(𝑥,𝑦)=0 and 𝜈_𝑃(𝑥,𝑦)<1).
The perfect antisymmetric of 𝑃 implies 𝜈_𝑃(𝑦,𝑥)=1. Since 𝑇 is a t-norm, 𝜈_𝑅(𝑦,𝑥)=𝑇(𝜈_𝐼(𝑥,𝑦),1)=𝜈_𝐼(𝑥,𝑦). This last equality and (i) of (2.3) imply 𝜈_𝑅(𝑥,𝑦)=𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦))=𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑃(𝑥,𝑦))≤𝜈_𝑅(𝑦,𝑥)=𝜈_𝐼(𝑥,𝑦). Thus, 𝜈_𝐼(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥)=max[𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥)].
(ii) Suppose that 𝜇_𝑃(𝑥,𝑦)=0 and 𝜈_𝑃(𝑥,𝑦)=1.
Since 𝑇 is a t-norm and 𝐼 is symmetric, 𝜈_𝑅(𝑥,𝑦)=𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦))=𝑇(𝜈_𝐼(𝑥,𝑦),1)=𝜈_𝐼(𝑥,𝑦). This equality and (i) of (2.3) imply 𝜈_𝑅(𝑦,𝑥)=𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑦,𝑥))≤𝜈_𝐼(𝑥,𝑦)=𝜈_𝑅(𝑥,𝑦). Thus, 𝜈_𝐼(𝑥,𝑦)=𝜈_𝑅(𝑥,𝑦)=max[𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥)].
The proof of the equality 𝜇_𝐼(𝑥,𝑦)=min(𝜇_𝑅(𝑥,𝑦),𝜇_𝑅(𝑦,𝑥)) is similar to the previous one.
(2) Let us show that 𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅(𝑦,𝑥)⇔𝜈_𝑃(𝑥,𝑦)=1.
Since 𝑅=𝐼∪_𝒥𝑃, thus 𝜈_𝑅(𝑥,𝑦)=𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦)) and 𝜈_𝑅(𝑦,𝑥)=𝑇(𝜈_𝐼(𝑦,𝑥),𝜈_𝑃(𝑦,𝑥)).
(⇒): Assume to the contrary that 𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅(𝑦,𝑥) and 𝜈_𝑃(𝑥,𝑦)<1.
Since 𝑃 is perfect antisymmetric and 𝜈_𝑃(𝑥,𝑦)<1, we have 𝜈_𝑃(𝑦,𝑥)=1. Thus 𝜈_𝑅(𝑦,𝑥)=𝑇(𝜈_𝐼(𝑦,𝑥),𝜈_𝑃(𝑦,𝑥))=𝜈_𝐼(𝑦,𝑥). Since 𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅(𝑦,𝑥), the previous result gives 𝜈_𝐼(𝑥,𝑦)=max(𝜈_𝑅(𝑥,𝑦),𝜈_𝑅(𝑦,𝑥))=𝜈_𝑅(𝑥,𝑦). Since 𝐼 is symmetric, the two previous equalities imply 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥). The 𝜋-symmetric of 𝑅 and the previous equality imply 𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥). Thus, since 𝑃 is simple, the two previous equalities imply 𝜈_𝑃(𝑥,𝑦)=𝜈_𝑃(𝑦,𝑥)=1, which contradicts the hypothesis 𝜈_𝑃(𝑥,𝑦)<1. Finally, 𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅(𝑦,𝑥)⇒𝜈_𝑃(𝑥,𝑦)=1.
(⇐): The proof of converse is obvious.
The proof of the equivalence 𝜇_𝑅(𝑥,𝑦)≤𝜇_𝑅(𝑦,𝑥)⇔𝜇_𝑃(𝑥,𝑦)=0 is analogous to the previous one.

Proof of Lemma 4.4. Suppose that 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥).
(i) Consider the function 𝑓 defined over [0,1] by 𝑓(𝑡)=𝑆(𝑡,𝜇_𝑅(𝑦,𝑥)), thus 𝑓(0)=𝜇_𝑅(𝑦,𝑥) and 𝑓(1)=1. 𝑓 is continuous and monotone because 𝑆 is continuous and monotone. Then 𝑓 takes all values between 𝜇_𝑅(𝑦,𝑥) and 1. In particular, 𝑓 has the value 𝜇_𝑅(𝑥,𝑦) for 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥). Thus (4.9)(𝐸₁) has at least a solution.
By considering the function 𝑔 defined over [0,1] by 𝑔(𝑡)=𝑇(𝑡,𝜈_𝑅(𝑦,𝑥)), we analogously show that (4.9)(𝐸₂) has at least a solution.
(ii) Let us show that each solution of (4.9)(𝐸₁) is strictly positive. Let 𝑡₁ be a solution of (4.9)(𝐸₁).
Assume to the contrary that 𝑡₁=0. Thus, 𝜇_𝑅(𝑥,𝑦)=𝑆(𝜇_𝑅(𝑦,𝑥),𝑡₁)=𝑆(𝜇_𝑅(𝑦,𝑥),0)=𝜇_𝑅(𝑦,𝑥), that is, 𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) which contradicts 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥). And we have 𝑡₁>0.
Let us show that each solution of (4.9)(𝐸₂) is least than 1. Let 𝑡₂ be a solution of (4.9)(𝐸₂). Assume to the contrary that 𝑡₂=1. Thus, 𝜈_𝑅(𝑥,𝑦)=𝑇(𝜈_𝑅(𝑦,𝑥),𝑡₂)=𝑇(𝜈_𝑅(𝑦,𝑥),1)=𝜈_𝑅(𝑦,𝑥), that is, 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥) which contradicts 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥). And we have 𝑡₂<1.
(iii) Assume that 𝒥 satisfies condition 𝐺. We can remark by (2) of Proposition 2.1 that 𝑆(𝜇_𝑅(𝑦,𝑥),𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)))=𝜇_𝑅(𝑥,𝑦) and 𝑇(𝜈_𝑅(𝑦,𝑥),𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦)))=𝜈_𝑅(𝑥,𝑦). Since 𝑅 is 𝜋-symmetric, 𝜇_𝑅(𝑥,𝑦)+𝜈_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥)+𝜈_𝑅(𝑦,𝑥). Because 𝒥 satisfies condition 𝐺, then 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥), and the previous equality imply 𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦))+𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦))≤1. Hence (𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)),𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦))) is a solution of (4.8).(iv) Assume that 𝒥 satisfies condition 𝐺, and let us show that 𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)) is the lowest solution for (4.9)(𝐸₁).
Consider 𝑡₁ another solution of (4.9)(𝐸₁). Thus we have 𝑆(𝑡₁,𝜇_𝑅(𝑦,𝑥))=𝜇_𝑅(𝑥,𝑦) which implies 𝑡₁∈{𝑡∈[0,1],𝑆(𝑡,𝜇_𝑅(𝑦,𝑥))≥𝜇_𝑅(𝑥,𝑦)}. Since 𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦))=min{𝑡∈[0,1],𝑆(𝑡,𝜇_𝑅(𝑦,𝑥))≥𝜇_𝑅(𝑥,𝑦)}, we deduce that 𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦))≤𝑡₁.
We analogously show that 𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦)) is the upper solution for (4.9)(𝐸₂).
(v) If 𝒥=(max,min), we easily show that (𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)),𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦))) is the unique solution of (4.8).
Suppose that 𝒥=(𝑆,𝑇) is a strict t-conorm on 𝐿^∗ satisfying condition 𝐺.
The previous functions 𝑓 and 𝑔 defined in (i) are, respectively, the bijections from [0,1] to [𝜇_𝑅(𝑦,𝑥),1] and from [0,1] to [0,𝜈_𝑅(𝑦,𝑥)]. We easily show that (𝑡₁,𝑡₂)=(𝐽_𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑅(𝑥,𝑦)),𝐼_𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑅(𝑥,𝑦))) is the unique solution of (4.8).

Proof of Theorem 4.5. (1)⇒(2): Lemma 4.2 implies (i).
Let us show (ii). ∀𝑥,𝑦∈𝑋, suppose that 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥).
(iii) of Lemma 4.4 implies that (4.8) has at least one solution. Set (𝑐_𝑥𝑦,𝑔_𝑥𝑦)∈𝐿^∗ one of these solutions. Thus 𝑐_𝑥𝑦 is a solution of (4.9)(𝐸₁), and 𝑔_𝑥𝑦 is a solution of (4.9)(𝐸₂). With (ii) of Lemma 4.4, we have 𝑐_𝑥𝑦>0 and 𝑔_𝑥𝑦<1. Since the equality 𝑅=𝐼∪_𝒥𝑃 is equivalent to equation (𝜇_𝑅(𝑥,𝑦),𝜈_𝑅(𝑥,𝑦))=𝒥[(𝜇_𝑅(𝑦,𝑥),𝜈_𝑅(𝑦,𝑥)),(𝜇_𝑃(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦))], hence 𝜇_𝑃(𝑥,𝑦)=𝑐_𝑥𝑦 and 𝜈_𝑃(𝑥,𝑦)=𝑔_𝑥𝑦.
For the case where 𝜇_𝑅(𝑥,𝑦)≤𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅(𝑦,𝑥), we easily show that Lemma 4.2 implies that 𝜇_𝑃(𝑥,𝑦)=0 and 𝜈_𝑃(𝑥,𝑦)=1.
(2)⇒(1): Let 𝑥,𝑦∈𝑋.
(i) implies that 𝜇_𝐼(𝑥,𝑦)=𝜇_𝐼(𝑦,𝑥), and 𝜈_𝐼(𝑥,𝑦)=𝜈_𝐼(𝑦,𝑥) which show that 𝐼 is symmetric.
Let us show that 𝑅=𝐼∪_𝒥𝑃, that is, 𝑆(𝜇_𝐼(𝑥,𝑦),𝜇_𝑃(𝑥,𝑦))=𝜇_𝑅(𝑥,𝑦) and 𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦))=𝜈_𝑅(𝑥,𝑦). Since 𝑅 is 𝜋-symmetric, we distinguish two cases.
(a)If 𝜇_𝑅(𝑥,𝑦)≤𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑥,𝑦)≥𝜈_𝑅(𝑦,𝑥), thus (i) implies 𝜇_𝐼(𝑥,𝑦)=𝜇_𝑅(𝑥,𝑦), and 𝜈_𝐼(𝑥,𝑦)=𝜈_𝑅(𝑥,𝑦), and the definition of 𝑃 gives 𝜇_𝑃(𝑥,𝑦)=0 and 𝜈_𝑃(𝑥,𝑦)=1. We have 𝑆(𝜇_𝐼(𝑥,𝑦),𝜇_𝑃(𝑥,𝑦))=𝑆(𝜇_𝑅(𝑥,𝑦),0)=𝜇_𝑅(𝑥,𝑦) and 𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦))=𝑇(𝜈_𝑅(𝑥,𝑦),1)=𝜈_𝑅(𝑥,𝑦).(b)If 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥), thus (ii) implies that 𝜇_𝑃(𝑥,𝑦) is a solution of (4.9)(𝐸₁), and 𝜈_𝑃(𝑥,𝑦) is a solution of (4.9)(𝐸₂); we have 𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑃(𝑥,𝑦))=𝜇_𝑅(𝑥,𝑦) and 𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑃(𝑥,𝑦))=𝜈_𝑅(𝑥,𝑦). Furthermore, as 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥), and 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥), (i) implies 𝜇_𝐼(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) and 𝜈_𝐼(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥). Then 𝑆(𝜇_𝐼(𝑥,𝑦),𝜇_𝑃(𝑥,𝑦))=𝑆(𝜇_𝑅(𝑦,𝑥),𝜇_𝑃(𝑥,𝑦))=𝜇_𝑅(𝑥,𝑦) and 𝑇(𝜈_𝐼(𝑥,𝑦),𝜈_𝑃(𝑥,𝑦))=𝑇(𝜈_𝑅(𝑦,𝑥),𝜈_𝑃(𝑥,𝑦))=𝜈_𝑅(𝑥,𝑦).
It is easy to show that P is simple and 𝑃 is perfect antisymmetric.

Proof of Corollary 4.7. The proof of this corollary is deduced from Theorem 4.5 and the last result of Lemma 4.4.

Proof of Corollary 4.8. The proof is deduced from Lemma 4.2, Theorem 4.5, and the last result of Proposition 3.9.

Proof of Proposition 5.5. Let 𝑥,𝑦,𝑧∈𝑋. Since 𝑅 is strongly complete, Corollary 3.11 implies 𝛼₃(𝑥,𝑦,𝑧)<𝛼₂(𝑥,𝑦,𝑧) and 𝛽₂(𝑥,𝑦,𝑧)<𝛽₃(𝑥,𝑦,𝑧).

Proof of Theorem 5.6. As 𝑅 is 𝜋-symmetric, (i) of (5.3) and (ii) of (5.3) are equivalent. Therefore, to show (5.3) means to show (i) of (5.3) or (ii) of (5.3).
(⇒): We distinguish two cases.
(1)Suppose that 𝑅 satisfies condition 𝐶^𝜇𝑅₁, and let us show that 𝑃 is pos-transitive. We show (i) of (5.3). Let 𝑥,𝑦,𝑧∈𝑋 such that 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥) and 𝜇_𝑅(𝑦,𝑧)>𝜇_𝑅(𝑧,𝑦). Let us show that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥). Since 𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧}, Lemma 3.13 implies 𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₄𝜇(𝑥,𝑦,𝑧)],_𝑅(𝑧,𝑥)∈[𝛼₁(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)]. (A.11) We distinguish two cases.(a)Suppose that 𝛼₃(𝑥,𝑦,𝑧)<𝛼₂(𝑥,𝑦,𝑧) or 𝜇_𝑅(𝑧,𝑥)<𝛼₂(𝑥,𝑦,𝑧)<𝛼₃(𝑥,𝑦,𝑧) or 𝛼₂(𝑥,𝑦,𝑧)<𝛼₃(𝑥,𝑦,𝑧)<𝜇_𝑅(𝑥,𝑧); thus (A.11) implies 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥).(b)Suppose that 𝛼₂(𝑥,𝑦,𝑧)≤𝛼₃(𝑥,𝑦,𝑧) and 𝜇_𝑅(𝑧,𝑥),𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)]; thus, since 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥), 𝜇_𝑅(𝑦,𝑧)>𝜇_𝑅(𝑧,𝑦) and 𝑅 satisfies condition 𝐶^𝜇𝑅₁, we deduce that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥).(2)With Lemma 4.2, the proof is analogous to the previous one.
(⇐) Suppose that 𝑃 is pos-transitive, and let us show that 𝑅 satisfies 𝐶^𝜇𝑅₁ or 𝐶^𝜈𝑅₁.
Since 𝑃 is pos-transitive, we have (5.3), that is, (i) of (5.3) or (ii) of (5.3). We distinguish two cases.
(a)Suppose (ii) of (5.3), and let us show that 𝑅 satisfies 𝐶^𝜈𝑅₁. Suppose that 𝑅 verifies 𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦),_𝑅(𝑧,𝑦)>𝜈_𝑅(𝑦,𝑧), (A.12) and 𝜈_𝑅(𝑥,𝑧),𝜈_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧)]. Let us show that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥). Since 𝑅 satisfies 𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦),_𝑅(𝑧,𝑦)>𝜈_𝑅(𝑦,𝑧), (A.13) the two previous inequalities and (ii) of (5.3) imply 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥).(b)With (i) of (5.3), we show analogously that 𝑅 satisfies 𝐶^𝜇𝑅₁.

Proof of Corollary 5.7. Since 𝑅 is strongly complete, then Proposition 5.5 implies that 𝑅 satisfies condition 𝐶^𝜇𝑅₁. Hence, Theorem 5.6 implies that 𝑃 is pos-transitive.

Proof of Lemma 5.8. To establish (5.15) is equivalent to establish (5.16).
We remark that as 𝑅 is 𝜋-symmetric, (i) of (5.16) and (ii) of (5.16) are equivalent.
(1)⇒(2): We distinguish two cases.
(1)Suppose that 𝑅 satisfies condition 𝐶^𝜇𝑅₂, and let us show (5.16). We show (i) of (5.16). Let 𝑥,𝑦,𝑧∈𝑋 such that 𝜇_𝑅(𝑦,𝑥)=𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)<𝜇_𝑅𝜇(𝑦,𝑧)or^𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜇_𝑅(𝑦,𝑧). (A.14) Let us show that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥). Since 𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧}, Lemma 3.13 implies (A.11). We distinguish two cases.(a)Suppose that 𝛼₃(𝑥,𝑦,𝑧)<𝛼₂(𝑥,𝑦,𝑧) or 𝜇_𝑅(𝑧,𝑥)<𝛼₂(𝑥,𝑦,𝑧)<𝛼₃(𝑥,𝑦,𝑧) or 𝛼₂(𝑥,𝑦,𝑧)<𝛼₃(𝑥,𝑦,𝑧)<𝜇_𝑅(𝑥,𝑧); thus (A.11) implies 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥).(b)Suppose that 𝛼₂(𝑥,𝑦,𝑧)≤𝛼₃(𝑥,𝑦,𝑧) and 𝜇_𝑅(𝑧,𝑥),𝜇_𝑅(𝑥,𝑧)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)]. With (A.14), the condition 𝐶^𝜇𝑅₂ implies that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥).(2)Suppose that 𝑅 satisfies condition 𝐶^𝜈𝑅₂, and let us show (5.16). We show (ii) of (5.16). Let 𝑥,𝑦,𝑧∈𝑋 such that 𝜈_𝑅(𝑦,𝑥)=𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)>𝜈_𝑅𝜈(𝑦,𝑧)or^𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦)_𝑅(𝑧,𝑦)=𝜈_𝑅(𝑦,𝑧). (A.15) Let us show that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥). Since 𝑅 is 𝒯-transitive on {𝑥,𝑦,𝑧}, Lemma 4.2 implies 𝜈_𝑅(𝑥,𝑧)∈[𝛽₄(𝑥,𝑦,𝑧),𝛽₂𝜈(𝑥,𝑦,𝑧)],_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₁(𝑥,𝑦,𝑧)]. (A.16) We distinguish two cases.(a)Suppose that 𝛽₃(𝑥,𝑦,𝑧)>𝛽₂(𝑥,𝑦,𝑧) or 𝜈_𝑅(𝑧,𝑥)>𝛽₂(𝑥,𝑦,𝑧)>𝛽₃(𝑥,𝑦,𝑧) or 𝛽₂(𝑥,𝑦,𝑧)>𝛽₃(𝑥,𝑦,𝑧)>𝜈_𝑅(𝑥,𝑧); thus (A.16) implies that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥).(b)Suppose that 𝛽₂(𝑥,𝑦,𝑧)≥𝛽₃(𝑥,𝑦,𝑧) and 𝜈_𝑅(𝑧,𝑥),𝜈_𝑅(𝑥,𝑧)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧)]. With (A.15), the condition 𝐶^𝜈𝑅₂ implies that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥).
(2)⇒(1) Suppose (5.16), and let us show that 𝑅 satisfies 𝐶^𝜇𝑅₂ or 𝐶^𝜈𝑅₂. We distinguish two cases.
(a)Suppose (i) of (5.16), and let us show that 𝑅 satisfies 𝐶^𝜇𝑅₂. Suppose (A.14), and 𝜇_𝑅(𝑥,𝑧),𝜇_𝑅(𝑧,𝑥)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)]. Let us show that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥). With (A.14), (i) of (5.16) implies 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥).(b)With (ii) of (5.16), we show analogously that 𝑅 satisfies 𝐶^𝜈𝑅₁.

Proof of Theorem 5.9. As 𝑅 is 𝜋-symmetric, Remark 5.2 implies that (5.4) and (5.5) are equivalent. Therefore, to show (5.4) means to show (5.4) or (5.5).
(⇒): We distinguish two cases.
(1)Suppose that 𝑅 satisfies condition 𝐶^𝜇𝑅₁ and 𝐶^𝜇𝑅₂, and let us show that 𝑃 is negative transitive. We show (5.4). Let 𝑥,𝑦,𝑧∈𝑋. We distinguish three cases.(a)Suppose 𝜇_𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦),_𝑅(𝑧,𝑦)<𝜇_𝑅(𝑦,𝑧), (A.17) and let us show that 𝜇_𝑅(𝑥,𝑧)≥𝜇_𝑅(𝑧,𝑥). Since 𝑅 satisfies condition 𝐶^𝜇𝑅₁, Theorem 5.6 implies that 𝑃 is pos-transitive. Thus, 𝜇_𝑅(𝑦,𝑥)<𝜇_𝑅𝜇(𝑥,𝑦),_𝑅(𝑧,𝑦)<𝜇_𝑅(𝑦,𝑧), (A.18) and the pos-transitivity of 𝑃 imply that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥). Hence 𝜇_𝑅(𝑥,𝑧)≥𝜇_𝑅(𝑧,𝑥).(b)Suppose (A.14) and let us show that 𝜇_𝑅(𝑥,𝑧)≥𝜇_𝑅(𝑧,𝑥). Since 𝑅 satisfies condition 𝐶^𝜇𝑅₂ and (A.14), then Lemma 5.8 implies that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥). Hence 𝜇_𝑅(𝑥,𝑧)≥𝜇_𝑅(𝑧,𝑥).(c)Suppose that 𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) and 𝜇_𝑅(𝑦,𝑧)=𝜇_𝑅(𝑧,𝑦), and let us show that 𝜇_𝑅(𝑥,𝑧)≥𝜇_𝑅(𝑧,𝑥). Assume to the contrary that 𝜇_𝑅(𝑥,𝑧)<𝜇_𝑅(𝑧,𝑥). Thus, 𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) and 𝜇_𝑅(𝑥,𝑧)<𝜇_𝑅(𝑧,𝑥), that is, 𝜇_𝑅(𝑥,𝑧)<𝜇_𝑅𝜇(𝑧,𝑥),_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥). (A.19) Since 𝑅 satisfies condition 𝐶^𝜇𝑅₂ and 𝜇_𝑅(𝑥,𝑧)<𝜇_𝑅𝜇(𝑧,𝑥),_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥), (A.20) then Lemma 5.8 implies that 𝜇_𝑅(𝑧,𝑦)>𝜇_𝑅(𝑦,𝑧) which contradicts the hypothesis 𝜇_𝑅(𝑦,𝑧)=𝜇_𝑅(𝑧,𝑦).(2)Suppose that 𝑅 satisfies condition 𝐶^𝜈𝑅₁ and 𝐶^𝜈𝑅₂, and let us show that 𝑃 is negative transitive. We show (5.5). Let 𝑥,𝑦,𝑧∈𝑋. We distinguish three cases.(a)Suppose that 𝑅 verifies 𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦),_𝑅(𝑧,𝑦)>𝜈_𝑅(𝑦,𝑧), (A.21) and let us show that 𝜈_𝑅(𝑥,𝑧)≤𝜈_𝑅(𝑧,𝑥). Since 𝑅 satisfies condition 𝐶^𝜈𝑅₁, Theorem 5.6 implies that 𝑃 is pos-transitive. Thus, 𝜈_𝑅(𝑦,𝑥)>𝜈_𝑅𝜈(𝑥,𝑦),_𝑅(𝑧,𝑦)>𝜈_𝑅(𝑦,𝑧), (A.22) and the pos-transitivity of 𝑃 imply that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥). Hence 𝜈_𝑅(𝑥,𝑧)≤𝜈_𝑅(𝑧,𝑥).(b)Suppose (A.15), and let us show that 𝜈_𝑅(𝑥,𝑧)≤𝜈_𝑅(𝑧,𝑥). Since 𝑅 satisfies condition 𝐶^𝜈𝑅₂ and (A.15), then Lemma 5.8 implies that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥). Hence 𝜈_𝑅(𝑥,𝑧)≤𝜈_𝑅(𝑧,𝑥).(c)Suppose that 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑦,𝑧)=𝜈_𝑅(𝑧,𝑦); and let us show that 𝜈_𝑅(𝑥,𝑧)≤𝜈_𝑅(𝑧,𝑥). Assume to the contrary that 𝜈_𝑅(𝑥,𝑧)>𝜈_𝑅(𝑧,𝑥). Thus, 𝜈_𝑅(𝑥,𝑧)>𝜈_𝑅𝜈(𝑧,𝑥),_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥). (A.23) Since 𝑅 satisfies condition 𝐶^𝜈𝑅₂ and 𝜈_𝑅(𝑥,𝑧)>𝜈_𝑅𝜈(𝑧,𝑥),_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥), (A.24) then Lemma 5.8 implies 𝜈_𝑅(𝑧,𝑦)<𝜈_𝑅(𝑦,𝑧) which contradicts hypothesis 𝜈_𝑅(𝑦,𝑧)=𝜈_𝑅(𝑧,𝑦).
(⇐) Suppose that 𝑃 is negative transitive, and let us show that 𝑅 satisfies (𝐶^𝜇𝑅₁ and 𝐶^𝜇𝑅₂) or (𝐶^𝜈𝑅₁ and 𝐶^𝜈𝑅₂).
Since 𝑃 is negative transitive, we have (5.4) and (5.5), that is, (5.4) or (5.5) as 𝑅 is 𝜋-symmetric. We distinguish two cases.
(i)Suppose (5.4), and let us show that 𝑅 satisfies 𝐶^𝜇𝑅₁ and 𝐶^𝜇𝑅₂. Suppose that 𝜇_𝑅(𝑥,𝑧),𝜇_𝑅(𝑧,𝑥)∈[𝛼₂(𝑥,𝑦,𝑧),𝛼₃(𝑥,𝑦,𝑧)] and (3.9). Let us show that 𝜇_𝑅(𝑥,𝑧)>𝜇_𝑅(𝑧,𝑥). In fact, (3.9) and (5.4) imply 𝜇_𝑅(𝑥,𝑧)≥𝜇_𝑅(𝑧,𝑥). Assume to the contrary that 𝜇_𝑅(𝑥,𝑧)=𝜇_𝑅(𝑧,𝑥). With (3.9), we distinguish two cases.(a)If 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥) and 𝜇_𝑅(𝑦,𝑧)≥𝜇_𝑅(𝑧,𝑦); thus we have 𝜇_𝑅(𝑥,𝑧)=𝜇_𝑅(𝑧,𝑥) and 𝜇_𝑅(𝑦,𝑧)≥𝜇_𝑅(𝑧,𝑦). Hence the negative transitivity of 𝑃 implies 𝜇_𝑅(𝑦,𝑥)≥𝜇_𝑅(𝑥,𝑦) which contradicts the hypothesis 𝜇_𝑅(𝑥,𝑦)>𝜇_𝑅(𝑦,𝑥).(b)If 𝜇_𝑅(𝑥,𝑦)=𝜇_𝑅(𝑦,𝑥) and 𝜇_𝑅(𝑦,𝑧)>𝜇_𝑅(𝑧,𝑦); thus we have 𝜇_𝑅(𝑥,𝑧)=𝜇_𝑅(𝑧,𝑥) and 𝜇_𝑅(𝑦,𝑧)=𝜇_𝑅(𝑧,𝑦). Hence the negative transitivity of 𝑃 implies 𝜇_𝑅(𝑧,𝑦)≥𝜇_𝑅(𝑦,𝑧) which contradicts the hypothesis 𝜇_𝑅(𝑦,𝑧)>𝜇_𝑅(𝑧,𝑦).(ii)Suppose (5.5) and let us show that 𝑅 satisfies 𝐶^𝜈𝑅₁ and 𝐶^𝜈𝑅₂. Suppose (3.11) and 𝜈_𝑅(𝑥,𝑧),𝜈_𝑅(𝑧,𝑥)∈[𝛽₃(𝑥,𝑦,𝑧),𝛽₂(𝑥,𝑦,𝑧)]. Let us show that 𝜈_𝑅(𝑥,𝑧)<𝜈_𝑅(𝑧,𝑥). In fact, (3.11) and (5.5) imply 𝜈_𝑅(𝑥,𝑧)≤𝜈_𝑅(𝑧,𝑥). Assume to the contrary that 𝜈_𝑅(𝑥,𝑧)=𝜈_𝑅(𝑧,𝑥). With (3.11), we distinguish two cases.(a)If 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑦,𝑧)≤𝜈_𝑅(𝑧,𝑦); thus we have 𝜈_𝑅(𝑥,𝑧)=𝜈_𝑅(𝑧,𝑥) and 𝜈_𝑅(𝑦,𝑧)≤𝜈_𝑅(𝑧,𝑦). Hence the negative transitivity of 𝑃 implies 𝜈_𝑅(𝑦,𝑥)≤𝜈_𝑅(𝑥,𝑦) which contradicts the hypothesis 𝜈_𝑅(𝑥,𝑦)<𝜈_𝑅(𝑦,𝑥).(b)If 𝜈_𝑅(𝑥,𝑦)=𝜈_𝑅(𝑦,𝑥) and 𝜈_𝑅(𝑦,𝑧)<𝜈_𝑅(𝑧,𝑦); thus we have 𝜈_𝑅(𝑥,𝑧)=𝜈_𝑅(𝑧,𝑥) and 𝜈_𝑅(𝑦,𝑧)=𝜈_𝑅(𝑧,𝑦). Hence the negative transitivity of 𝑃 implies 𝜈_𝑅(𝑧,𝑦)≤𝜈_𝑅(𝑦,𝑧) which contradicts the hypothesis 𝜈_𝑅(𝑦,𝑧)<𝜈_𝑅(𝑧,𝑦).

Proof of Corollary 5.10. The proof is deduced from Proposition 5.5 and Theorem 5.9.

Acknowledgments

The authors are thankful to the members of Laboratoire MASS and those of CREM for their help and advice. The second author thanks French Governement and AUF (Agence Universitaire de la Francophonie). This work was achieved when he was student at Université de Caen Basse-Normandie France, under the research grants “Bourse de formation à la recherche de la Francophonie 2007-2008” and “Bourse de doctorat du SCAC-Yaoundé 2007-2008”.