New results on the measures of transitivity

Abstract

In this paper, we present some new results in the form of inequalities connecting transitivity of the given fuzzy preference relation with its consistent behaviour. The additive and multiplicative generators of t-norms and t-conorms play the key role in establishing these results.

Keywords

Fuzzy relation Triangular norm Triangular conorm Implication

1 Introduction

Decision making (DM) is an intellectual procedure which is used to select the best option(s) amongst several different options, it initiates when we have to do something but do not know what. Every individual faces DM situations in his/her daily life: common examples for these situations are shopping, to choose what to eat, and deciding whom or what to vote for in an election or referendum, and can be categorized in several different groups according to certain characteristics as the source(s) for the information and the preference representation formats that are used to solve the decision problem.

In DM, experts compare a finite set of alternatives and construct preference relations where consistency is an important issue to accept when data are provided by the experts and is associated with transitivity property. Though various versions of transitivity have been used in large number of especial applications of preference, but all these definitions have also been censured and the space for newer models of transitivity is still there. In [3], Beg and Ashraf proposed a new definition of degree or measure of a fuzzy relation for being fuzzy T-transitive called ϵ-transitive fuzzy relation. Later, the same notion was studied by Caiping in [8]. This new notion was based upon fuzzy set theoretic operators, in particular the fuzzy implication. In the former literature, different definitions of fuzzy implications have been developed by different authors. Two types of fuzzy implications were proposed by Trillas & Valverde in [16, 17] which were defined with the help of fuzzy negation, conjunction and disjunction. In [3], Beg and Ashraf obtained several novel results using these forms of implications in the definition of ϵ-transitivity.

The notion of transitivity and consistency of a fuzzy relation are synonymous in some sense. The major inconsistency raises in a crisp preference relation when according to some experts’ opinion: a is preferable to b and b is preferable to c but a is not preferable to c. So nothing could be concluded from the given relation as a structure of preference. In fuzzy preference relations, the similar phenomenon arises but in the form of degrees of preference. As far as FPRs are concerned, there is a list of consistency conditions required from degrees of preference in the form of transitivity conditions. A summarized list for such conditions has been given in literature, the detailed study could be found in [4 , 13].

Any of these conditions may be used to discuss the transitivity of a given preference relation, but all these conditions make a strict decision that the given relation is transitive or not. While in real life situations, we must be able to talk about degree of transitivity of a fuzzy relation i.e., it may be more or less transitive. This paper makes use of the degrees of transitivity and preference to discuss the mathematical results that reflect the effect of being more or less transitive for a fuzzy preference relation and consequently being more or less consistent. Several new inequalities have been obtained establishing relationships between the preference degrees and consistency degrees. To our knowledge this is the first attempt to work out the mathematical formulations representing the nature of preference data while given the degree of being transitive.

The rest of paper is organized as follows: In Section 2, we focus on some basic preliminaries used in this paper. In Section 3, we explored the situations in which the degree for being T-transitive is considered under the use of R-implication and S-implication with additive and multiplicative generators of triangular norm and triangular conorm. In Section 4, some situations for fuzzy preference relations being S-transitive are explored using R-implication and S-implication. Last section includes some conclusions.

2 Preliminaries

In 1965, Zadeh introduced the notion of fuzzy set theory by presenting his formative paper "Fuzzy Sets" [19], designated with a number between 0 and 1. This notion demonstrates that an object links more or less to the specific group we want to adjust it to; that was how the idea of defining the membership of an element to a set not on the Aristotelian pair {0, 1} any more but on the continuous interval [0, 1] was born. After the emergence of fuzzy set theory in 1965, the simple task of looking at relations was initiated by Lotfi A. Zadeh [20] himself in 1971. In that seminal paper, he introduced the concept of a fuzzy relation, defined the notion of similarity as a generalization of the notion of equivalence and presented the concept of fuzzy ordering.

Definition 2.1. Fuzzy Relation [20]: A fuzzy relation R on a universe X is a X × X ⟶ [0, 1] mapping. The degree of relationship between x and y is denoted by R _xy for all x, y ∈ X .

Definition 2.2. Fuzzy Preference Relation [15]: A preference relation R on a set X of alternatives is characterized by a membership function R : X × X → [0, 1], where every value r _ij represents the preference degree of alternative x _i over x _j: $r_{ij} = \frac{1}{2}$ indicates indifference between alternatives x _i and x _j (x _i ∼x _j); r _ij = 1 specifies that alternative x _i is completely preferred to x _j; $r_{ij} > \frac{1}{2}$ indicates that alternative x _i is preferred to x _j (x _i ≻x _j).

Definition 2.3. Triangular Norm and Triangular Conorm [13]: The triangular norm (t-norm) T and triangular conorm (t-conorm) S are the [0, 1] ² ⟶ [0, 1], increasing, associative and commutative mappings satisfying: T (1, x) = x and S (x, 0) = x for all x ∈ X.

Definition 2.4. Additive Generator of t-norm [13]: An additive generator f of a t-norm T is a strictly decreasing [0, 1] → [0, ∞) function which is also right continuous in 0 and satisfies f (1) =0, such that: $T (x, y) = f^{- 1} ((f (x) + f (y)) \land f (0)) for all x, y \in X,$ the t-norm T is strictly monotone if and only if f (0) =∞ and nilpotent if and only if f (0)< ∞.

Definition 2.5. Multiplicative Generator of t-norm [13]: A multiplicative generator g of a t-norm T is a strictly increasing [0, 1] → [0, 1] function which is right continuous in 0 and satisfies g (1) =1, such that: $T (x, y) = g^{- 1} ((g (x) \cdot g (y)) \lor g (0)) for all x, y \in [0, 1],$ where T is nilpotent if and only if g (0) >0 and is strict if and only if g (0) =0.

Definition 2.6. Additive Generator of t-conorm[14]: An additive generator s of a t-conorm S is a strictly increasing [0, 1] → [0, ∞) function which is left continuous in 1 and satisfies s (0) =0, such that: $S (x, y) = s^{- 1} [s (1) \land (s (x) + s (y))] for any x, y \in [0, 1],$ where S is nilpotent if s (1)> ∞ and is strict if s (1) = ∞.

Definition 2.7. Multiplicative Generator of t-conorm[14]: A multiplicative generator θ of a t-conorm S is a strictly decreasing [0, 1] → [0, 1] function which is left continuous in 1 and satisfies θ (0) =1, such that: $S (x, y) = θ^{- 1} (θ (x) \cdot θ (y) \lor θ (1)) for all x, y \in [0, 1],$ and S is nilpotent if θ (1) >0 and is strict if θ (1) =0 .

Definition 2.8. [12] Triplet: A triplet (T, S, N), where T is a t-norm, S is a t-conorm and N is a strict negation, is called a De Morgan triplet if for all x, y ∈ [0, 1]:T (x, y) = N ^-1 (S (N (x), N (y))); S (x, y) = N ^-1 (T (N (x), N (y))) .

Definition 2.9. Fuzzy implication[12]: A fuzzy implication is a binary operation on [0, 1] with order reversing first partial mappings and order preserving second partial mappings such that: I (0, 0) = I (0, 1) = I (1, 1) =1, and I (1, 0) =0 .

Some important implications are given as:

Kleene-Dienes: I _b (x, y) = max(1 - x, y);

Reichenbach: I _r (x, y) =1 - x + xy;

Łukasiewicz: I _L (x, y) = min(1 - x + y, 1);

Gödel: $I_{g} (x, y) = {\begin{matrix} [c] c 1, if x \leq y \\ y, if x > y \end{matrix}$ .

Continuing with implications, Trillas & Valverde proposed in [16, 17] two types of fuzzy implications defined with the help of fuzzy negation, conjunction and disjunction.

Definition 2.10. R-implication: An R-implication generated from a t-norm T is defined by: $I (x, y) = sup {t \in [0, 1] ∣ T (x, t) \leq y}$ for all x, y ∈ [0, 1]. If T is a left-continuous, then above implication can be rewritten as: $I (x, y) = \max {t \in [0, 1] ∣ T (x, t) \leq y}$ for all x, y ∈ [0, 1]. Some of the R-implications based on different t-norms are I _g (x, y), I _L (x, y) defined above.

Remark 2.1. If T is a continuous Archimedean t-norm and f : [0, 1] → [0, ∞] is an additive generator of T, then the R-implication associated with T can be obtained for all x, y ∈ X by [14]: $I_{T} (x, y) = f^{- 1} (max (f (y) - f (x), 0))$

Remark 2.2. If T is continuous Archimedean t-norm and g is a multiplicative generator of T, then the R-implication associated with T can be obtained for all x, y ∈ [0, 1] by [14]: $I_{T} (x, y) = g^{- 1} (min (\frac{g (y)}{g (x)}, 1)) .$

Definition 2.11. S-implication: Let S be a t-conorm and N be a fuzzy negation. An S-implication (also known as (S, N)-implication) is defined as: $I (x, y) = S (N (x), y) for all x, y \in [0, 1] .$ Some of the S-implications based on different t-conorms are I _b (x, y), I _r (x, y) defined above.

Remark 2.3. If s is an additive generator of an Archimedean conorm S, then its S-implication can be represented for all x, y ∈ [0, 1] by [14]: $I_{S} (x, y) = s^{- 1} (min (s (N (x)) + s (y), s (1))) .$

Remark 2.4. If g is the multiplicative generator of Archimedean t-conorm S, then its S-implication can be represented for all x, y ∈ [0, 1] as [14]: $I_{S} (x, y) = θ^{- 1} [max {θ (N (x)) \cdot θ (y), θ (1)}] .$

Definition 2.12. Fuzzy Ternary Relation and ϵ-Fuzzy Transitive Relation [3]: Let R be a fuzzy relation on X. The fuzzy set of transitivity tr ^I,T (R) is a fuzzy ternary relation on X for all x, y, z ∈ X and is defined as: ${tr}^{I, T} (R_{xyz}) = I (T (R_{xy}, R_{yz}), R_{xz}) .$ (1) The transitivity function so defined assigns a degree of transitivity to the relation at each point of X × X × X. For a given fuzzy relation R on a universe X, a t-norm T and any fuzzy implication I, the measure of transitivity of R for all x, y, z ∈ X is given by: ${Tr}^{I, T} (R) = inf_{x, y, z \in X} (I (T (R_{xy}, R_{yz}), R_{xz}))$ (2) if Tr (R) = ϵ, then the relation R is called ϵ-fuzzy transitive, R is non-transitive if ϵ = 0, weak fuzzy transitive if ϵ < 0.5 and strong fuzzy transitive if ϵ ≥ 0.5 .

Definition 2.13. T -transitivity[18]: A mapping R : X × X → [0, 1] is said to be T-transitive if ∀ x, y, z ∈ X it holdsT (R _xy, R _yz) ≤ R _xz .

Definition 2.14. S -transitivity[6]: A mapping R : X × X → [0, 1] is called S-transitive such that for all x, y, z ∈ X it holdsS (R _xy, R _yz) ≥ R _xz .

3 Measures of T-transitivity

In this section, we shall focus our attention on exploring the relationship between the degree of being T-transitive and consistent behavior of the fuzzy preference relation. This will be done in two ways: first we will explore the situation in which the degree for being T-transitive is considered while an R-implication is used to calculate this degree then in the next subsection we shall investigate the results using an S-implication in the same definition. To dispense with all possible situations we shall work for both; the additive generators and the multiplicative generators.

3.1 The inequalities related to R-implications

In [3], Beg and Ashraf proposed a new definition of degree or measure of a fuzzy relation for being fuzzy T-transitive called ϵ-transitive relation. This new notion was based upon fuzzy set theoretic operations, in particular the fuzzy implication. In the former literature, different definitions of fuzzy implications have been developed by different authors [1 , 17]. Beg and Ashraf obtained several novel results using these forms of implications in the definition of ϵ-transitivity.

Theorem 3.1. If an R-implication I associated with a continuous Archimedean t-norm T with f as its additive generator, is used in the definition of fuzzy transitivity of an ϵ-fuzzy transitive relation R, then for all x, y, z ∈ X there exists c ∈ (R _xz, T (R _xy, R _yz)) such that

$either T (R_{xy}, R_{yz}) \leq R_{xz} or T (R_{xy}, R_{yz}) \leq R_{xz} + \frac{- f (ϵ)}{f (c)} .$

Proof. Given R is an ϵ-fuzzy transitive relation so,

$inf_{x, y, z \in X} I (T (R_{xy}, R_{yz}), R_{xz}) = ϵ$ for all x, y, z ∈ X,

it implies that I (T (R _xy, R _yz), R _xz) ≥ ϵ > 0 for all x, y, z ∈ X .

Using representation of I (Remark 2.1), we get for all x, y, z ∈ X $f^{- 1} (max (f (R_{xz}) - f (T (R_{xy}, R_{yz})), 0)) \geq ϵ .$ Applying f on both sides with strictly decreasing nature $max (f (R_{xz}) - f (T (R_{xy}, R_{yz})), 0) \leq f (ϵ),$ it further implies that $f (R_{xz}) - f (T (R_{xy}, R_{yz})) \leq f (ϵ) and 0 \leq f (ϵ)$ (3) for all x, y, z ∈ X. Here arise two cases:

(a) T (R _xy, R _yz) ≤ R _xz i.e., the given relation is T-transitive. As proved in [3], the T-transitive relations are 1-fuzzy transitive. Hence (4) takes the form f (R _xz) - f (T (R _xy, R _yz)) ≤0 . Which trivially gives T (R _xy, R _yz) ≤ R _xz, i.e., the assumption.

(b) R _xz < T (R _xy, R _yz) i.e., f (R _xz) > f (T (R _xy, R _yz)) due to decreasing nature of f . So (4) takes the form f (T (R _xy, R _yz)) - f (R _xz) ≥ - f (ϵ) . Due to Lagrange’s mean value theorem and properties of f, there exists a c ∈] R _xz, T (R _xy, R _yz) [such that

f (c) (T (R _xy, R _yz) - R _xz) ≥ - f (ϵ) .

Since f (c) <0, due to strictly decreasing nature of f, we get $T (R_{xy}, R_{yz}) - R_{xz} \leq \frac{- f (ϵ)}{f (c)}$ ,

hence $T (R_{xy}, R_{yz}) \leq R_{xz} + \frac{- f (ϵ)}{f (c)} for all x, y, z \in X .$ (4) Combining (4) with the assumption for the case (a), we finally get for all x, y, z ∈ X $\begin{matrix} either T (R_{xy}, R_{yz}) & \leq R_{xz} or other wise \\ T (R_{xy}, R_{yz}) & \leq R_{xz} + \frac{- f (ϵ)}{f (c)} . \end{matrix}$ This gives us a relationship which help us determine the least value of R _xz given the degree of consistency of the given relation R _xy, and R _yz. Since f is a monotonically decreasing function f (ϵ) decreases with increase in ϵ. Moreover, $f ´ (c)$ is a negative quantity. These facts clearly indicate that as ϵ increases, T (R _xy, R _yz) approaches R _xz i.e. the relation moves towards T-transitivity.

Corollary 3.1. If the requirements of Lagrange’s Mean Value Theorem could not be met, then the inequality (4) may be reworked to get another view of the transitivity scenario. For an ϵ-equivalence relation R, if an R-implication I is used in the definition of fuzzy transitivity related to an Archimedean t-norm T, then for all x, y, z ∈ X f (R _xz) - f (R _xy) - f (R _yz) ≤ f (ϵ)

and f (R _xz) ≤ f (0) + f (ϵ),

where f is an additive generator of T. From (4b) it follows that f (R _xz) - f (T (R _xy, R _yz)) ≤ f (ϵ) and 0 ≤ f (ϵ) . Using the fact that f is an additive generator of T then for all x, y, z ∈ X f (R _xz) - min(f (R _xy) + f (R _yz), f (0)) ≤ f (ϵ),

$max (f (R_{xz}) - f (R_{xy}) - f (R_{yz}), f (R_{xz}) - f (0)) \leq f (ϵ) .$ Hence

$\begin{matrix} f (R_{xz}) - f (R_{xy}) - f (R_{yz}) & \leq & f (ϵ) and f (R_{xz}) \\ \leq & f (ϵ) + f (0) . \end{matrix}$ (5)

Example 3.1. We consider relations with lower and higher degrees of transitivity. For this, we consider Lukaseiwicz t-norm W along with it’s additive generator f (x) =1 - x . The inequality (4c) can be used to conclude following results:

Degree of transitivity	R_xy	and	R_yz		region forR_xz
ϵ =0.4,	0.8	and	0.9	implies	Rxz≥ 0.1
ϵ =0.6,	0.8	and	0.9	implies	R_xz ≥ 0.3
ϵ =0.9,	0.8	and	0.9	implies	R_xz ≥ 0.6

Now we consider product t-norm P, along with its additive generator t(x)=-lnx so so (4d) takes the form: -ln(R _xz)+ln(R _xy)+ln(R _yz)≤-ln(ϵ)and-ln(R _xz)≤ -ln(ϵ)-ln(0), ln(R _xz)≥+ln(R _xy)+ln(R _yz)+ln(ϵ)and - ln(R _xz)≤+∞. impliesln(R _xz)≥ln(R _xy.R _yz.ϵ)i.e., R _xz≥R _xy.R _yz.ϵ.

Degree of transitivity	R_xy	and	R_yz		region forR_xz
ϵ =0.4,	0.8	and	0.9	implies	Rxz≥ 0.288
ϵ =0.6,	0.8	and	0.9	implies	R_xz ≥ 0.432
ϵ =0.9,	0.8	and	0.9	implies	R_xz ≥ 0.4248

We conclude that the implicit relationship between degree of consistency of an FPR and its degrees of preference is explicit now. Moreover the different forms of transitivities behave less or more stringent.

Theorem 3.2. If an R-implication I is used in the definition of fuzzy transitivity of an ϵ-transitive relation R with a continuous Archimedean t-norm T with a multiplicative generator g, then for all x, y, z ∈ X $g (R_{xy}) \cdot g (R_{yz}) \leq \frac{g (R_{xz})}{g (ϵ)} .$ Proof. By using Remark 2.2, as Remark 2.1 in previous theorem, ϵ-transitivity implies that $g^{- 1} (min (\frac{g (R_{xz})}{g (T (R_{xy}, R_{yz}))}, 1)) \geq ϵ for all x, y, z \in X .$ Applying g on both sides with increasing nature $min (\frac{g (R_{xz})}{g (T (R_{xy}, R_{yz}))}, 1) \geq g (ϵ) for all x, y, z \in X .$ It implies that $\frac{g (R_{xz})}{g (T (R_{xy}, R_{yz}))} \geq g (ϵ) for all x, y, z \in X .$ g (R _xz) ≥ g (T (R _xy, R _yz)) · g (ϵ) forall x, y, z ∈ X .

Since g is a multiplicative generator, it further implies thatg (R _xz) ≥ max(g (R _xy) · g (R _yz), g (0)) · g (ϵ)

for all x, y, z ∈ X, and hence $g (R_{xy}) \cdot g (R_{yz}) \leq \frac{g (R_{xz})}{g (ϵ)} for all x, y, z \in X .$

3.2 The inequalities For S-implications

Theorem 3.3. Let R be an ϵ-transitive relation which is not T-transitive. If an S-implication I is used in the definition of fuzzy transitivity related to continuous Archimedean t-conorm S with an additive generator s, then for all x, y, z ∈ X $R_{x, z} \geq s^{- 1} (s (ϵ) - s (N (R_{x, y})) - s (N (R_{y, z}))) .$ Proof. Using Remark 2.3 in the definition of ϵ-transitivity implies that $s^{- 1} (min (s (N (T (R_{x, y}, R_{y, z}))) + s (R_{x, z}), s (1))) \geq ϵ$ for all x, y, z ∈ X. Increasing nature of s implies that $min (s (N (T (R_{x, y}, R_{y, z}))) + s (R_{x, z}), s (1)) \geq s (ϵ)$ for all x, y, z ∈ X. Applying Definition 2.8 it further implies that $min (s (S (N (R_{x, y}), N (R_{y, z}))) + s (R_{x, z}), s (1)) \geq s (ϵ),$ and $s (S (N (R_{x, y}), N (R_{y, z}))) + s (R_{x, z}) \geq s (ϵ)$ for all x, y, z ∈ X. Using Remark 2.3 again $min (s (N (R_{x, y})) + s (N (R_{y, z})), s (1)) + s (R_{x, z}) \geq s (ϵ),$ and $s (N (R_{x, y})) + s (N (R_{y, z})) + s (R_{x, z}) \geq s (ϵ)$ (6) for all x, y, z ∈ X. (4f) can be written for all x, y, z ∈ X as $R_{x, z} \geq s^{- 1} (s (ϵ) - s (N (R_{x, y})) - s (N (R_{y, z}))) .$

Theorem 3.4. If an S-implication I is used in the definition of fuzzy transitivity related to continuous Archimedean t-conorm S with a multiplicative generator θ, then for all x, y, z ∈ X $θ (R_{xz}) \leq \frac{θ (ϵ)}{θ (N (R_{xy})) \cdot θ (N (R_{yz}))} .$ Proof. Choosing Remark 2.4 in (4a), we have for all x, y, z ∈ X $θ^{- 1} [max {θ (N (T (R_{x, y}, R_{y, z}))) \cdot θ (R_{x, z}), θ (1)}] \geq ϵ .$ Applying θ with decreasing nature $max [θ (N (T (R_{x, y}, R_{y, z}))) \cdot θ (R_{x, z}), θ (1)] \leq θ (ϵ),$ which further implies for all x, y, z ∈ X that $max [θ (S (N (R_{x, y}), N (R_{y, z}))) \cdot θ (R_{x, z}), θ (1)] \leq θ (ϵ),$ (7) and $θ (S (N (R_{x, y}), N (R_{y, z}))) \cdot θ (R_{x, z}) \leq θ (ϵ) .$ (8) Now using Definition 2.7 $max {θ (1), θ (N (R_{xy})) \cdot θ (N (R_{yz}))} \cdot θ (R_{xz}) \leq θ (ϵ),$ and hence, for all x, y, z ∈ X $θ (R_{xz}) \leq \frac{θ (ϵ)}{θ (N (R_{xy})) \cdot θ (N (R_{yz}))} .$

4 Measures of S-transitivity

This section focus on exploring the relationship between the degree of being S-transitive and consistent behavior of the fuzzy preference relation. The central aim of the work is twofold: firstly it explores the situation in which the degree for being S-transitive is considered while an R-implication is used to calculate this degree, and secondly we investigate the results using the degree for an S-implication in the same definition.

Definition 4.1. β-Fuzzy Transitive Relation: A fuzzy relation R is called S-transitive to the degree β or β-fuzzy transitive if: $β = inf_{y \in X} I (R_{xz}, S (R_{xy}, R_{yz}))$ for all x, y, z ∈ X .

4.1 β-Transitivity Related to R-implications

Theorem 4.1. If R is a β-fuzzy transitive relation and an R-implication I is used in the definition of an FPR, then there exists a ∈ (R _xz, S (R _xy, R _yz)) such that $R_{xz} \geq S (R_{xy}, R_{yz}) - \frac{f (β)}{f ´ (a)} for all x, y, z \in X,$ where f is an additive generator of continuous Archim-edeant-norm T.

Proof. If R is a β-transitive relation, then $\begin{matrix} inf_{y \in X} I (R_{xz}, S (R_{xy}, R_{yz})) & = β, \\ I (R_{xz}, S (R_{xy}, R_{yz})) & \geq β \end{matrix}$

for all x, y, z ∈ X. Now, using Remark 2.1 $f^{- 1} (max (f (S (R_{xy}, R_{yz})) - f (R_{xz}), 0)) \geq β .$ Using f, it implies $max (f (S (R_{xy}, R_{yz})) - f (R_{xz}), 0) \leq f (β),$ which further implies that $f (S (R_{xy}, R_{yz})) - f (R_{xz}) \leq f (β) .$ Since the nature of f is strictly decreasing, therefore by Taylor’s mean value theorem there exists a real number a ∈] R _xz, S (R _xy, R _yz) [such that $f (S (R_{xy}, R_{yz})) - f (R_{xz}) = f (a) [S (R_{xy}, R_{yz}) - R_{xz}] .$ It implies now $f ´ (a) [S (R_{xy}, R_{yz}) - R_{xz}] \leq f (β),$ and $R_{xz} \geq S (R_{xy}, R_{yz}) - \frac{f (β)}{f ´ (a)} for all x, y, z \in X .$ Theorem 4.2. If R is a β-fuzzy transitive relation and an R-implication I is used in the definition of fuzzy transitivity related to continuous Archimedean t-norm with a multiplicative generator g, then $g (R_{xz}) \geq \frac{g (R_{xy}) \cdot g (R_{yz})}{g (β)} for all x, y, z \in X .$ Proof. Using Definition 4.1 $inf_{y \in X} I (R_{xz}, S (R_{xy}, R_{yz})) = β,$ and $I (R_{xz}, S (R_{xy}, R_{yz})) \geq β for all x, y, z \in X .$ By using Remark 2.2 $g^{- 1} (min (\frac{g (S (R_{xy}, R_{yz}))}{g (R_{xz})}, 1)) \geq β$ for all x, y, z ∈ X. Applying g on both sides with increasing nature $min (\frac{g (S (R_{xy}, R_{yz}))}{g (R_{xz})}, 1) \geq g (β) for all x, y, z \in X .$ Hence, it implies for all x, y, z ∈ X that $\frac{g (S (R_{xy}, R_{yz}))}{g (R_{xz})} \geq g (β)$ ,

andg (S (R _xy, R _yz)) ≥ g (R _xz) · g (β).

Since g is a multiplicative generator, it implies that $max (g (R_{xy}) \cdot g (R_{yz}), g (1)) \leq g (R_{xz}) \cdot g (β),$ and $g (R_{xz}) \geq \frac{g (R_{xy}) \cdot g (R_{yz})}{g (β)} for all x, y, z \in X .$

4.2 β-Transitivity Related to S-implications

Theorem 4.3. Let R be a fuzzy preference relation. If an S-implication I is used in the definition of fuzzy transitivity related to continuous Archimedean t-conorm S with an additive generator s, then for all x, y, z ∈ X $N (R_{x, z}) \geq s^{- 1} (s (ϵ) - s (R_{x, y}) - s (R_{y, z})) .$ Proof. Given R is an β-fuzzy transitive relation so, $inf_{y \in X} I (R_{xz}, S (R_{xy}, R_{yz})) = β,$ i.e., $I (R_{xz}, S (R_{xy}, R_{yz})) \geq β for all x, y, z \in X .$ By using Remark 2.3 $s^{- 1} (min (s (N (R_{x, z})) + s (S (R_{x, y}, R_{y, z})), s (1))) \geq β$ for all x, y, z ∈ X. Increasing nature of s implies for all x, y, z ∈ X that $min (s (N (R_{x, z})) + s (S (R_{x, y}, R_{y, z})), s (1)) \geq s (β),$ and $s (N (R_{x, z})) + s (S (R_{x, y}, R_{y, z})) \geq s (β) .$ Again by using Remark 2.3 $min (s (R_{x, y}) + s (R_{y, z}), s (1)) + s (N (R_{x, z})) \geq s (β),$ $s (R_{x, y}) + s (R_{y, z}) + s (N (R_{x, z})) \geq s (β),$ and hence, for all x, y, z ∈ X $N (R_{x, z}) \geq s^{- 1} (s (ϵ) - s (R_{x, y}) - s (R_{y, z})) .$

5 Conclusion

This paper presents some new results in the form of inequalities connecting transitivity of the given fuzzy relation with its clustering nature, here ϵ-transitivity has been used to formulate new inequalities interconnecting with T-transitivity and S-transitivity. Additive and multiplicative generators of t-norms and t-conorms play a prime role for these inequalities. The forms obtained open new research direction. The results based on the measure of the considered transitivity conditions ensure coherence among the objects.

Footnotes

Acknowledgments

The authors appreciated the very constructive comments from anonymous reviewers that have helped a lot to enhance the value of this manuscript.

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