Closing reciprocal relations w.r.t. stochastic transitivity

Available online at www.sciencedirect.com

Fuzzy Sets and Systems 241 (2014) 2 – 26 www.elsevier.com/locate/fss

Closing reciprocal relations w.r.t. stochastic transitivity S. Fresona , B. De Baetsb , H. De Meyera,∗ a Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, B-9000 Gent, Belgium b Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium

Available online 20 February 2013

Abstract We equip the set of reciprocal relations with an appropriate order relation capturing the bipolar semantics, turning this set into a complete join-semilattice. We also introduce the notion of a compatible family of reciprocal relations and show that such a family has an infimum. Moreover, we discuss the one-to-one correspondence between the set of 3-valued reciprocal relations and the set of complete crisp relations. We refine the theorem of Bandler and Kohout concerning the existence of closures of elements of a poset w.r.t. some given property, rendering it applicable to the above join-semilattice. The paper is solely dedicated to the transitivity property. Although uniquely defined for 3-valued reciprocal relations, general reciprocal relations can exhibit various types of transitivity. We characterize the mappings g and h for which the corresponding g-stochastic and h-isostochastic transitive closure exists for any reciprocal relation. In particular, it follows that weak and strong stochastic transitive closures always exist, while this is not the case for moderate stochastic transitivity. Moreover, max-isostochastic transitivity turns out to be the only practically relevant type of isostochastic transitivity. Finally, we provide algorithms realizing each of these transitive closures. © 2013 Elsevier B.V. All rights reserved. Keywords: Stochastic transitivity; Reciprocal relation; Transitive closure

1. Introduction One of the key concepts in fuzzy set theory is that of a fuzzy relation, which typically exhibits a unipolar semantics. Another type of [0,1]-valued relations, also often used in fuzzy set theory, are reciprocal relations Q, satisfying the constraint Q(x, y) + Q(y, x) = 1, thus exhibiting a bipolar semantics. Reciprocal relations are popular tools for representing graded preferences [3,4,21,25,26,29,30,32,33], often in the context of group decision making, although they do not accommodate for incomparability of alternatives in contrast to the more general fuzzy preference structures [24,35]. Reciprocal relations are more predominant in probabilistic settings, where they appear in the context of binary choice probabilities [22], when expressing winning probabilities between the components of a random vector [9,16,18] or as mutual rank probability relations in the context of poset ranking [10,14]. Whereas for fuzzy relations, T-transitivity, with T a (left-continuous) triangular norm, is widely accepted, for reciprocal relations a variety of types of transitivity has been introduced, such as g-stochastic transitivity [27], h-isostochastic transitivity [11], FG-transitivity [31], cycle transitivity [11], and so on. Surprisingly, while for fuzzy relations the existence of T-transitive closures has been studied in depth [7,24], and many algorithms have been proposed (see e.g. [28]), ∗ Corresponding author. Tel.: +32 9 2644810; fax: +32 9 2644995.

E-mail address: [email protected] (H. De Meyer). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.01.014

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

3

sometimes even focusing on a single t-norm (usually the minimum operator) [17], there appears to be no literature on the existence of transitive closures of reciprocal relations. The purpose of this paper is to fill this lacune. It also contributes to the growing body of work contributing to a better understanding of the distinction between unipolar and bipolar semantics [19,20]. The existence of crisp or T-transitive closures easily follows from the theorem of Bandler and Kohout [1] concerning the existence of closures of all elements of a poset w.r.t. some given property. In view of their bipolar semantics, it makes no sense to apply the usual subsethood-based ordering of fuzzy relations to reciprocal relations as well. Fortunately, equipping the set of reciprocal relations with an appropriate order relation allows to turn this set into a complete joinsemilattice, as discussed in Section 3, thus generalizing the corresponding observation for the set of complete relations in Section 2. Moreover, by relaxing one of the conditions of the theorem of Bandler and Kohout, it now becomes possible to characterize the existence of transitive closures of reciprocal relations for a variety of types of transitivity, as expounded in Sections 4–6. In particular, we provide a necessary and sufficient condition (implying left continuity) on the mapping g guaranteeing the existence of g-stochastic transitive closures. In essence, such mappings g turn out to be distorted maxima. When applied to the mapping h, the same condition guarantees the existence of h-isostochastic transitive closures, albeit further reducing the set of suitable h. Finally, in Section 7, we provide an algorithm realizing this closure for each of the types of transitivity considered. Whereas for the weak stochastic transitive closure, a minor adaptation of the classical Floyd–Warshall algorithm [23,36] suffices, algorithmic realizations for the other types were inspired by our former work on T-transitive closures of fuzzy relations [12,28], in particular the weight-driven strategy they were built upon. We conclude with some suggestions for future work. 2. Crisp relations 2.1. Basic notions A crisp relation R ∈ P(X 2 ) (where P stands for powerset) on a universe X can be identified with its characteristic mapping R : X 2 → {0, 1}. If (x, y) ∈ R, also denoted as xRy, then R(x, y) = 1, else R(x, y) = 0. In the case of a finite universe X = {x1 , . . . , xn }, the relation R can be represented by an n × n-matrix R with elements R(xi , x j ) ∈ {0, 1}. A relation R on a universe X can also be (graphically) represented by a directed graph G R = (V R , E R ), with V R = X the set of nodes and E R = {(xi , x j ) ∈ V R2 |R(xi , x j ) = 1} the set of arcs. P(X 2 ) is equipped with a standard order relation, union and intersection. When using the functional representation, R1 ∈ P(X 2 ) is smaller than R2 ∈ P(X 2 ), or R2 is larger than R1 , denoted as R1 ⊆ R2 , if (∀(x, y) ∈ X 2 )(R1 (x, y) ≤ R2 (x, y)).

(1)

If R1 ⊆ R2 or R2 ⊆ R1 , then the relations are called comparable. (P(X 2 ), ⊆) forms a complete lattice, bounded by the greatest element R = X 2 and the smallest element R⊥ = ∅. The union and intersection of two relations R1 , R2 ∈ P(X 2 ) are given by R1 ∪ R2 (x, y) = max(R1 (x, y), R2 (x, y))

(2)

R1 ∩ R2 (x, y) = min(R1 (x, y), R2 (x, y)),

(3)

and

respectively. Both definitions are easily extended to the union and intersection of a family of relations. (P(X 2 ), ∪, ∩) is the algebraic equivalent of the complete lattice (P(X 2 ), ⊆) and the connection between the union and intersection and the order relation is given by R1 ∩ R2 ⊆ Ri ⊆ R1 ∪ R2 , i = 1, 2.

(4)

2.2. Complete relations A crisp relation R on a universe X is complete when (∀(x, y) ∈ X 2 )(R(x, y) = 1 ∨ R(y, x) = 1).

(5)

4

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

We denote the set of all complete crisp relations on a universe X by C X , i.e. C X = {R ∈ P(X 2 )|R is complete}.

(6)

By definition, a complete relation is reflexive, i.e. R(x, x) = 1 for all x ∈ X . Since C X is a subset of P(X 2 ), it inherits definitions (1)–(3). Proposition 1. If R ∈ C X , R  ∈ P(X 2 ) and R ⊆ R  , then R  ∈ C X . Clearly, R is a complete relation and is the greatest element of C X . On the other hand, R⊥ is not complete. The set of minimal complete relations on X is given by R⊥ = {R ∈ C X |(∀(x, y) ∈ X 2 )(R(x, y) = 1 ⇒ (R(y, x) = 0 ∨ x = y))}. Note that R⊥ consists of the antisymmetric relations in C X . If X is finite, i.e. |X | = n, then there are relations.

(7) 2n(n−1)/2

such

Example 1. For X = {x, y, z}, the minimal complete relations are x y z

x 1 0 0

y 1 1 0

z 1 1 1

x y z

x 1 1 0

y 0 1 0

z 1 1 1

x y z x y z

x 1 0 0 x 1 1 0

y 1 1 1 y 0 1 1

z 1 0 1 z 1 0 1

x y z x y z

x 1 0 1 x 1 1 1

y 1 1 0 y 0 1 0

z 0 1 1 z 0 1 1

x y z x y z

x 1 0 1 x 1 1 1

y 1 1 1 y 0 1 1

z 0 0 1 z 0 0 1

Clearly, the intersection of complete relations is not necessarily complete. Hence, the structure (C X , ⊆) is only a complete join-semilattice [6] with greatest element R , algebraically equivalent to (C X , ∪). In order to identify the situations in which intersection is meaningful, we introduce the notion of compatibility. Definition 1. Two relations R1 , R2 ∈ C X are said to be compatible, denoted by R1 ≈ R2 , if there exists a relation R ∈ C X such that R ⊆ R1 and R ⊆ R2 .

(8)

A family of complete relations is said to be compatible if there exists a complete relation that is smaller than all relations in the family. The compatibility relation ≈ on C X is reflexive and symmetric, making it a tolerance relation [37]. Note that compatibility of a family of complete relations is stronger than pairwise compatibility of any two members of that family. In a finite setting, the number of complete relations compatible with a given complete relation R increases with the number of pairs {x, y} ⊂ X for which R(x, y) = R(y, x) = 1. Proposition 2. Let |X | = n. A complete relation R ∈ C X is compatible with 21/2(n

2 −r )

31/2(r −n)

complete relations, where r = |{(x, y) ∈ X 2 |R(x, y) = R(y, x) = 1}|. Proof. Let X = {x1 , . . . , xn } and consider two elements xi and x j with i < j. We distinguish two situations: (i) R(xi , x j )  R(x j , xi ): any R  that is compatible with R either satisfies R  (xi , x j ) = R(xi , x j )  R(x j , xi ) = R  (x j , xi )

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

5

or R  (x j , xi ) = R  (x j , xi ) = 1. (ii) R(xi , x j ) = R(x j , xi ): any R  that is compatible with R satisfies one of following three conditions: (a) R  (xi , x j ) = 1, R  (x j , xi ) = 0, (b) R  (xi , x j ) = 0, R  (x j , xi ) = 1, (c) R  (xi , x j ) = 1, R  (x j , xi ) = 1. Since there are 21 (n 2 − r ) pairs {xi , x j } with i < j and R(xi , x j )  R(x j , xi ), and 21 (r − n) pairs {xi , x j } with i < j and R(xi , x j ) = R(x j , xi ), the proposition follows immediately.  Example 2. Consider the complete relation R on X = {x, y, z} defined as R x y z

x 1 0 1

y 1 1 0

z 0 1 1

Since n = 3 and r = 3, R is compatible with 23 30 = 8 complete relations. Apart from R and R itself, these are R1

x y z

R2

x y z

R3

x y z

x y z

1 1 0 1 1 1 1 0 1

x y z

1 1 1 0 1 1 1 0 1

x y z

1 1 0 0 1 1 1 1 1

R4

x y z

R5

x y z

R6

x y z

x y z

1 1 1 0 1 1 1 1 1

x y z

1 1 1 1 1 1 1 0 1

x y z

1 1 0 1 1 1 1 1 1

In this case, the family consisting of the relations R, R1 , R2 , R3 , R4 , R5 , R6 , R is compatible. Example 3. Consider the complete relation R on X = {x, y, z} defined as R x y z

x 1 1 0

y 1 1 0

z 1 1 1

Since n = 3 and r = 5, R is compatible with 22 31 = 12 complete relations. Apart form R and R itself, these are R1

x y z

R2

x y z

R3

x y z

R4

x y z

x y z

1 1 1 0 1 1 0 0 1

x y z

1 0 1 1 1 1 0 0 1

x y z

1 1 1 0 1 1 1 1 1

x1 y z

1 0 1 1 1 1 1 1 1

R5

x y z

R6

x y z

R7

x y z

R8

x y z

x y z

1 1 1 0 1 1 0 1 1

x1 x2 x3

1 0 1 1 1 1 0 1 1

x y z

1 0 1 1 1 1 1 0 1

x y z

1 1 1 0 1 1 1 0 1

R9

x y z

 R10

x y z

x y z

1 1 1 1 1 1 1 0 1

x y z

1 1 1 1 1 1 0 1 1

6

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

 ,R In this case, for example, R1 and R2 are not compatible. Hence, the family consisting of the relations R, R1 , . . . , R10  is also not compatible.

It follows easily from the definition that comparable relations are compatible. Proposition 3. For any R1 , R2 ∈ C X , it holds that R1 ⊆ R2 implies R1 ≈ R2 . Example 4. The relation R is compatible with any other complete relation on the same universe, since R ⊆ R for 2 all R ∈ C X . As r = n 2 , we count 3n −n/2 = |C X | complete relations compatible with R . However, the converse is not true. Example 5. It is clear that R, R6 ∈ C X as defined in Example 3, although compatible, are not comparable. The notion of compatibility provides a necessary and sufficient condition for the intersection of two complete relations to be complete. Proposition 4. For any R1 , R2 ∈ C X , it holds that R1 ≈ R2 if and only if R1 ∩ R2 ∈ C X . Proof. First assume that R1 ≈ R2 . If R1 ∩R2 (x, y) = 0, then there exists i ∈ {1, 2} such that Ri (x, y) = 0, Ri (y, x) = 1. For j = 3 − i, since R1 ≈ R2 , either R j (x, y) = R j (y, x) = 1 or R j (x, y) = Ri (x, y) and R j (y, x) = Ri (y, x). Both situations imply that R j (y, x) = 1. Hence, R1 ∩ R2 (y, x) = min(R1 (y, x), R2 (y, x)) = 1. Conversely, if R1 ∩ R2 ∈ C X , the compatibility of R1 and R2 follows immediately from R1 ∩ R2 ⊆ Ri for i = 1, 2.  Proposition 4 can easily be generalized to a family of complete relations. Proposition 5. A family (Ri )i∈I in C X is compatible if and only if

 i∈I

Ri ∈ C X .

Proposition 5 implies that for any R ∈ C X , it holds that ([R, R ], ⊆) is a complete lattice. Corollary 1. For any R0 ∈ C X and any compatible family (Ri )i∈I in C X , it holds that R0 ≈ extended family (Ri )i∈I ∪{0} is compatible.

 i∈I

Ri if and only if the

 Proof. Accordingto Proposition  4, the compatibility of R0 and i∈I Ri is equivalent to the completeness of the intersection R0 ∩ i∈I Ri = i∈I ∪{0} Ri , which on its turn, due to Proposition 5, is equivalent to the compatibility of the extended family.  3. Reciprocal relations 3.1. Basic notions A reciprocal [0,1]-valued relation is a mapping Q : X 2 → [0, 1] that satisfies (∀(x, y) ∈ X 2 )(Q(x, y) + Q(y, x) = 1).

(9)

We denote the set of all reciprocal [0,1]-valued relations, or in short reciprocal relations, on a universe X by Q X . An important subset of Q X consists of those reciprocal relations taking values in {0, 21 , 1} only, called 3-valued reciprocal relations. We denote this subset by Q∗X . In the case of a finite universe X, a reciprocal relation can be represented by a square matrix with elements in [0,1] or also as a weighted directed graph. The standard order relation in (1) is not suitable for generalization to reciprocal relations, as it would render all of them incomparable. An appropriate definition is given next.

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

7

Definition 2. Let Q 1 , Q 2 ∈ Q X . Q 1 is smaller than Q 2 , or Q 2 is larger than Q 1 , denoted as Q 1  Q 2 , if (∀(x, y) ∈ X 2 )((Q 1 (x, y) ≥ Q 2 (x, y) ≥ 21 ) ∨ (Q 1 (x, y) ≤ Q 2 (x, y) ≤ 21 )).

(10)

This definition bears a striking resemblance with the sharper than relation between fuzzy sets, often used in the context of entropy measures of fuzzy sets, such as in the work of De Luca and Termini [15]. Proposition 6. (Q X , ) is a partially ordered set, i.e.  is reflexive, antisymmetric and transitive. Proof. The reflexivity and antisymmetry follow from the reflexivity and antisymmetry of ≤ on [0,1]. We verify the transitivity. Assume that Q 1  Q 2 and Q 2  Q 3 . Consider (x, y) ∈ X 2 . If Q 1 (x, y) ≥ Q 2 (x, y) ≥ 21 and Q 2 (x, y) ≥ Q 3 (x, y) ≥ 21 , we find Q 1 (x, y) ≥ Q 3 (x, y) ≥ 21 . If Q 1 (x, y) ≥ Q 2 (x, y) ≥ 21 and Q 2 (x, y) ≤ Q 3 (x, y) ≤ 21 , then necessarily Q 2 (x, y) = 21 = Q 3 (x, y), which again leads to Q 1 (x, y) ≥ Q 3 (x, y) ≥ 21 . The remaining two cases lead to Q 1 (x, y) ≤ Q 3 (x, y) ≤ 21 .  The greatest reciprocal relation Q  on X is given by Q  (x, y) = 21 for all (x, y) ∈ X 2 . There is no smallest reciprocal relation on X, but rather a set of minimal ones, which turn out to be 3-valued as well. This set of minimal reciprocal relations on X is given by Q∗⊥ = {Q ∈ Q∗X |(∀(x, y) ∈ X 2 )(Q(x, y) =

1 2

⇒ x = y)}.

If X is finite, i.e. |X | = n, then there are again 2n(n−1)/2 such relations. Using the interval notation, i.e. for Q 0 , Q 1 ∈ Q X , the set {Q ∈ Q X |Q 0  Q  Q 1 } is denoted by [Q 0 , Q 1 ], the set Q X can be written as the (non-disjoint) union of intervals  QX = [Q, Q  ]. (11) Q∈Q∗⊥

The following proposition expresses that any non-empty family of reciprocal relations has a supremum, i.e. a smallest upper bound. Hence, (Q X , ) is a complete join-semilattice [6]. Proposition 7. The supremum of a non-empty family (Q i )i∈I of reciprocal relations, denoted by i∈I Q i , is the reciprocal relation given by ⎧ 1 ⎪ ⎪ inf Q i (x, y), if inf Q i (x, y) ≥ ⎪ ⎪ i∈I 2 ⎪ ⎨ i∈I  (12) Q i (x, y) = sup Q (x, y), if sup Q (x, y) ≤ 1 i i ⎪ ⎪ 2 i∈I ⎪ i∈I i∈I ⎪ ⎪ ⎩1 otherwise 2, In case of two relations, the supremum is commonly called the join. In the present situation, the join of Q 1 , Q 2 ∈ Q X , denoted by Q 1  Q 2 , is the reciprocal relation given by ⎧ 1 ⎪ ⎨ min(Q 1 (x, y), Q 2 (x, y)), if min(Q 1 (x, y), Q 2 (x, y)) ≥ 2 Q 1  Q 2 (x, y) = max(Q 1 (x, y), Q 2 (x, y)), if max(Q 1 (x, y), Q 2 (x, y)) ≤ 21 (13) ⎪ ⎩1 otherwise 2, We will use the term union to refer to the supremum of reciprocal relations. Note that the union of reciprocal relations is commutative and associative, i.e. for Q 1 , Q 2 , Q 3 ∈ Q X , Q 1  Q 2 = Q 2  Q 1 and (Q 1  Q 2 )  Q 3 = Q 1  (Q 2  Q 3 ). To investigate the existence of a greatest lower bound for a non-empty family of reciprocal relations, we extend the notion of compatibility to Q X .

8

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Definition 3. Two reciprocal relations Q 1 , Q 2 ∈ Q X are said to be compatible, denoted by Q 1 ≈ Q 2 , if there exists a reciprocal relation Q ∈ Q X such that Q  Q 1 and Q  Q 2 . A non-empty family of reciprocal relations is said to be compatible if there exists a reciprocal relation that is smaller than all relations in the family. The notion of compatibility provides a necessary and sufficient condition for the existence of an infimum, i.e. a greatest lower bound. Proposition 8. A non-empty family (Q i )i∈I of reciprocal relations has a greatest lower bound if and only if it is a compatible family. If the family (Q i )i∈I is compatible, then this infimum, denoted by i∈I Q i , is the reciprocal relation given by ⎧ 1 ⎪ Q i (x, y) ≥ ⎪ ⎨ sup Q i (x, y), if inf  i∈I 2 i∈I Q i (x, y) = . (14) 1 ⎪ ⎪ i∈I ⎩ inf Q i (x, y), if sup Q i (x, y) ≤ i∈I 2 i∈I Proof. Obviously, if a family (Q i )i∈I of reciprocal relations has a greatest lower bound, then the latter is a reciprocal relation that is smaller than all reciprocal relations in the family, and the family is compatible. Conversely, if a family is compatible then for any (x, y) it either holds that (∀i ∈ I )(Q i (x, y) ≥ 21 ) or (∀i ∈ I )(Q i (x, y) ≤ 21 ), i.e. inf i∈I Q i (x, y) ≥ 21 or supi∈I Q i (x, y) ≤ 21 . It is then clear that (14) results in a reciprocal relation, which is then easily seen to yield the greatest lower bound.  The infimum of two compatible reciprocal relations Q 1 and Q 2 , also called their meet and denoted by Q 1  Q 2 , is the reciprocal relation given by max(Q 1 (x, y), Q 2 (x, y)), if min(Q 1 (x, y), Q 2 (x, y)) ≥ 21 Q 1  Q 2 (x, y) = . (15) min(Q 1 (x, y), Q 2 (x, y)), if max(Q 1 (x, y), Q 2 (x, y)) ≤ 21 We will use the term intersection to refer to the reciprocal relation given by (14) or (15). It is easily seen that the intersection of compatible relations is commutative. Associativity follows from (15) together with the following proposition, which ensures compatibility. Proposition 9. For any Q 0 ∈ Q X and any compatible family (Q i )i∈I in Q X , it holds that Q 0 ≈ i∈I Q i if and only if the extended family (Q i )i∈I ∪{0} is compatible. Proof. The proof is similar to the proof of Corollary 1.  The above results imply for any Q ∈ Q X , the interval ([Q, Q  ], ) is a complete bounded lattice, algebraically equivalent to ([Q, Q  ], , ). In view of (11), this means that (Q X , ) is the union of the complete bounded lattices ([Q, Q  ], ), with Q ∈ Q∗⊥ . 3.2. 3-valued reciprocal relations As mentioned earlier, a special subset of the set of reciprocal relations is the set of all 3-valued reciprocal relations. Note that (Q∗X , ) is a complete join-semilattice as well, with the same greatest element Q  and the set of minimal elements Q∗⊥ as (Q X , ). 3.2.1. An order isomorphism The mapping  : C X → Q∗X defined by (R)(x, y) =

1 + R(x, y) − R(y, x) , 2

(16)

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

9

maps any complete relation R ∈ C X to a 3-valued reciprocal relation Q R = (R) ∈ Q∗X . Vice versa, the mapping  : Q∗X → C X defined by 1, if Q(x, y) ≥ 21 , (17) (Q)(x, y) = 0, if Q(x, y) < 21 maps any 3-valued reciprocal relation Q ∈ Q∗X to a complete relation R Q = (Q) ∈ C X . Note that the advantage of using the 3-valued representation (R) of a complete relation R is that the information concerning both R(x, y) and R(y, x) is contained in (R)(x, y) alone or, equivalently, in (R)(y, x) alone. One easily verifies that (R ) = Q  and (Q  ) = R . Proposition 10. The mappings  and  are each other’s inverse, i.e.  ◦  = idC X and  ◦  = idQ∗X . Proof. First consider a complete relation R ∈ C X and x, y ∈ X . We distinguish three cases: (i) R(x, y) = R(y, x) = 1: then (R)(x, y) = (R)(y, x) = 21 , whence ((R))(x, y) = ((R))(y, x) = 1. (ii) R(x, y) = 1 and R(y, x) = 0: then (R)(x, y) = 1 and (R)(x, y) = 0, whence ((R))(x, y) = 1 and ((R))(x, y) = 0. (iii) R(x, y) = 0 and R(y, x) = 1: similar to case (ii). Hence,  ◦  = idC X . The proof of the second identity is a matter of direct verification as well.  It is easy to see that the mapping  is an order isomorphism between (C X , ⊆) and (Q∗X , ). Proposition 11. For any R1 , R2 ∈ C X and any Q 1 , Q 2 ∈ Q∗X , it holds that R1 ⊆ R2 ⇔ (R1 )  (R2 ), Q 1  Q 2 ⇔ (Q 1 ) ⊆ (Q 2 ). Corollary 2. If for R1 , R2 ∈ C X it holds that R1 ≈ R2 , then (R1 ) ≈ (R2 ). Similarly, if for Q 1 , Q 2 ∈ Q∗X it holds that Q 1 ≈ Q 2 , then (Q 1 ) ≈ (Q 2 ). The following proposition is a matter of direct verification. Proposition 12. For any R1 , R2 ∈ C X , it holds that (i) (R1 ∪ R2 ) = (R1 )  (R2 ), (ii) If R1 ≈ R2 , then (R1 ∩ R2 ) = (R1 )  (R2 ). For any Q 1 , Q 2 ∈ Q∗X , it holds that (iii) (Q 1  Q 2 ) = (Q 1 ) ∪ (Q 2 ), (iv) If Q 1 ≈ Q 2 , then (Q 1  Q 2 ) = (Q 1 ) ∩ (Q 2 ). 3.2.2. Reducing Q X to Q∗X By formally, but not semantically, regarding a reciprocal relation Q ∈ Q X as a fuzzy relation, we can consider its alpha-cut (Q) : 1, if Q(x, y) ≥  (Q) (x, y) = . (18) 0, if Q(x, y) <  For  = 21 , the resulting cut is a complete relation, i.e. for any Q ∈ Q X it holds that (Q)1/2 ∈ C X . Obviously, the 21 -cut for reciprocal relations is an extension of  to Q X . From here on, we therefore consider  : Q X → C X .

10

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

It is readily seen that “equality of 21 -cuts” is an equivalence relation on Q X and that any equivalence class in Q X includes exactly one Q ∈ Q∗X . The equivalence classes are given by [Q] = {Q  ∈ Q X |(Q  )1/2 = (Q)1/2 }.

(19)

Note that [Q  ] = {Q  }. Given a reciprocal relation Q ∈ Q X , the only 3-valued reciprocal relation belonging to the same equivalence class is ((Q)), further denoted by Q ∗ . The following observations are straightforward. Proposition 13. For any Q ∈ Q X , it holds that Q ∗  Q. Furthermore, Q ∗ = Q if and only if Q ∈ Q∗X . Proposition 13 allows us to reformulate the definition of compatibility requiring a 3-valued relation as lower bound. It also implies that any equivalence class as given by (19) is a compatible family of Q X . Corollary 3. For any family (Q i )i∈I of reciprocal relations, the following hold: (i) (Q i )i∈I is compatible if and only if there exists some Q ∈ Q∗X such that Q  Q i for all i ∈ I . (ii) If Q i∗ = Q ∗j for all i, j ∈ I , then (Q i )i∈I is compatible. Note that the converse of (ii) does not hold, as Q  is compatible with any complete relation Q, while it does not hold that Q ∗ = Q  for any such Q. Proposition 14. For any Q 1 , Q 2 ∈ Q X , if it holds that Q 1  Q 2 , then Q ∗1  Q ∗2 . Proposition 15. For any Q 1 , Q 2 ∈ Q X , it holds that (Q 1  Q 2 ) = (Q 1 ) ∪ (Q 2 ), ((Q 1 ) ∪ (Q 2 )) = Q ∗1  Q ∗2 . If Q 1 ≈ Q 2 , then (Q 1  Q 2 ) = (Q 1 ) ∩ (Q 2 ), ((Q 1 ) ∩ (Q 2 )) = Q ∗1  Q ∗2 . Proof. All equalities can be proven by direct verification.  4. Closures It is well known that the transitive closure of a given crisp relation, as the smallest transitive relation containing the given relation, always exists. The same notion has been investigated in the context of fuzzy relations, in particular for the T-transitivity property, with T a t-norm [7]. The focus can be directed toward other properties as well, as done extensively by Bandler and Kohout [1]. More generally, the notion of closure of a given mathematical object w.r.t. some property of interest makes sense as soon as an appropriate partial order relation can be defined on the set of objects. Definition 4. Consider a property P that elements of a poset (S, ≤S ) can satisfy or fail to satisfy. If for a given s ∈ S, an element s  ∈ S satisfies (i) s  satisfies P, (ii) s ≤S s  , (iii) for any s  ∈ S, if s  satisfies P and s ≤S s  , then s  ≤S s  , then s  is called a P-closure of s. Obviously, if a P-closure of an object s exists, it must be unique. In that case, we denote the P-closure by s P . Hence, an object s has property P if and only if its P-closure coincides with s. For many properties P, the P-closure exists for

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

11

some but not all objects. The interesting properties are those for which every object has a P-closure. In these cases only, one can define a P-closure operator on (S, ≤S ). The necessary and sufficient conditions for this to occur have been established by Bandler and Kohout [1] in the context of crisp as well as fuzzy relations, recalled hereafter for crisp relations. Theorem 1. The P-closure exists for all elements of (P(X 2 ), ⊆) if and only if (i) R satisfies P. (ii) The intersection of every non-empty family of relations, all of which satisfy P, satisfies P. Corollary 4. Let P be a property such that the P-closure exists for all elements of (P(X 2 ), ⊆). For any R ∈ P(X 2 ), its P-closure is given by P

R =

{R  ∈ P(X 2 )|R ⊆ R  ∧ (R  has pr oper t y P)}.

(20)

Together with Proposition 5, the preceding corollary implies that the P-closure operator preserves completeness. Corollary 5. Let P be a property such that the P-closure exists for all elements of (P(X 2 ), ⊆). The P-closure of any R ∈ C X is complete as well and is given by P

R =

{R  ∈ C X |R ⊆ R  ∧ (R  has pr oper t y P)}.

(21)

Theorem 1 can be generalized as follows. The proof is a simple adaptation of that of Bandler and Kohout in [1]. Theorem 2. Let (S, ≤S ) be a poset with greatest element s , such that for every non-empty lower bounded subset a greatest lower bound exists. The P-closure exists for all elements in (S, ≤S ) if and only if (i) s satisfies P; (ii) For every non-empty, lower bounded subset in S, all elements of which satisfy P, its greatest lower bound satisfies P. Proof. We first prove the necessity of (i) and (ii). According to the definition, s ≤S s P , while s P ≤S s , since s is the greatest element of S. Thus, since s P = s , s satisfies P. Consider a lower bounded subset A ⊆ S, the elements of which all satisfy P. Let s be the greatest lower bound of A. According to the definition, s P is a lower bound of A and therefore s P ≤S s. It follows that s = s P and that s satisfies P. Next, we prove the sufficiency of (i) and (ii). Consider s ∈ S and let A be the subset of all s  ∈ S that are greater than or equal to s and satisfy P. Clearly, A contains s . As A is lower bounded by s, it has a greatest lower bound s0 , which, according to the assumptions, satisfies P. Finally, it is easily seen that s0 is the P-closure of s.  Corollary 6. Let P be a property such that the P-closure exists for all elements in (S, ≤S ). For any s ∈ S, its P-closure is given by s P = inf{s  ∈ S|s ≤S s  ∧ (s  has pr oper t y P)}.

(22)

Corollary 7. The poset (C X , ⊆) trivially satisfies the conditions of Theorem 2. If for a given property P, the P-closure exists for all elements in (P(X 2 ), ⊆), then it also exists for all elements in (C X , ⊆). Moreover, the P-closure of R ∈ C X in the smaller poset (C X , ⊆) coincides with that in the larger one (P(X 2 ), ⊆). 5. Transitivity of 3-valued reciprocal relations Transitivity of crisp relations is a well-known property. A crisp relation R ∈ P(X 2 ) is said to be transitive if (∀(x, y, z) ∈ X 3 )((R(x, y) = 1 ∧ R(y, z) = 1) ⇒ R(x, z) = 1).

(23)

12

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Fig. 1. Illustration of transitive complete relations Ra , Rb , Rc and Rd on the universe X = {x, y, z}.

Obviously, the greatest crisp relation R is transitive and transitivity is preserved by intersection on P(X 2 ). Theorem 1 then guarantees that any crisp relation has a transitive closure. Moreover, the transitive closure R of a given crisp relation R is given as

R= {R  ∈ P(X 2 )|R ⊆ R  and R  is transitive}. (24) Note that when applied to a complete relation R, R is complete as well, as is also expressed by Corollary 5. As the definition of transitivity applies to complete relations as well, it can serve as inspiration for the definition of transitivity of 3-valued reciprocal relations. Definition 5. A relation Q ∈ Q∗X is called transitive if (Q) is transitive. The following proposition is easily verified. Proposition 16. A 3-valued reciprocal relation Q ∈ Q∗X is transitive if and only if  (∀x, y, z ∈ X ) Q(x, y) ≥ 21 ∧ Q(y, z) ≥ 21 ⇒ Q(x, z) ≥ 21 .

(25)

A relation R ∈ C X is transitive if and only if any three elements of X can be labeled x, y and z such that one of the following four situations is obtained (the graphical representations of the four situations are shown in Fig. 1): Ra x y z

x 1 0 0

y 1 1 0

z 1 1 1

Rb x y z

x 1 0 1

y 1 1 1

z 1 0 1

x 1 1 1

Rc x y z

y 0 1 0

z 1 1 1

Rd x y z

x 1 1 1

y 1 1 1

z 1 1 1

(26)

Equivalently, a relation Q ∈ Q∗X is transitive if and only if any three elements in X can be labeled x, y and z such that one of the following situations is obtained, where Q a = (Ra ), Q b = (Rb ), Q c = (Rc ) and Q d = (Rd ). Qa

x y z

x

1 2

1 1

y

0

1 2

z

0 0

Qb

x y z

x

1 2

1

1 2

1

y

0

1 2

1 2

z

1 2

1

Qc

x y z

x

1 2

0

y

1 2

z

Qd

0

1 2

x

1

1 2

1

y

1 2

0

1 2

z

x y z 1 2 1 2 1 2

1 2 1 2 1 2

1 2 1 2 1 2

(27)

The above observations can be translated to the 3-valued representation, i.e. the poset (Q∗X , ). For any Q ∈ Q∗X , its transitive closure exists and is given by Q = {Q  ∈ Q∗X |Q  Q  and (Q  is transitive)}.

(28)

Note that condition (ii) of Theorem 3 here expresses that intersection preserves the transitivity of a family of compatible relations. 6. Stochastic transitivity of reciprocal relations 6.1. Basic notions As reciprocal relations take values in the entire unit interval, there is no unique definition of transitivity, and many proposals have been suggested in the literature. A common type of transitivity of reciprocal relations is stochastic transitivity. A general definition for g-stochastic transitivity was given in [8,11].

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

13

Definition 6. Let g : [ 21 , 1]2 → [0, 1] be an increasing mapping such that g( 21 , 21 ) ≤ 21 . A reciprocal relation Q ∈ Q X is called g-stochastic transitive (or gS-transitive) if for any (x, y, z) ∈ X 3 it holds that (Q(x, y) ≥

1 2

∧ Q(y, z) ≥ 21 ) ⇒ Q(x, z) ≥ g(Q(x, y), Q(y, z)).

(29)

Example 6. The previous definition generalizes the standard types of stochastic transitivity, namely: • • • •

Weak stochastic transitivity [27] (WS-transitivity) when g = 21 , Moderate stochastic transitivity [27] (MS-transitivity) when g = min, Strong stochastic transitivity [27] (SS-transitivity) when g = max, -transitivity [2], with  ∈ [0, 1], when g =  max +(1 − ) min.

Clearly, SS-transitivity implies -transitivity, which implies MS-transitivity, which in turn implies WS-transitivity. Note that the condition g( 21 , 21 ) ≤ 21 arises from the following minimal requirement we insist to impose: the reciprocal representation (R) of any transitive complete relation R should be gS-transitive. In particular, Q  satisfies all types of gS-transitivity. A maximal requirement could be to insist that the only 3-valued reciprocal relations that are gS-transitive are precisely the reciprocal representations of transitive complete relations. In that case, the additional condition 0 < g( 21 , 21 ) should be imposed. Hence, if 0 < g( 21 , 21 ) ≤ 21 , then gS-transitivity of 3-valued reciprocal relations coincides with the transitivity of 3-valued relations. In view of Corollary 7, any gS-transitive closure of a 3-valued reciprocal relation then coincides with its transitive closure. Note that the mappings g of Example 6 satisfy an even stronger condition, namely g( 21 , 21 ) = 21 ; the corresponding types of transitivity all imply WS-transitivity. Example 7. Consider the mapping g : [ 21 , 1]2 → [0, 1] defined by g(a, b) = a + b − 1. Clearly, g is increasing and g( 21 , 21 ) = 0 ≤ 21 . The 3-valued reciprocal relation Q 1 ∈ Q X , with X = {x, y, z}, given as Q1 x y z

x y z 0.5 0.5 1 0.5 0.5 0.5 0 0.5 0.5

satisfies condition (29), while it is not a transitive member of Q∗X . A different requirement one could imagine is to demand that for any gS-transitive reciprocal relation Q it holds that Q ∗ (or, equivalently, (Q)) is transitive as well. This leads to the additional condition g( 21 , 21 ) ≥ 21 , which combined with the prevailing condition yields g( 21 , 21 ) = 21 . Example 8. Consider the mapping g : [ 21 , 1]2 → [0, 1] defined by g(a, b) = ab. Again, condition (29) is satisfied. The reciprocal relation Q 2 ∈ Q X , with X = {x, y, z}, given as Q2 x y z

x y z 0.5 0.6 0.4 0.4 0.5 0.6 0.6 0.4 0.5

is gS-transitive. However, the corresponding relation Q ∗2 , given as Q ∗2 x y z

x y z 0.5 1 0 0 0.5 1 1 0 0.5

is not a transitive member of Q∗X . In [8,11], a stricter type of stochastic transitivity was introduced.

14

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Definition 7. Let h : [ 21 , 1]2 → [0, 1] be an increasing mapping such that h( 21 , 21 ) = 21 and h( 21 , 1) = h(1, 21 ) = 1. A reciprocal relation Q ∈ Q X is called h-isostochastic transitive (or hIS-transitive) if for any (x, y, z) ∈ X 3 it holds that  Q(x, y) ≥ 21 ∧ Q(y, z) ≥ 21 ⇒ Q(x, z) = h(Q(x, y), Q(y, z)). (30) The conditions imposed on the mapping h are more restrictive than those imposed on g in Definition 6. Again, these conditions are needed to guarantee that hIS-transitivity is a generalization of transitivity of complete crisp relations. Note that any mapping h satisfying the conditions of Definition 7 satisfies the conditions of Definition 6 and that hIS-transitivity implies gS-transitivity if h = g. All mappings g considered in Example 6 satisfy the conditions of Definition 7. 6.2. The existence of g-stochastic transitive closures In order to investigate the existence of gS-transitive closures of reciprocal relations, it is instructive to specify Theorem 2 to the present setting. Note that in this setting, a non-empty, lower bounded subset is nothing else but a non-empty family of compatible reciprocal relations, of which the greatest lower bound is just the intersection. Theorem 3. Consider the poset (Q X , ) and a type of gS-transitivity. The gS-transitive closure exists for all reciprocal relations if and only if (i) Q  is g-stochastic transitive; (ii) The intersection of every non-empty compatible family of gS-transitive reciprocal relations is gS-transitive. A necessary and sufficient condition guaranteeing the existence of gS-transitive closures can be established in case g( 21 , 21 ) = 21 . In view of its monotonicity, the mapping g can then be considered a [ 21 , 1]2 → [ 21 , 1] mapping. Recall that a mapping g : [ 21 , 1]2 → [ 21 , 1] is called left continuous if it is left continuous in each argument. For an increasing mapping g, left-continuity in the first argument is equivalent with

 sup g(ai , b) = g sup ai , b i∈I

i∈I

for any family (ai )i∈I and b in [ 21 , 1]. Theorem 4. Let g : [ 21 , 1]2 → [ 21 , 1] be an increasing mapping such that g( 21 , 21 ) = exists for all Q ∈ Q X if and only if g satisfies 

sup g(ai , bi ) = g sup ai , sup bi i∈I

i∈I

1 2.

The gS-transitive closure

(31)

i∈I

for any families (ai )i∈I and (bi )i∈I in [ 21 , 1]. Proof. We verify the conditions of Theorem 3. As mentioned before, Q  is gS-transitive. First, suppose that condition (31) holds. Let (Q i )i∈I be a compatible family of gS-transitive relations. Consider (x, y, z) ∈ X 3 and suppose that   Q i (x, y) ≥ 21 and Q i (y, z) ≥ 21 . (32) i∈I

It then holds that Q i (x, y) ≥

i∈I 1 2

and Q i (y, z) ≥

Q i (x, z) ≥ g(Q i (x, y), Q i (y, z)) ≥

1 2

1 2

for any i ∈ I . Since Q i is gS-transitive, it holds that

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

15

and hence  Q i (x, z) = sup Q i (x, z) i∈I

i∈I

≥ sup g(Q i (x, y), Q i (y, z)) i∈I



= g sup Q i (x, y), sup Q i (y, z)  =g

i∈I



i∈I

Q i (x, y),

i∈I



 Q i (y, z)

i∈I

which proves the gS-transitivity of i∈I Q i . Conversely, consider (ai )i∈I and (bi )i∈I in [ 21 , 1]. Consider (x, y, z) ∈ X 3 and construct the family (Q i )i∈I of reciprocal relations such that Q i (x, y) = ai , Q i (y, z) = bi and Q i (x, z) = g(ai , bi ), for any i ∈ I , and taking value 1/2 for all other couples in X 2 . Clearly, this is a compatible family of gS-transitive relations. By assumption, its intersection is gS-transitive as well, i.e.  sup g(ai , bi ) = Q i (x, z) i∈I

i∈I



≥g



Q i (x, y),

i∈I



 Q i (y, z)

i∈I



= g sup ai , sup bi . i∈I

i∈I

Since g is increasing, also the converse inequality holds, and (31) follows.  Note that condition (31) implies left continuity, but not conversely, as the following example shows. Example 9. The mapping g =  max +(1 − ) min satisfies (31) if and only if  = 1. Indeed, it is trivially satisfied by g = max. However, for a1 = b2 = 21 , a2 = b1 = 1 and  < 1, it holds that max(g ( 21 , 1), g (1, 21 )) = 21 (1 + ) < g (1, 1) = 1. When applied to  = 0, this means that reciprocal relations do not have a MS-transitive closure in general. Condition (31) identifying the mappings g for which gS-transitive closures exist is rather restrictive, as it is only satisfied by distortions of the maximum. This is expressed in the following representation theorem. For the sake of brevity, it is expressed for symmetric mappings g, but a similar theorem holds in the non-symmetric case as well. Theorem 5. Let g : [ 21 , 1]2 → [ 21 , 1] be an increasing mapping such that g( 21 , 21 ) = 21 . Then g is symmetric and fulfills (31) if and only if there exists a left-continuous increasing mapping  : [ 21 , 1] → [ 21 , 1], with ( 21 ) = 21 , such that g =  ◦ max .

(33)

Proof. Suppose that g is symmetric and equality (31) is fulfilled. In particular, for I = {1, 2}, it holds that max(g(a1 , b1 ), g(a2 , b2 )) = g(max(a1 , a2 ), max(b1 , b2 )). Let a1 = a0 , a2 = b0 , b1 = b0 and b2 = a0 , with a0 < b0 . The symmetry of g implies that g(a0 , b0 ) = max(g(a0 , b0 ), g(a0 , b0 )) = g(max(a0 , b0 ), max(a0 , b0 )). Let  : [ 21 , 1] → [ 21 , 1] be the mapping defined by (t) = g(t, t), then g can be written as g(a, b) = (max(a, b)). Obviously,  is increasing and satisfies ( 21 ) = 21 . Since g is left continuous (which is equivalent to saying that g is lower semi-continuous, in view of its monotonicity), the mapping  is left continuous as well.

16

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Conversely, suppose that g is of the given form, then it is obviously symmetric. Let (ai )i∈I and (bi )i∈I be two families in [ 21 , 1], then we need to show that

 sup (max(ai , bi )) =  sup max(ai , bi ) . i∈I

i∈I

Since  is increasing and left continuous, equality (31) readily follows.  Example 10. The family of mappings u : [ 21 , 1] → [ 21 , 1], with u ∈ [1, ∞[, defined by u (t) =

1 2

+ (t − 21 )u ,

(34)

satisfies the conditions of Theorem 5. The corresponding family of mappings gu : [ 21 , 1]2 → [ 21 , 1] defined by gu = u ◦ max, includes g1 = max with 1 = id and g∞ = 21 with ∞ = 21 , corresponding to SS- and WS-transitivity. Example 11. Another class of suitable mappings  can be obtained as follows. Consider an automorphism  of the unit interval, then the mapping  : [ 21 , 1] → [ 21 , 1] defined by  (t) =

(2t − 1) + 1 2

is increasing and satisfies  ( 21 ) = 2t 2 − 2t + 1.

1 2

and  (1) = 1. For instance, considering (t) = t 2 , one obtains  (t) =

Proposition 17. Let  : [ 21 , 1] → [ 21 , 1] be a left-continuous increasing mapping such that ( 21 ) = corresponding mapping g =  ◦ max. For any gS-transitive reciprocal relation Q it holds that Q(x, y) ≥

1 2

1 2

and consider the

⇒ Q(x, y) ≥ (Q(x, y))

for any (x, y) ∈ X 2 . Corollary 8. Under the conditions of Proposition 17, if (t0 ) > t0 for some t0 ∈] 21 , 1[, for any gS-transitive reciprocal relation Q it holds that Q(x, y) ∈ / [t0 , (t0 )[ for any (x, y) ∈ X 2 . Hence, the use of a mapping  exceeding the identity mapping poses a restriction on the values gS-transitive reciprocal relations can take. A similar phenomenon has occurred in the quest for a suitable definition of the concept of a fuzzy preference structure [13,34]. From a practical point of view, it might not be acceptable that even a single Q(x, y) cannot be chosen freely. This can lead to extreme situations, as is illustrated in the following example. Example 12. Consider the mapping  of Example 11 with (t) = √ 2t − 1 + 1 .  (t) = 2

√

t, i.e.

It holds that  (t) > t for all t ∈] 21 , 1[. If follows from Corollary 8 that Q ∈ Q X is gS-transitive, with g =  ◦ max, only if Q ∈ Q∗X and Q is transitive in Q∗X . Note that, with the mapping ∗ defined by 1 , if t = 21 ∗  (t) = 2 , (35) 1, if t > 21 g ∗ S-transitivity, with g ∗ = ∗ ◦ max, is equivalent to the above type of transitivity.

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

17

Example 13. Consider the mapping  of Example 11 with (t) = 4t 3 − 6t 2 + 3t, i.e.  (t) = 16t 3 − 36t 2 + 27t − 6. It holds that  (t) > t for all t ∈ ] 21 , 43 [. The corresponding gS-transitivity, with g =  ◦max, only allows gS-transitive reciprocal relations to take values in { 21 } ∪ [ 43 , 1]. This type of gS-transitivity is equivalent to the one corresponding to ⎧1 if t = 21 ⎪ ⎨ 2, (t) = 43 , if t ∈] 21 , 43 ] . ⎪ ⎩  (t), if t ∈] 43 , 1]

(36)

6.3. The existence of h-isostochastic transitive closures The investigation of the existence of hIS-transitive closures of reciprocal relations leads to the same condition as for gS-transitivity. The proof of Theorem 4 remains valid here. Hence, for a mapping h satisfying h( 21 , 1) = h(1, 21 ) = h(1, 1) = 1, hIS-transitive closures exist if and only if hS-transitive closures exist. In particular, h = max satisfies these conditions. Theorem 6. Let h : [ 21 , 1]2 → [ 21 , 1] be an increasing mapping such that h( 21 , 21 ) = h(1, 1) = 1. The hIS-transitive closure exists for all Q ∈ Q X if and only if h satisfies 

sup h(ai , bi ) = h sup ai , sup bi , i∈I

i∈I

1 2

and h( 21 , 1) = h(1, 21 ) =

(37)

i∈I

for any families (ai )i∈I and (bi )i∈I in [ 21 , 1]. Example 14. Recall that a reciprocal relation Q ∈ Q X is called multiplicatively transitive [5,33] if for any (x, y, z) ∈ X 3 it holds that Q(x, z) Q(x, y) Q(y, z) = · . Q(z, x) Q(y, x) Q(z, y) We have shown that this type of transitivity is equivalent to h-isostochastic transitivity [11] with h T defined by h T (x, y) =

xy . x y + (1 − x)(1 − y)

Note that the latter formulation is also more appropriate as it avoids division by zero. For a1 = b2 = x ∈] 21 , 1[, it holds that

1 2

and a2 = b1 =

 

x2 1 1 , x , h T x, =x 2 = h T (x, x). max h T 2 2 x + (1 − x)2 This means that reciprocal relations do not have a multiplicative transitive closure in general. Theorem 7. Let h : [ 21 , 1]2 → [ 21 , 1] be an increasing mapping such that h( 21 , 21 ) = 21 and h( 21 , 1) = h(1, 21 ) = h(1, 1) = 1. Then h is symmetric and fulfills (37) if and only if there exists a left-continuous increasing mapping  : [ 21 , 1] → [ 21 , 1], with ( 21 ) = 21 and (1) = 1, such that h =  ◦ max . Note that the mappings  considered in Example 11 all satisfy the conditions of Theorem 7.

(38)

18

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Proposition 18. Let  : [ 21 , 1] → [ 21 , 1] be a left-continuous increasing mapping such that ( 21 ) = 21 and (1) = 1 and consider the corresponding mapping h =  ◦ max. For any hIS-transitive reciprocal relation Q it holds that Q(x, y) ≥

1 2

⇒ Q(x, y) = (Q(x, y))

for any (x, y) ∈ X 2 . Hence, the use of a mapping  different from the identity mapping poses a restriction on the values hIS-transitive reciprocal relations can take. Consider for instance an hIS-transitive reciprocal relation Q such that Q(x, y) ≥ 1/2, Q(y, z) ≥ 1/2 and Q(x, z) ≥ 1/2. Then it holds that Q(x, u) = h(Q(x, y), Q(y, u)) = h(Q(x, y), g(Q(y, z), Q(z, u))) and Q(x, u) = h(Q(x, z), Q(z, u)) = h(Q(x, y), g(Q(y, z), Q(z, u))) whence at least for the triplet (Q(x, y), Q(y, z), Q(z, u)) the mapping h is associative [11]. Hence, if we do not want to restrict the range of hIS-transitive reciprocal relations, associativity naturally appears as an additional condition on h. However, in that case, the mapping  behaves as the identity mapping on its range. Theorem 8. Let  : [ 21 , 1] → [ 21 , 1] be a left-continuous increasing mapping such that ( 21 ) = 21 and (1) = 1, and consider the corresponding mapping h =  ◦ max. The mapping h is associative if and only if  = 2 . Moreover, if  is continuous, then h is associative if and only if  = id. Proof. Expressing the associativity of h leads to (max(t, (max(u, v)))) = (max((max(t, u)), v)). Letting u = v = 21 , it follows that (t) = 2 (t) for any t ∈ [ 21 , 1]. Conversely, if (t) = 2 (t), then h(t, h(u, v)) = (max(t, (max(u, v)))) = max((t), 2 (u), 2 (v)) = max((t), (u), (v)) = h(h(t, u), v). Now let  be continuous. Suppose that  is not strictly increasing. Consider a maximal interval [t1 , t2 ] on which  takes a constant value, say a. If a > t2 , then (a) > a, i.e. ((t2 )) > (t2 ), a contradiction. If a < t2 , then let u ∈] max(a, t1 ), t2 [. Due to the continuity of , there exists t ∈]t2 , 1] such that (t) = u. Hence a = (u) = ((t)) = (t), whence t ∈ [t1 , t2 ], a contradiction. Finally, if a = t2 , then let u ∈]t1 , t2 [. Due to the continuity of , there exists t ∈ [ 21 , t1 [ such that (t) = u. Hence, a = (u) = ((t)) = (t), whence t ∈ [t1 , t2 ], once again a contradiction. We conclude that  is strictly increasing. Since  is continuous, it has an inverse −1 and (t) = 2 (t) is equivalent to t = (t). 

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

19

Hence, max-isostochastic transitivity is the only type of hIS-transitivity, with h associative, that allows for closures and for which the full range is available for at least some Q(x, y). Example 15. Under the conditions of Theorem 8, it holds for any t ∈ [ 21 , 1] such that (t) < t, that  is constant on [(t), t]. In particular, if  ≤ id, there exists a disjoint family [ai , bi ]i∈I of subintervals of [ 21 , 1] such that (t) = ai , if t ∈ [ai , bi ] for some i ∈ I , and (t) = t otherwise. An example of such a mapping  is given by ⎧ 6 t, if 21 ≤ t ≤ 10 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6 6 7 ⎪ , if 10 ≤ t ≤ 10 ⎪ ⎪ ⎨ 10 7 8 (t) = t, if 10 . < t ≤ 10 ⎪ ⎪ ⎪ ⎪ 8 8 9 ⎪ ⎪ 10 , if 10 ≤ t ≤ 10 ⎪ ⎪ ⎪ ⎩ 9 t, if 10
(39)

satisfies the conditions of Theorem 7. For  = 21 , the corresponding h equals max. 7. Algorithms In this section, we investigate how to compute the closure of a given reciprocal relation for those types of gS- and hIS-transitivity for which closures exist. We consider a finite universe X = {x1 , . . . , xn }. 7.1. Transitive closures of crisp and 3-valued reciprocal relations An algorithm for computing the transitive closure of a crisp relation was proposed by Warshall [36]. This algorithm is listed in Algorithm 1 and is called TCCrisp. Algorithm 1. TCCrisp. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

Data: X, R ∈ P(X 2 ) Result: R ← R for all x ∈ X do for all y ∈ X do for all z ∈ X do if (R(y, x) = 1) ∧ (R(x, z) = 1) ∧ (R(y, z) = 0) then R(y, z) ← 1 end if end for end for end for

Due to Corollary 5, given a complete relation R, algorithm TCCrisp will produce the transitive closure R of R. Note that at any time during the nested iteration loops, when starting from a complete relation R, completeness is preserved. In the language of 3-valued reciprocal relations, Algorithm 1 is easily translated into Algorithm 2 called TC3Reciproc that returns the transitive closure of a given reciprocal relation Q ∈ Q∗X .

20

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Algorithm 2. TC3Reciproc. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

Data: X, Q ∈ Q∗X Result: Q ← Q for all x ∈ X do for all y ∈ X do for all z ∈ X do if (Q(y, x) ≥ 1/2) ∧ (Q(x, z) ≥ 1/2) ∧ (Q(y, z) < 1/2) then Q(y, z) ← 1/2 Q(z, y) ← 1/2 end if end for end for end for

7.2. Weakly stochastic transitive closures As mentioned before, the WS-transitive closure always exists. The following proposition provides an easy way of computing it. Lemma 1. For any Q ∈ Q X , it holds that Q ∗  Q(x, y) ≥

1 2

if and only if Q ∗ (x, y) ≥ 21 .

Proof. Let x, y ∈ X such that Q ∗  Q(x, y) ≥ 21 . We distinguish the following cases: (i) If Q ∗  Q(x, y) > 21 , then the definition of  implies that min(Q ∗ (x, y), Q(x, y)) > 21 , and hence Q ∗ (x, y) > 21 . (ii) If Q ∗  Q(x, y) = 21 , then the definition of  and the compatibility of Q ∗ and Q imply that Q ∗ (x, y) = Q(x, y) = 21 . However, if Q(x, y) = 21 , then Q ∗ (x, y) = 21 and hence also Q ∗ (x, y) = 21 .

1 2

and/or

Summarizing, it holds that Q ∗  Q(x, y) ≥ 21 implies Q ∗ (x, y) ≥ 21 . Similarly, one can show that Q ∗  Q(x, y) < implies Q ∗ (x, y) < 21 . Combined, this leads to the more elegant equivalence stated.  Theorem 9. For any Q ∈ Q X , its WS-transitive closure Q Q

WS

WS

1 2

is given by

= Q ∗  Q. WS

Proof. Obviously, Q  Q . Next, we show that Q ∗  Q is WS-transitive. Indeed, suppose that Q ∗  Q(x, y) ≥ 21 and Q ∗  Q(y, z) ≥ 21 , or, equivalently, according to Lemma 1, Q ∗ (x, y) ≥ 21 and Q ∗ (y, z) ≥ 21 , whence Q ∗ (x, z) ≥ 21 , or, equivalently, Q ∗  Q(x, z) ≥ 21 . Finally, suppose that for a given WS-transitive Q 1 such that Q  Q 1 it does not hold that Q always identify x, y ∈ X such that Q(x, y) < 21 and

WS

 Q 1 , then we can

Q 1 (x, y) < max(Q ∗ (x, y), Q(x, y)) ≤ 21 . The latter implies that Q ∗ (x, y) = 21 > Q(x, y) and there exists z ∈ X such that Q(x, z) ≥ 21 and Q(z, y) ≥ 21 . However, it then also holds that Q 1 (x, z) ≥ 21 and Q 1 (z, y) ≥ 21 , while Q 1 (x, y) < 21 , which contradicts the WStransitivity of Q 1 . 

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

21

One way of computing the WS-transitive closure of Q would be to compute Q ∗ using Algorithm TC3Reciproc and to subsequently take the union with the original Q. However, this can be realized in a single pass, as is expressed in Algorithm 3, called WSTCReciproc. Note that this algorithm is formally the same as TC3Reciproc, only differing in the fact that it now also takes general reciprocal relations as input. Its correctness is easily verified. Algorithm 3. WSTCReciproc. 1: Data: X, Q ∈ Q X WS

2: Result: Q ← Q 3: for all x ∈ X do 4: for all y ∈ X do 5: for all z ∈ X do 6: if (Q(y, x) ≥ 1/2) ∧ (Q(x, z) ≥ 1/2) ∧ (Q(y, z) < 1/2) then 7: Q(y, z) ← 1/2 8: Q(z, y) ← 1/2 9: end if 10: end for 11: end for 12: end for

7.3. Strongly stochastic transitive closures Computing the SS-transitive closure of a reciprocal relation Q requires the appropriate lowering of some Q(x, y) > 21 , some of which remain strictly greater than 21 , while others become 21 . The latter are exactly the same as the ones that have to be adapted in order to compute the WS-transitive closure. As these adaptations are indispensable, we propose to follow a two-step procedure: first we compute Q in Algorithm 4, called SSTCReciproc. Algorithm 4. SSTCReciproc. 1: Data: X, Q ∈ Q X 2: Result: Q ← Q 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:

WS

SS

Q←Q S ← {(x, y) ∈ X 2 | Q(x, y) > 21 } repeat Get (x, y) ∈ S with minimal Q(x, y) for all z ∈ X do if (Q(x, z) ≥ 21 ) ∧ (Q(z, y) ≥ 21 ) then Q(x, z) ← min(Q(x, z), Q(x, y)) Q(z, x) ← 1 − Q(x, z) Q(z, y) ← min(Q(z, y), Q(x, y)) Q(y, z) ← 1 − Q(z, y) end if end for S ← S \ {(x, y)} until S is empty

WS

, then we compute Q

WS

SS

SS

, i.e. Q . This procedure is expressed

22

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

For a given reciprocal relation, the algorithm first computes its WS-transitive closure (line 3), which can be done by invoking WSTCReciproc. The remainder of the algorithm consists of one main loop (lines 5–16) considering all arcs having a weight strictly greater than 21 . Moreover, these arcs are considered in increasing order of their actual weights, an ordering step successfully used in other transitive closure algorithms [12,28]. Note that each pass of the main loop can change the order of pending arcs. If at line 6 different arcs have the same minimal weight, one is chosen arbitrarily. In lines 8–13, given (x, y) with current minimal weight Q(x, y) > 21 , for all nodes z ∈ X the weights Q(x, z) and Q(z, y) are adjusted so that the triangular subgraph with nodes x, y, z is made strongly stochastic transitive. This is done by lowering Q(x, z) and Q(z, y) if necessary. Theorem 10. SSTCReciproc generates the SS-transitive closure for any Q ∈ Q X . Proof. We can assume that Q is WS-transitive. Let us call Q SS-transitive at (x, y) ∈ X 2 if (∀z ∈ X )(min(Q(x, z), Q(z, y)) ≥

1 2

⇒ Q(x, y) ≥ max(Q(x, z), Q(z, y))).

Consider the instance (x, y) = (xi , yi ) of the main loop of the algorithm. We denote by Q i−1 the intermediate result when entering and by Q i the result when leaving the corresponding pass. Note that Q i and Q i−1 only differ in the weights of (xi , z) and (z, yi ). For every z ∈ X we find that if Q i−1 (xi , z) ≥ 21 and Q i−1 (z, yi ) ≥ 21 , then max(Q i (xi , z), Q i (z, yi )) ≤ max(Q i−1 (xi , yi ), Q i−1 (xi , yi )) = Q i−1 (xi , yi ) = Q i (xi , yi ). Hence, Q i is SS-transitive at (xi , yi ). Moreover, the adjustments from Q i−1 to Q i are the minimal ones necessary for guaranteeing the SS-transitivity at (xi , yi ). Now assume that Q i−1 is SS-transitive at (x j , y j ) and that the instance (x, y) = (x j , y j ) has been treated in a previous pass of the main loop. Then, necessarily Q i−1 (xi , yi ) ≥ Q i−1 (x j , y j ). SS-transitivity at (x j , y j ) can only be violated if x j = xi or y j = yi . However, if x j = xi , then Q i (x j , y j ) = min(Q i−1 (x j , y j ), Q i−1 (xi , yi )) = Q i−1 (x j , y j ) and analogously if y j = yi . Hence, Q i is SS-transitive at (x j , y j ). As in every pass the necessary minimal adjustments are put in place, SSTCReciproc generates the smallest SStransitive relation greater than or equal to Q, i.e. the SS-transitive closure of Q.  7.4. g-Stochastic transitive closures The results of the previous section can be generalized to the case of gS-transitivity for the mappings g considered in Theorem 5, i.e. g =  ◦ max, with  : [ 21 , 1] → [ 21 , 1] a left-continuous increasing mapping such that ( 21 ) = 21 . Additionally, we demand that  ≤ id, excluding the mappings mentioned in Corollary 8. In the following, we use the pseudoinverse (−1) of  defined by (−1) (t) = sup{t  ∈ [ 21 , 1]|(t  ) < t}.

(40)

If  is strictly increasing, then (−1) = −1 . Note that the considered class includes  = id corresponding with SS-transitivity. For any mapping g in the considered class, we have g ≥ 21 . Hence, it holds that gS-transitivity is stronger than WS-transitivity and we find WS

gS

Q  Q  Q . Again, computing the gS-transitive closure of a given Q can be done in two steps. Algorithm 5, called GSTCReciproc, is an adaptation of SSTCReciproc.

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

23

Algorithm 5. GSTCReciproc. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:

Data: X, Q ∈ Q X , g, (−1) gS Result: Q ← Q WS Q←Q S ← {(x, y) ∈ X 2 |Q(x, y) > 21 } repeat Get (x, y) ∈ S with minimal Q(x, y) for all z ∈ X do if (Q(x, z) ≥ 21 ) ∧ (Q(z, y) ≥ 21 ) then Q(x, z) ← min(Q(x, z), (−1) (Q(x, y))) Q(z, x) ← 1 − Q(x, z) Q(z, y) ← min(Q(z, y), (−1) (Q(x, y))) Q(y, z) ← 1 − Q(z, y) end if end for S ← S\{(x, y)} until S is empty

Theorem 11. Let  : [ 21 , 1] → [ 21 , 1] be an increasing left-continuous mapping such that ( 21 ) = 21 and consider the corresponding mapping g =  ◦ max. If  ≤ id, then GSTCReciproc generates the gS-transitive closure for any Q ∈ QX . Proof. We can assume that Q is WS-transitive. Let us call Q gS-transitive at (x, y) ∈ X 2 if (∀z ∈ X )(min(Q(x, z), Q(z, y)) ≥

1 2

⇒ Q(x, y) ≥ g(Q(x, z), Q(z, y))).

Consider the instance (x, y) = (xi , yi ) of the main loop of the algorithm. We denote by Q i−1 the intermediate result when entering and by Q i the result when leaving the corresponding pass. Note that Q i and Q i−1 only differ in the weights of (xi , z) and (z, yi ). For every z ∈ X we find that if Q i−1 (xi , z) ≥ 21 and Q i−1 (z, yi ) ≥ 21 , then g(Q i (xi , z), Q i (z, yi )) ≤ (max((−1) (Q i−1 (xi , yi )), (−1) (Q i−1 (xi , yi )))) = Q i−1 (xi , yi ) = Q i (xi , yi ). Hence, Q i is gS-transitive at (xi , yi ). Moreover, the adjustments from Q i−1 to Q i are the minimal ones necessary for guaranteeing the gS-transitivity at (xi , yi ). Now assume that Q i−1 is gS-transitive at (x j , y j ) and that the instance (x, y) = (x j , y j ) has been treated in a previous pass of the main loop. Then, necessarily Q i−1 (xi , yi ) ≥ Q i−1 (x j , y j ). gS-transitivity at (x j , y j ) can only be violated if x j = xi or y j = yi . Since (t) ≤ t for any t ∈ [ 21 , 1], it follows that (−1) (t) ≥ t for any t ∈ [ 21 , 1]. Hence, if x j = xi , then Q i (x j , y j ) = min(Q i−1 (x j , y j ), (−1) (Q i−1 (xi , yi ))) = Q i−1 (x j , y j ), and analogously if y j = yi . Hence, Q i is gS-transitive at (x j , y j ). As in every pass the necessary minimal adjustments are put in place, GSTCReciproc generates the gS-transitive closure of Q.  7.5. h-Isostochastic transitive closures We have shown that in case of hIS-transitivity, associativity of h is a natural condition and implies that h =  ◦ max, with  : [ 21 , 1] → [ 21 , 1] a left-continuous and increasing mapping such that ( 21 ) = 21 , (1) = 1 and  = 2 . Additionally, we again demand that  ≤ id.

24

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

Due to the additional restrictions on h as compared to the restrictions on g for gS-transitivity, the latter is implied by hIS-transitivity when h = g. The computation of the hIS-transitive closure of a given reciprocal relation Q can therefore hS be achieved by computing the hIS-transitive closure of the hS-transitive closure Q of Q. This is realized in Algorithm 6, called HISTCReciproc. Taking into account that GSTCReciproc is a two-step algorithm, HISTCReciproc can be considered a three-step algorithm. Algorithm 6. HISTCReciproc. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

Data: X, Q ∈ Q X , h hIS Result: Q ← Q hS Q←Q for all x ∈ X do for all y ∈ X do for all z ∈ X do if (Q(y, x) ≥ 1/2) ∧ (Q(x, z) ≥ 1/2) then Q(y, z) ← h(Q(y, x), Q(x, z)) Q(z, y) ← 1 − Q(y, z) end if end for end for end for

Theorem 12. Let  : [ 21 , 1] → [ 21 , 1] be an increasing left-continuous mapping such that ( 21 ) = 21 and (1) = 1, and consider the corresponding mapping g =  ◦ max. If  ≤ id and h is associative, then HISTCReciproc generates the hIS-transitive closure for any Q ∈ Q X . Proof. We can assume that Q is hS-transitive. Let us call Q hIS-transitive at x ∈ X if (∀(y, z) ∈ X 2 )(min(Q(y, x), Q(x, z)) ≥

1 2

⇒ Q(y, z) = h(Q(y, x), Q(x, z))).

Consider the instance x = xi of the outer for-loop. We denote by Q i−1 the intermediate result when entering and by Q i the result when leaving this pass. For every (y, z) ∈ X 2 , if Q i−1 (y, xi ) ≥ 21 and Q i−1 (xi , z) ≥ 21 , then h(Q i (y, xi ), Q i (xi , z)) = h(Q i−1 (y, xi ), Q i−1 (xi , z)) = Q i (y, z). Hence, Q i is hIS-transitive at xi . Moreover, the adjustments made are minimal. Now assume that Q i−1 is hIS-transitive at x j , i  j, and that the instance x = x j has been treated in a previous pass. Consider (y, z) ∈ X 2 such that Q i−1 (y, x j ) ≥ 21 and Q i−1 (x j , z) ≥ 21 , then we need to show that h(Q i (y, x j ), Q i (x j , z)) = Q i (y, z), knowing that h(Q i−1 (y, x j ), Q i−1 (x j , z)) = Q i−1 (y, z). Obviously, we only have to consider the case Q i−1 (y, xi ) ≥

1 2

and Q i−1 (xi , z) ≥ 21 . Two cases need to be distinguished.

1. If Q i−1 (x j , xi ) ≥ 21 , then we have Q i−1 (y, xi ) = h(Q i−1 (y, x j ), Q i−1 (x j , xi )). The following adjustments will be made Q i (y, z) = h(Q i−1 (y, xi ), Q i−1 (xi , z)), Q i (x j , z) = h(Q i−1 (x j , xi ), Q i−1 (xi , z)).

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

25

The associativity of h implies that h(Q i (y, x j ), Q i (x j , z)) = h(Q i−1 (y, x j ), h(Q i−1 (x j , xi ), Q i−1 (xi , z))) = h(h(Q i−1 (y, x j ), Q i−1 (x j , xi )), Q i−1 (xi , z)) = h(Q i−1 (y, xi ), Q i−1 (xi , z)) = Q i (y, z). 2. If Q(xi , x j ) ≥ made

1 2,

then we have Q i−1 (xi , z) = h(Q i−1 (xi , x j ), Q i−1 (x j , z)). The following adjustments will be

Q i (y, z) = h(Q i−1 (y, xi ), Q i−1 (xi , z)), Q i (y, x j ) = h(Q i−1 (y, xi ), Q i−1 (xi , x j )). The associativity of h again implies that h(Q i (y, x j ), Q i (x j , z)) = h(h(Q i−1 (y, xi ), Q i−1 (xi , x j )), Q i−1 (x j , z)) = h(Q i−1 (y, xi ), h(Q i−1 (xi , x j ), Q i−1 (x j , z))) = h(Q i−1 (y, xi ), Q i−1 (xi , z)) = Q i (y, z). Hence, Q i is hIS-transitive at x j . As in every pass the necessary minimal adjustments are put in place, HISTCReciproc generates the hIS-transitive closure of Q.  8. Conclusion A first contribution of this paper lies in a refinement of the theorem of Bandler and Kohout [1] concerning the existence of closures of elements of a poset w.r.t. some given property. This refinement exposes the most general setting in which this existence can be characterized. However, it is still only concerned with the existence for all elements. It might be interesting to investigate weaker settings guaranteeing the existence of closures for some important subsets of elements only. The theorem presented here can be applied in particular to the poset of three-valued reciprocal relations or the poset of [0,1]-valued reciprocal relations, structures that have been given ample attention in this paper as well. We have focused our attention on different types of transitivity of reciprocal relations, and have provided a necessary and sufficient condition on g guaranteeing the existence of gS-transitive closures. When applied to h, the same condition guarantees the existence of hIS-transitive closures. In particular, we have shown that WS- and SS-transitive closures exist, while this is not the case for MS-transitivity. The existence of gS-transitive closures turns out to be limited to mappings g belonging to a class of distorted maxima, while the existence of hIS-transitive closures further restricts the suitable mappings h. Moreover, we have provided algorithms realizing these transitive closures for mappings g and h bounded by the identity mapping, a condition shown to exclude practically irrelevant situations anyhow. In future work, we will investigate a similar refinement of the theorem of Bandler and Kohout concerning the existence of openings of elements of a poset w.r.t. some given property. However, in the context of the poset of reciprocal relations, subtleties will arise due to the specific structure laid bare in this paper. The present work was also restricted to the most common types of transitivity, while also other types of transitivity (such as FG-transitivity [31] or cycle transitivity [11]), or even other properties of reciprocal relations, are worth investigating. Finally, the algorithms presented consist of two or three steps. Although at present we do not see how a single-step algorithm could be realized, this issue might deserve further attention. References [1] [2] [3] [4] [5]

W. Bandler, L.J. Kohout, Special properties, closures and interiors of crisp and fuzzy relations, Fuzzy Sets Syst. 26 (1988) 317–331. K. Basu, Fuzzy revealed preference theory, J. Econ. Theory 32 (1984) 212–227. J. Bezdek, B. Spillman, R. Spillman, A fuzzy relation space for group decision theory, Fuzzy Sets Syst. 1 (1978) 255–268. J. Bezdek, B. Spillman, R. Spillman, Fuzzy relation spaces for group decision theory: an application, Fuzzy Sets Syst. 2 (1978) 5–14. F. Chiclana, E. Herrera-Viedma, S. Alonso, F. Herrera, Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity, IEEE Trans. Fuzzy Syst. 17 (2009) 14–23. [6] B.A. Davey, H.A. Priestly, Introduction to Lattices and Order, Cambridge Mathematical Textbooks, 1990.

26

S. Freson et al. / Fuzzy Sets and Systems 241 (2014) 2 – 26

[7] B. De Baets, H. De Meyer, On the existence and construction of T-transitive closures, Inf. Sci. 152 (2003) 167–179. [8] B. De Baets, H. De Meyer, Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity, Fuzzy Sets Syst. 152 (2005) 249–270. [9] B. De Baets, H. De Meyer, On the cycle-transitive comparison of artificially coupled random variables, Internat. J. Approximate Reasoning 47 (2008) 306–322. [10] B. De Baets, H. De Meyer, K. De Loof, On the cycle-transitivity of the mutual rank probability relation of a poset, Fuzzy Sets Syst. 161 (2010) 2695–2708. [11] B. De Baets, H. De Meyer, B. De Schuymer, S. Jenei, Cyclic evaluation of transitivity of reciprocal relations, Soc. Choice Welfare 26 (2006) 217–238. [12] B. De Baets, H. De Meyer, H. Naessens, A top-down algorithm for generating the Hasse tree of a fuzzy preorder closure, IEEE Trans. Fuzzy Syst. 12 (2004) 838–848. [13] B. De Baets, B. Van de Walle, E. Kerre, A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. Part 2: the identity case, Fuzzy Sets Syst. 99 (1998) 303–310. [14] K. De Loof, B. De Baets, H. De Meyer, R. Brüggemann, A hitchhiker’s guide to poset ranking, Combinatorial Chem. High Throughput Screening 11 (2008) 734–744. [15] A. De Luca, S. Termini, Definition of nonprobabilistic entropy in setting of fuzzy sets theory, Inf. Control 20 (1972) 301–312. [16] H. De Meyer, B. De Baets, B. De Schuymer, On the transitivity of the comonotonic and countermonotonic comparison of random variables, J. Multivar. Anal. 98 (2007) 177–193. [17] H. De Meyer, H. Naessens, B. De Baets, Algorithms for computing the min-transitive closure and associated partition tree of a symmetric fuzzy relation, Eur. J. Oper. Res. 155 (2004) 226–238. [18] B. De Schuymer, H. De Meyer, B. De Baets, Cycle-transitive comparison of independent random variables, J. Multivar. Anal. 96 (2005) 352–373. [19] D. Dubois, H. Prade, An introduction to bipolar representations of information and preference, Internat. J. Intelligent Syst. 23 (2008) 866–877. [20] D. Dubois, H. Prade, Gradualness, uncertainty and bipolarity: making sense of fuzzy sets, Fuzzy Sets Syst. 192 (2012) 3–24. [21] M. Fedrizzi, M. Fedrizzi, R.A. Marques Pereira, Soft consensus and network dynamics in group decision making, Internat. J. Intelligent Syst. 14 (1999) 63–77. [22] P.C. Fishburn, Binary choice probabilities: on the varieties of stochastic transitivity, J. Math. Psychol. 10 (1972) 327–352. [23] R.W. Floyd, Algorithm 97: shortest path, Commun. ACM 5 (1962) 345. [24] J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers, 1994. [25] J. Kacprzyk, M. Fedrizzi, H. Nurmi, Group decision-making and consensus under fuzzy preferences and fuzzy majority, Fuzzy Sets Syst. 49 (1992) 21–31. [26] D. Martinetti, I. Montes, S. Diaz, S. Montes, A study on the transitivity of probabilistic and fuzzy relations, Fuzzy Sets Syst. 184 (2011) 156–170. [27] B. Monjardet, A generalization of probabilistic consistency: linearity conditions for valued preference relations, in: J. Kacprzyk, M. Roubens (Eds.), Non-Conventional Preference Relations in Decision Making, Lecture Notes in Economics and Mathematical Systems, vol. 301, 1988, pp. 37–53. [28] H. Naessens, H. De Meyer, B. De Baets, Algorithms for the computation of T-transitive closures, IEEE Trans. Fuzzy Syst. 104 (2002) 541–551. [29] H. Nurmi, Approaches to collective decision making with fuzzy preference relations, Fuzzy Sets Syst. 6 (1981) 249–259. [30] M. Rademaker, B. De Baets, Aggregation of monotone reciprocal relations with application to group decision making, Fuzzy Sets Syst. 184 (2011) 29–51. [31] Z. Switalski, General transitivity conditions for fuzzy reciprocal preference matrices, Fuzzy Sets Syst. 137 (2003) 85–100. [32] T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets Syst. 12 (1984) 117–131. [33] T. Tanino, Fuzzy preference relations in group decision making, in: J. Kacprzyk, M. Roubens (Eds.), Non-Conventional Preference Relations in Decision Making, Lecture Notes in Economics and Mathematical Systems, vol. 301, Springer-Verlag, Berlin, 1988, pp. 54–71. [34] B. Van de Walle, B. De Baets, E. Kerre, A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. Part 1: general argumentation, Fuzzy Sets Syst. 97 (1998) 349–359. [35] B. Van de Walle, B. De Baets, E. Kerre, Characterizable fuzzy preference structures, Ann. Oper. Res. 80 (1998) 105–136. [36] S. Warshall, A theorem on boolean matrices, J. ACM 9 (1962) 11–12. [37] E.C. Zeeman, The topology of the brain and visual perception, in: M.K. Fort (Ed.), The Topology of 3-Manifolds and Related Topics, PrenticeHall, Englewood Cliffs, NJ, 1962, pp. 240–256.