Hesitant fuzzy relations and some of their operations are defined. A first definition of T-transitive property and its closure is defined.Peer ReviewedPostprint (author's final draft

Garmendia Salvador, Luis

González del Campo, Ramón

Recasens Ferrés, Jorge

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Hesitant fuzzy relations and their transitive closure  L.Garmendia 1, R. González del Campo 1, J. Recasens2  1 F. Informática, Universidad Complutense de Madrid, Spain  lgarmend@fdi.ucm.es, rgonzale@estad.ucm.es 2 Universitat Politecnica de Catalunya, Spain  j.recasens@upc.edu Abstract  The hesitant fuzzy relations and some of their operations are defined. Two definition of T-transitive property and their closures are defined.  Introduction The hesitant fuzzy sets are a generalization of fuzzy sets that allow to assign few or several membership degrees to the elements of a finite universe, introducing the concept of possible hesitation of their membership degrees.  Operations and partial order ≤H on Lists.  Let L <[0, 1]> be a finite sorted list of membership degrees in [0, 1], from now just noted L. Sorted list are Abstract Data Types (ADT) well know in computer programming sciences that allow at least to ask its length, ask whether it is empty, and add and remove elements keeping the order of the elements. Definition 1.  Let L be a sorted list of l membership degrees in [0, 1], let n be a natural number then L x n is a sorted list of l x n membership degrees adding very membership degree n times.  Example: {0.4, 0.8, 1} x 3 = {0.4, 0.4, 0.4, 0.8, 0.8, 0.8, 1, 1, 1} Definition 2.  Let L be a sorted list of l membership degrees in [0, 1], then L[i] is the ith element of the list L, for i = 1..l. Definition 3.  Let L1 and L2 be lists of length l1 and l2 . Then L1 ≤H L2  if and only if   L1x(l/l1)[i] ≤ L2x(l/l2)[i] for all i= 1.. mcm(l1, l2) = l.  ≤H is a partial order on the set of Lists. Definition 4. Let T be a triangular norm. The intersection TH  of 2 lists L1 and L2 of length l1 and l2 is a list TH(L1, L2)  of length l = mcm(l1, l2) such that TH(L1, L2) [i] = T(L1x(l/l1)[i], L2x(l/l2)[i]) for all i= 1.. l  Hesitant fuzzy sets and hesitant fuzzy sets bags  Let E be a finite universe of discourse. Definition 5.  A hesitant fuzzy set µH on a universe E is a mapping µH: E → L where L is a sorted list of membership degrees in [0, 1].  Let li be the length of the list Li = µH(ei). So li is the number of membership degrees of ei.  Let lH be the mcm of all li for all i in 1.card(E). Definition 6. A hesitant fuzzy set bag Bµ of a hesitant fuzzy set µH on a universe E is a mapping  Bµ: E → LB where LB is a sorted list of length lH and Bµ(ei) = Li x (lH/li) Example: Let E = {e1, e2}, Let µH = {{0.4, 0.6}, {0.5, 0.7, 1}} Then Bµ = {{0.4, 0.4, 0.4, 0.6, 0.6, 0.6}, {0.5, 0.5, 0.7, 0.7, 1, 1} } Note that{0.4, 0.6}≤H{0.5, 0.7, 1}cause{0.4, 0.4, 0.4, 0.6, 0.6, 0.6}≤{0.5, 0.5, 0.7, 0.7, 1, 1}  Hesitant fuzzy relations and hesitant fuzzy relations bags  Definition 7.  A hesitant fuzzy relation RH on a universe E is a mapping RH: ExE→ L where L is a sorted list of membership degrees in [0, 1].  Let lij be the length of the list Lij = RH(ei, ei).  Let lR be the mcm of all lij for all i, j in 1..card(E). Definition 8. A hesitant fuzzy relation bag BR of a hesitant fuzzy relation RH is a mapping  BR: E x E→ LB where LB is a sorted list of length lR and BR(ei, ei) = Lij x (lR/lij) Example:  Let E ={e1, e2}, Let µH = { {0.4, 0.8}, {0.5, 0.7, 1} } Let RH =  0}8.0,4.0{}1,7.0,5.0{1      Then  BR = }0,0,0,0,0,0{}8.0,8.0,8.0,4.0,4.0,4.0{}1,1,7.0,7.0,5.0,5.0{}1,1,1,1,1,1{  T-Transitive property and T-transitive closure of Hesitant fuzzy relations  Definition 9.  Let T be a triangular norm. A hesitant fuzzy relation RH on a universe E is T-transitive if and only if  TH(RH(a, b), RH(b, c) ) ≤H RH(a, c) for all a, b, c in E. Definition 10.  Let T be a triangular norm. Let RH be hesitant fuzzy relation on a universe E . The T transitive closure of  RH is the  ≤H-lowest relation RHT that is T-transitive and ≤H -contains  RH  Theorem 1.  The T-transitive closure of a hesitant fuzzy relation always exists and it is unique  Example: Let E = {e1, e2}, Let RH =  0}8.0,4.0{}1,7.0,5.0{1      Then  BR = }0,0,0,0,0,0{}8.0,8.0,8.0,4.0,4.0,4.0{}1,1,7.0,7.0,5.0,5.0{}1,1,1,1,1,1{ Then  RHMin , RHProd , RHW (closures for the t-norms min, product and Lucasiewicz) are RHMin = }8.0,8.0,7.0,4.0,4.0,4.0{}8.0,8.0,8.0,4.0,4.0,4.0{}1,1,7.0,7.0,5.0,5.0{}1,1,1,1,1,1{  RHProd = }8.0,8.0,56.0,28.0,2.0,2.0{}8.0,8.0,8.0,4.0,4.0,4.0{}1,1,7.0,7.0,5.0,5.0{}1,1,1,1,1,1{  RHW = }8.0,8.0,5.0,1.0,0,0{}6.0,6.0,6.0,4.0,4.0,4.0{}1,1,7.0,7.0,5.0,5.0{}1,1,1,1,1,1{ References    [1] J. Recasens. Indistinguishability Operators. Modelling Fuzzy Equallities and Fuzzy Equivalence Relations. Studies in Fuzziness and Soft Computing. Springer. 2011.    [2] V. Torra. Hesitat fuzzy sets. International Journal of Intelligent Systems 25-6  (2010) 529–539.   [3] R.R.Yager.  On the theory of bags. International Journal of General Systems, 13-1 (1986) 23–37. ______________________________       We thank the support of the research proyect  TIN2012-32482. 

Hesitant fuzzy relations and their transitive closure

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Hesitant fuzzy relations and their transitive closure

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