Transitive Closure of Interval-valued Fuzzy Relations

In this paper are introduced some concepts of interval-valued fuzzy relations and some of their properties: reflexivity, symmetry, T-transitivity, composition and local reflexivity. The existence and uniqueness of T-transitive closure of interval-valued fuzzy relations is proved. An algorithm to compute the T-transitive closure of finite interval-valued fuzzy relations is showed. Some properties and some examples is given for t-representable and t-pseudo representable generalized t-norms.

[1]  Francisco Herrera,et al.  Fuzzy Sets and Their Extensions: Representation, Aggregation and Models , 2008 .

[2]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning - II , 1975, Inf. Sci..

[3]  Etienne Kerre,et al.  On the cuts of intuitionistic fuzzy compositions , 2003 .

[4]  Etienne E. Kerre,et al.  On the composition of intuitionistic fuzzy relations , 2003, Fuzzy Sets Syst..

[5]  Humberto Bustince,et al.  Intuitionistic fuzzy relations (Part I). , 1995 .

[6]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[7]  Bernard De Baets,et al.  On the existence and construction of T-transitive closures , 2003, Inf. Sci..

[8]  Chris Cornelis,et al.  Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application , 2004, Int. J. Approx. Reason..

[9]  J. Goguen.  L-fuzzy sets , 1967 .

[10]  Humberto Bustince,et al.  Structures on intuitionistic fuzzy relations , 1996, Fuzzy Sets Syst..

[11]  Ivor Grattan-Guinness,et al.  Fuzzy Membership Mapped onto Intervals and Many-Valued Quantities , 1976, Math. Log. Q..

[12]  Bernard De Baets,et al.  Algorithms for the computation of T-transitive closures , 2002, IEEE Trans. Fuzzy Syst..

[13]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[14]  Chris Cornelis,et al.  Advances and challenges in interval-valued fuzzy logic , 2006, Fuzzy Sets Syst..

[15]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[16]  Humberto Bustince,et al.  On the relevance of some families of fuzzy sets , 2007, Fuzzy Sets Syst..

[17]  Humberto Bustince,et al.  Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets , 1996, Fuzzy Sets Syst..

[18]  Radko Mesiar,et al.  Triangular norms on product lattices , 1999, Fuzzy Sets Syst..

[19]  Barbara Pekala,et al.  Intuitionistic fuzzy preference relations , 2011, EUSFLAT Conf..

[20]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[21]  Glad Deschrijver,et al.  Arithmetic operators in interval-valued fuzzy set theory , 2007, Inf. Sci..

[22]  K. Jahn.  Intervall‐wertige Mengen , 1975 .

[23]  Barbara Pekala,et al.  Properties of Interval-Valued Fuzzy Relations, Atanassov's Operators and Decomposable Operations , 2010, IPMU.