On the correspondence between reciprocal relations and strongly complete fuzzy relations

Abstract Fuzzy relations and reciprocal relations are two popular tools for representing degrees of preference. It is important to note that they carry a different semantics and cannot be equated directly. We propose a simple transformation based on implication operators that allows to establish a one-to-one correspondence between both formalisms. It sets the basis for a common framework in which properties such as transitivity can be studied and definitions belonging to different formalisms can be compared. As a byproduct, we propose a new family of upper bound functions for cycle-transitivity. Finally, we unveil some interesting equivalences between types of transitivity that were left uncompared till now.

[1]  Luciano Basile Deleting inconsistencies in nontransitive preference relations , 1996 .

[2]  Bernard De Baets,et al.  Cyclic Evaluation of Transitivity of Reciprocal Relations , 2006, Soc. Choice Welf..

[3]  J. Kacprzyk,et al.  Group decision making and consensus under fuzzy preferences and fuzzy majority , 1992 .

[4]  D. Dubois,et al.  An introduction to bipolar representations of information and preference , 2008 .

[5]  Balasubramaniam Jayaram,et al.  On the continuity of residuals of triangular norms , 2010 .

[6]  Francisco Herrera,et al.  Some issues on consistency of fuzzy preference relations , 2004, Eur. J. Oper. Res..

[7]  B. De Baets,et al.  Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity , 2005, Fuzzy Sets Syst..

[8]  Bernard De Baets,et al.  Basic properties of implicators in a residual framework. , 1999 .

[9]  Bernard De Baets,et al.  Characterizable fuzzy preference structures , 1998, Ann. Oper. Res..

[10]  Sándor Jenei,et al.  Continuity of left-continuous triangular norms with strong induced negations and their boundary condition , 2001, Fuzzy Sets Syst..

[11]  Bernard De Baets,et al.  A frequentist view on cycle-transitivity of reciprocal relations , 2015, Fuzzy Sets Syst..

[12]  Zbigniew Switalski,et al.  General transitivity conditions for fuzzy reciprocal preference matrices , 2003, Fuzzy Sets Syst..

[13]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[14]  Michal Baczynski,et al.  (S, N)- and R-implications: A state-of-the-art survey , 2008, Fuzzy Sets Syst..

[15]  Humberto Bustince,et al.  A review of the relationships between implication, negation and aggregation functions from the point of view of material implication , 2016, Inf. Sci..

[16]  Bernard De Baets,et al.  On the cycle-transitive comparison of artificially coupled random variables , 2008, Int. J. Approx. Reason..

[17]  B. Baets,et al.  Residual implicators of continuous t-norms , 1996 .

[18]  B. De Baets,et al.  On the transitivity of the comonotonic and countermonotonic comparison of random variables , 2007 .

[19]  B. De Baets,et al.  On the Cycle-Transitivity of the Dice Model , 2003 .

[20]  P. Fishburn Binary choice probabilities: on the varieties of stochastic transitivity , 1973 .

[21]  Bernard De Baets,et al.  Closing reciprocal relations w.r.t. stochastic transitivity , 2014, Fuzzy Sets Syst..

[22]  Bernard De Baets,et al.  Transitivity Bounds in Additive Fuzzy Preference Structures , 2007, IEEE Transactions on Fuzzy Systems.

[23]  Bernard De Baets,et al.  On the cycle-transitivity of the mutual rank probability relation of a poset , 2010, Fuzzy Sets Syst..

[24]  B. De Baets,et al.  Cycle-transitive comparison of independent random variables , 2005 .

[25]  Susana Montes,et al.  A study on the transitivity of probabilistic and fuzzy relations , 2011, Fuzzy Sets Syst..

[26]  T. Tanino Fuzzy preference orderings in group decision making , 1984 .