Generation of linear orders for intervals by means of aggregation functions

The problem of choosing an appropriate total order is crucial for many applications that make use of extensions of fuzzy sets. In this work we introduce the concept of an admissible order as a total order that extends the usual partial order between intervals. We propose a method to build these admissible orders in terms of two aggregation functions and we prove that some of the most used examples of total orders that appear in the literature are specific cases of our construction.

[1]  R. Mesiar,et al.  ”Aggregation Functions”, Cambridge University Press , 2008, 2008 6th International Symposium on Intelligent Systems and Informatics.

[2]  Zeshui Xu,et al.  Choquet integrals of weighted intuitionistic fuzzy information , 2010, Inf. Sci..

[3]  Humberto Bustince,et al.  A class of aggregation functions encompassing two-dimensional OWA operators , 2010, Inf. Sci..

[4]  L. A. ZADEH,et al.  The concept of a linguistic variable and its application to approximate reasoning - I , 1975, Inf. Sci..

[5]  Radko Mesiar,et al.  Aggregation functions on bounded partially ordered sets and their classification , 2011, Fuzzy Sets Syst..

[6]  Humberto Bustince,et al.  Interval-Valued Fuzzy Sets Applied to Stereo Matching of Color Images , 2011, IEEE Transactions on Image Processing.

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

[8]  Humberto Bustince,et al.  Interval-valued fuzzy sets constructed from matrices: Application to edge detection , 2009, Fuzzy Sets Syst..

[9]  Francisco Herrera,et al.  A genetic tuning to improve the performance of Fuzzy Rule-Based Classification Systems with Interval-Valued Fuzzy Sets: Degree of ignorance and lateral position , 2011, Int. J. Approx. Reason..

[10]  Jean-Pierre Bourguignon,et al.  Mathematische Annalen , 1893 .

[11]  Z. S. Xu,et al.  The uncertain OWA operator , 2002, Int. J. Intell. Syst..

[12]  Humberto Bustince,et al.  On averaging operators for Atanassov's intuitionistic fuzzy sets , 2011, Inf. Sci..

[13]  Zeshui Xu,et al.  Some geometric aggregation operators based on intuitionistic fuzzy sets , 2006, Int. J. Gen. Syst..

[14]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[15]  Lotfi A. Zadeh,et al.  Fuzzy sets and systems , 1990 .

[16]  Humberto Bustince,et al.  Consensus in multi-expert decision making problems using penalty functions defined over a Cartesian product of lattices , 2014, Inf. Fusion.

[17]  L. Brouwer.  Über Abbildung von Mannigfaltigkeiten , 1911 .