Axiomatic characterizations of (S, T)-fuzzy rough approximation operators

Axiomatic characterizations of approximation operators are of importance in the study of rough set theory. In this paper axiomatic characterizations of relation-based (S, T)-fuzzy rough approximation operators are investigated. By employing a triangular conorm S and a triangular norm T on 0, 1, we first introduced the constructive definitions of S-lower and T-upper fuzzy rough approximation operators with their essential properties. We then propose an operator-oriented characterization of (S, T)-fuzzy rough sets, that is, fuzzy set-theoretic operators defined by axioms guarantee the existence of different types of fuzzy relations which produce the same operators. We show that the S-lower (and, respectively, T-upper) fuzzy rough approximation operators generated by a generalized fuzzy relation can be described by only one axiom. We further show that (S, T)-fuzzy rough approximation operators corresponding to special types of fuzzy relations, such as serial, reflexive, symmetric, and T-transitive ones as well as any of their compositions, can also be characterized by single axioms.

[1]  Xia Wang,et al.  Independence of axiom sets characterizing formal concepts , 2013, Int. J. Mach. Learn. Cybern..

[2]  Wei-Zhi Wu,et al.  Some Mathematical Structures of Generalized Rough Sets in Infinite Universes of Discourse , 2011, Trans. Rough Sets.

[3]  Bijan Davvaz,et al.  Axiomatic systems for rough set-valued homomorphisms of associative rings , 2013, Int. J. Approx. Reason..

[4]  Guilong Liu,et al.  Generalized rough sets over fuzzy lattices , 2008, Inf. Sci..

[5]  Wei-Zhi Wu,et al.  On Some Mathematical Structures of T-Fuzzy Rough Set Algebras in Infinite Universes of Discourse , 2011, Fundam. Informaticae.

[6]  Fei-Yue Wang,et al.  Reduction and axiomization of covering generalized rough sets , 2003, Inf. Sci..

[7]  Yan-Lan Zhang,et al.  On minimization of axiom sets characterizing covering-based approximation operators , 2011, Inf. Sci..

[8]  Witold Pedrycz,et al.  The Development of Fuzzy Rough Sets with the Use of Structures and Algebras of Axiomatic Fuzzy Sets , 2009, IEEE Transactions on Knowledge and Data Engineering.

[9]  Tsau Young Lin,et al.  Rough Approximate Operators: Axiomatic Rough Set Theory , 1993, RSKD.

[10]  Wei-Zhi Wu,et al.  Constructive and axiomatic approaches of fuzzy approximation operators , 2004, Inf. Sci..

[11]  Guilong Liu,et al.  Axiomatic systems for rough sets and fuzzy rough sets , 2008, Int. J. Approx. Reason..

[12]  Tong-Jun Li,et al.  The minimization of axiom sets characterizing generalized approximation operators , 2006, Inf. Sci..

[13]  Anna Maria Radzikowska,et al.  A comparative study of fuzzy rough sets , 2002, Fuzzy Sets Syst..

[14]  Yiyu Yao,et al.  Two views of the theory of rough sets in finite universes , 1996, Int. J. Approx. Reason..

[15]  Yiyu Yao,et al.  Generalization of Rough Sets using Modal Logics , 1996, Intell. Autom. Soft Comput..

[16]  Ming-Wen Shao,et al.  Generalized fuzzy rough approximation operators determined by fuzzy implicators , 2013, Int. J. Approx. Reason..

[17]  Xiaodong Liu,et al.  Nearness approximation space based on axiomatic fuzzy sets , 2012, Int. J. Approx. Reason..

[18]  Wei-Zhi Wu,et al.  Generalized fuzzy rough sets , 2003, Inf. Sci..

[19]  Xiao-Ping Yang,et al.  Minimization of axiom sets on fuzzy approximation operators , 2007, Inf. Sci..

[20]  Xizhao Wang,et al.  On the generalization of fuzzy rough sets , 2005, IEEE Transactions on Fuzzy Systems.

[21]  Yee Leung,et al.  Generalized fuzzy rough sets determined by a triangular norm , 2008, Inf. Sci..

[22]  Guilong Liu,et al.  Using one axiom to characterize rough set and fuzzy rough set approximations , 2013, Inf. Sci..

[23]  Guo-Jun Wang,et al.  An axiomatic approach of fuzzy rough sets based on residuated lattices , 2009, Comput. Math. Appl..

[24]  Yiyu Yao,et al.  Constructive and Algebraic Methods of the Theory of Rough Sets , 1998, Inf. Sci..

[25]  Yu Yang,et al.  Independence of axiom sets on intuitionistic fuzzy rough approximation operators , 2012, International Journal of Machine Learning and Cybernetics.

[26]  Wei-Zhi Wu,et al.  On axiomatic characterizations of three pairs of covering based approximation operators , 2010, Inf. Sci..

[27]  H. Thiele.  On axiomatic characterizations of fuzzy approximation operators. I. The fuzzy rough set based case , 2001 .

[28]  杨晓平,et al.  An axiomatic characterization of probabilistic rough sets , 2014 .

[29]  Helmut Thiele,et al.  On axiomatic characterization of fuzzy approximation operators. II. The rough fuzzy set based case , 2001, Proceedings 31st IEEE International Symposium on Multiple-Valued Logic.

[30]  Helmut Thiele,et al.  On axiomatic characterisations of crisp approximation operators , 2000, Inf. Sci..

[31]  Nehad N. Morsi,et al.  Axiomatics for fuzzy rough sets , 1998, Fuzzy Sets Syst..

[32]  Wen-Xiu Zhang,et al.  An axiomatic characterization of a fuzzy generalization of rough sets , 2004, Inf. Sci..

[33]  Jerzy W. Grzymala-Busse,et al.  Rough Sets , 1995, Commun. ACM.

[34]  Yee Leung,et al.  On characterizations of (I, J)-fuzzy rough approximation operators , 2005, Fuzzy Sets Syst..

[35]  Jinjin Li,et al.  Some minimal axiom sets of rough sets , 2015, Inf. Sci..