Using one axiom to characterize L-fuzzy rough approximation operators based on residuated lattices

Abstract Axiomatic characterization of approximation operators plays an important role in the study of rough set theory. Different axiom sets of abstract operators can illustrate different classes of rough set systems. In this paper, we are devoted to searching for a single axiom to characterize L-fuzzy rough approximation operators based on residuated lattices. Axioms of L-fuzzy set theoretic operators make sure of the existence of certain types of L-fuzzy relations which produce the same operators. We demonstrate that the lower (upper) L-fuzzy rough approximation operators generated by a generalized L-fuzzy relation can be characterized by only one axiom. Furthermore, we also use one axiom to characterize L-fuzzy rough approximation operators produced by the L-fuzzy serial, reflexive, symmetric and T -transitive relations as well as any of their compositions.

[1]  Bao Qing Hu,et al.  Fuzzy rough sets based on generalized residuated lattices , 2013, Inf. Sci..

[2]  Daniel S. Yeung,et al.  Rough approximations on a complete completely distributive lattice with applications to generalized rough sets , 2006, Inf. Sci..

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

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

[5]  Z. Pawlak.  Rough Sets: Theoretical Aspects of Reasoning about Data , 1991 .

[6]  Wei-Zhi Wu,et al.  Using One Axiom to Characterize Fuzzy Rough Approximation Operators Determined by a Fuzzy Implication Operator , 2015, Fundam. Informaticae.

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

[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]  Bao Qing Hu,et al.  Granular variable precision L-fuzzy rough sets based on residuated lattices , 2016, Fuzzy Sets Syst..

[10]  Yiyu Yao,et al.  The two sides of the theory of rough sets , 2015, Knowl. Based Syst..

[11]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[12]  E. Turunen.  Mathematics Behind Fuzzy Logic , 1999 .

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

[14]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[15]  Jing-Yu Yang,et al.  Constructive and axiomatic approaches to hesitant fuzzy rough set , 2014, Soft Comput..

[16]  Yunqiang Yin,et al.  Fuzzy Roughness in Hyperrings Based on a Complete Residuated Lattice , 2011 .

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

[18]  Guoyin Wang,et al.  Axiomatic characterizations of (S, T)-fuzzy rough approximation operators , 2016, Inf. Sci..

[19]  Yee Leung,et al.  Generalized fuzzy rough approximation operators based on fuzzy coverings , 2008, Int. J. Approx. Reason..

[20]  D. Dubois,et al.  ROUGH FUZZY SETS AND FUZZY ROUGH SETS , 1990 .

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

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

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

[24]  Witold Pedrycz,et al.  Kernelized Fuzzy Rough Sets and Their Applications , 2011, IEEE Transactions on Knowledge and Data Engineering.

[25]  P. Dao.  The Characterization of Residuated Lattices and Regular Residuated Lattices , 2002 .

[26]  Anna Maria Radzikowska,et al.  Fuzzy Rough Sets Based on Residuated Lattices , 2004, Trans. Rough Sets.

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

[28]  Liwen Ma,et al.  Two fuzzy covering rough set models and their generalizations over fuzzy lattices , 2016, Fuzzy Sets Syst..

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

[30]  Chun Yong Wang,et al.  Single axioms for lower fuzzy rough approximation operators determined by fuzzy implications , 2017, Fuzzy Sets Syst..

[31]  Bao Qing Hu,et al.  L -fuzzy multigranulation rough set based on residuated lattices , 2016, J. Intell. Fuzzy Syst..

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

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

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

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

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

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

[38]  Zhudeng Wang,et al.  Some properties of L-fuzzy approximation spaces based on bounded integral residuated lattices , 2014, Inf. Sci..

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

[40]  Jianming Zhan,et al.  Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice , 2011, Neural Computing and Applications.

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

[42]  Bin Yang,et al.  A fuzzy covering-based rough set model and its generalization over fuzzy lattice , 2016, Inf. Sci..

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

[44]  Jan Pavelka,et al.  On Fuzzy Logic I Many-valued rules of inference , 1979, Math. Log. Q..

[45]  Didier Dubois,et al.  Putting Rough Sets and Fuzzy Sets Together , 1992, Intelligent Decision Support.

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

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

[48]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

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