The axiomatic characterizations on L-fuzzy covering-based approximation operators

Abstract Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations or L-fuzzy coverings that reproduce the approximation operators. Axiomatic characterizations of approximation operators based on L-fuzzy coverings have not been fully explored, although those based on L-fuzzy relations have been studied thoroughly. Focusing on three pairs of widely used L-fuzzy covering-based approximation operators, we establish an axiom set for each of them, and their independence is examined. It should be noted that the axiom set of each L-fuzzy covering-based approximation operator is different from its crisp counterpart, with an either new or stronger axiom included in the L-fuzzy version.

[1]  C.J.H. Mann,et al.  Fuzzy Relational Systems: Foundations and Principles , 2003 .

[2]  Radim Belohlávek,et al.  Concept lattices of isotone vs. antitone Galois connections in graded setting: Mutual reducibility revisited , 2012, Inf. Sci..

[3]  S.P. Tiwari,et al.  Fuzzy rough sets, fuzzy preorders and fuzzy topologies , 2013, Fuzzy Sets Syst..

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

[5]  Bao Qing Hu,et al.  Fuzzy variable precision rough sets based on residuated lattices , 2015, Int. J. Gen. Syst..

[6]  R. Löhner.  Approximation of Operators , 2008 .

[7]  H. Priestley,et al.  Distributive Lattices , 2004 .

[8]  William Zhu,et al.  On Three Types of Covering-Based Rough Sets , 2014, IEEE Transactions on Knowledge and Data Engineering.

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

[10]  Xiao-Ping Yang.  The Minimization of Axiom Sets Characterizing Generalized Fuzzy Rough Approximation Operators , 2005, FSKD.

[11]  Yiyu Yao,et al.  Neighborhood systems and approximate retrieval , 2006, Inf. Sci..

[12]  Fei-Yue Wang,et al.  On three types of covering-based rough sets via definable sets , 2007, 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[13]  Fang Jinming,et al.  Stratified L-ordered convergence structures , 2010, Fuzzy Sets Syst..

[14]  Ying Sai,et al.  A comparison of two types of rough sets induced by coverings , 2009, Int. J. Approx. Reason..

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

[16]  Jinming Fang,et al.  Relationships between L-ordered convergence structures and strong L-topologies , 2010, Fuzzy Sets Syst..

[17]  R. Belohlávek.  Fuzzy Closure Operators , 2001 .

[18]  Qiu Jin,et al.  One-axiom characterizations on lattice-valued closure (interior) operators , 2016, J. Intell. Fuzzy Syst..

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

[20]  Dexue Zhang,et al.  Fuzzy preorder and fuzzy topology , 2006, Fuzzy Sets Syst..

[21]  Radim Belohlávek,et al.  Fuzzy closure operators II: induced relations, representation, and examples , 2002, Soft Comput..

[22]  Zheng Pei,et al.  Approximation operators on complete completely distributive lattices , 2013, Inf. Sci..

[23]  Bao Qing Hu,et al.  Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets , 2013, Inf. Sci..

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

[25]  Radim Belohlávek,et al.  Fuzzy interior operators , 2004, Int. J. Gen. Syst..

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

[27]  Guilong Liu,et al.  The relationship among different covering approximations , 2013, Inf. Sci..

[28]  William Zhu,et al.  Relationship between generalized rough sets based on binary relation and covering , 2009, Inf. Sci..

[29]  Kai Hu,et al.  The lower and upper p-topological (p-regular) modifications for lattice-valued convergence spaces , 2016, Fuzzy Sets Syst..

[30]  Qingguo Li,et al.  Construction of rough approximations in fuzzy setting , 2007, Fuzzy Sets Syst..

[31]  Qingguo Li,et al.  On enriched L-topologies: Base and subbase , 2015, J. Intell. Fuzzy Syst..

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

[33]  Yu-Ru Syau,et al.  Neighborhood systems and covering approximation spaces , 2014, Knowl. Based Syst..

[34]  Changqing Li,et al.  Relationships between generalized rough sets based on covering and reflexive neighborhood system , 2015, Inf. Sci..

[35]  Dexue Zhang,et al.  An enriched category approach to many valued topology , 2007, Fuzzy Sets Syst..

[36]  Qingguo Li,et al.  The relationship between L-fuzzy rough set and L-topology , 2011, Fuzzy Sets Syst..

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

[38]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

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

[40]  Guoyin Wang,et al.  Generalized approximations defined by non-equivalence relations , 2012, Inf. Sci..

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

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

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

[44]  Qiu Jin,et al.  On stratified L-convergence spaces: Pretopological axioms and diagonal axioms , 2012, Fuzzy Sets Syst..

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

[46]  Yiyu Yao,et al.  Covering based rough set approximations , 2012, Inf. Sci..

[47]  Radim Bělohlávek,et al.  Fuzzy Relational Systems: Foundations and Principles , 2002 .

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

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

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

[51]  Fangfang Zhao,et al.  Reduction of Rough Set Based on Generalized Neighborhood System Operator , 2016 .

[52]  Qingguo Li,et al.  Homomorphisms-based attribute reduction of dynamic fuzzy covering information systems , 2015, Int. J. Gen. Syst..

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

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

[55]  Stanislav Krajci,et al.  Formal concept analysis of higher order , 2013, CLA.

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

[57]  Theresa Beaubouef,et al.  Rough Sets , 2019, Lecture Notes in Computer Science.

[58]  Yong Chan Kim,et al.  Join-meet approximation operators and fuzzy preorders , 2015, J. Intell. Fuzzy Syst..

[59]  Radim Belohlávek,et al.  Closure Spaces of Isotone Galois Connections and Their Morphisms , 2011, Australasian Conference on Artificial Intelligence.

[60]  Qionghai Dai,et al.  A novel approach to fuzzy rough sets based on a fuzzy covering , 2007, Inf. Sci..

[61]  Eduard Bartl,et al.  Residuated lattices of block relations: size reduction of concept lattices , 2016, Int. J. Gen. Syst..

[62]  Andrei Popescu,et al.  Non-dual fuzzy connections , 2004, Arch. Math. Log..