On Rough Fuzzy Set Algebras

A rough fuzzy set is a pair of fuzzy sets resulting from the approximation of a fuzzy set in a crisp approximation space. A rough fuzzy set algebra is a fuzzy set algebra with added dual pair of rough fuzzy approximation operators. In this paper, structures of rough fuzzy set algebras are studied. It is proved that if a system $({\cal F}(U), \cap, \cup, \sim, L, H)$ is a (a serial, a reflexive, a symmetric, a transitive, an Euclidean, a Pawlak, respectively) rough fuzzy set algebra then the derived system $({\cal F}(U), \cap, \cup, \sim, LL, HH)$ is a (a serial, a reflexive, a symmetric, a transitive, an Euclidean, a Pawlak, respectively) rough fuzzy set algebra. Properties of rough fuzzy approximation operators in different types of rough fuzzy set algebras are also examined.

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

[2]  Wei-Zhi Wu,et al.  A Study on Relationship Between Fuzzy Rough Approximation Operators and Fuzzy Topological Spaces , 2005, FSKD.

[3]  Yiyu Yao.  Combination of Rough and Fuzzy Sets Based on α-Level Sets , 1997 .

[4]  C. J. V. Rijsbergen,et al.  Rough Sets, Fuzzy Sets and Knowledge Discovery , 1994, Workshops in Computing.

[5]  Yee Leung,et al.  Connections between rough set theory and Dempster-Shafer theory of evidence , 2002, Int. J. Gen. Syst..

[6]  Tsau Young Lin,et al.  Rough Sets and Data Mining: Analysis of Imprecise Data , 1996 .

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

[8]  Masahiro Inuiguchi.  Generalizations of Rough Sets: From Crisp to Fuzzy Cases , 2004, Rough Sets and Current Trends in Computing.

[9]  S. Nanda,et al.  Fuzzy rough sets , 1992 .

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

[11]  Stephen D. Comer,et al.  An algebraic approach to the approximation of information , 1991, Fundamenta Informaticae.

[12]  Daniel Vanderpooten,et al.  A Generalized Definition of Rough Approximations Based on Similarity , 2000, IEEE Trans. Knowl. Data Eng..

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

[14]  Zongben Xu,et al.  Rough Set Models on Two universes , 2004, Int. J. Gen. Syst..

[15]  A. Wiweger.  On topological rough sets , 1989 .

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

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

[18]  Marek Chuchro,et al.  On Rough Sets in Topological Boolean Algebras , 1993, RSKD.

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

[20]  Wei-Zhi Wu,et al.  Neighborhood operator systems and approximations , 2002, Inf. Sci..

[21]  Yee Leung,et al.  An uncertainty measure in partition-based fuzzy rough sets , 2005, Int. J. Gen. Syst..

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

[23]  Lech Polkowski,et al.  Rough Sets in Knowledge Discovery 2 , 1998 .

[24]  Yiyu Yao,et al.  Relational Interpretations of Neigborhood Operators and Rough Set Approximation Operators , 1998, Inf. Sci..

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

[26]  A. Skowron,et al.  Methodology and applications , 1998 .

[27]  J. Recasens,et al.  UPPER AND LOWER APPROXIMATIONS OF FUZZY SETS , 2000 .

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

[29]  Yiyu Yao,et al.  Interpretation of Belief Functions in The Theory of Rough Sets , 1998, Inf. Sci..

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

[31]  Sadaaki Miyamoto,et al.  Rough Sets and Current Trends in Computing , 2012, Lecture Notes in Computer Science.

[32]  Dimiter Vakarelov,et al.  A model logic for similarity relations in pawlak knowledge representation systems , 1991, Fundam. Informaticae.

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