Algebraic Structures for Rough Sets

Using as example an incomplete information system with support a set of objects X, we discuss a possible algebraization of the concrete algebra of the power set of X through quasi BZ lattices. This structure enables us to define two rough approximations based on a similarity and on a preclusive relation, with the second one always better that the former. Then, we turn our attention to Pawlak rough sets and consider some of their possible algebraic structures. Finally, we will see that also Fuzzy Sets are a model of the same algebras. Particular attention is given to HW algebra which is a strong and rich structure able to characterize both rough sets and fuzzy sets.

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

[2]  Andrzej Skowron,et al.  New Directions in Rough Sets, Data Mining, and Granular-Soft Computing , 1999, Lecture Notes in Computer Science.

[3]  Ewa Orlowska,et al.  Relational Methods for Computer Science Applications , 2001 .

[4]  D. Vanderpooten.  Similarity Relation as a Basis for Rough Approximations , 1995 .

[5]  Z. Pawlak.  Rough sets and fuzzy sets , 1985 .

[6]  Gianpiero Cattaneo,et al.  Generalized Rough Sets (Preclusivity Fuzzy-Intuitionistic (BZ) Lattices) , 1997, Stud Logica.

[7]  Beloslav Riečan,et al.  Sets and fuzzy sets , 1997 .

[8]  Gianpiero Cattaneo,et al.  Heyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets , 2002, Rough Sets and Current Trends in Computing.

[9]  D Ciucci,et al.  BZW algebras for an abstract approach to roughness and fuzziness , 2002 .

[10]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[11]  Andrzej Skowron,et al.  Discovery of Data Patterns with Applications to Decomposition and Classification Problems , 1998 .

[12]  Salvatore Greco,et al.  Fuzzy Similarity Relation as a Basis for Rough Approximations , 1998, Rough Sets and Current Trends in Computing.

[13]  Ivo Düntsch,et al.  Beyond modalities: sufficiency and mixed algebras , 2001 .

[14]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[15]  Piero Pagliani,et al.  Rough Sets and Nelson Algebras , 1996, Fundam. Informaticae.

[16]  Alexis Tsoukiàs,et al.  On the Extension of Rough Sets under Incomplete Information , 1999, RSFDGrC.

[17]  Zdzisław Pawlak,et al.  Rough sets: a new approach to vagueness , 1992 .

[18]  A. Monteiro.  Sur les algèbres de Heyting symétriques , 1980 .

[19]  J. Kacprzyk,et al.  Incomplete Information: Rough Set Analysis , 1997 .

[20]  Gianpiero Cattaneo,et al.  BZMVdM algebras and stonian MV-algebras (applications to fuzzy sets and rough approximations) , 1999, Fuzzy Sets Syst..

[21]  Gianpiero Cattaneo,et al.  A Quantitative Analysis of Preclusivity vs. Similarity Based Rough Approximations , 2002, Rough Sets and Current Trends in Computing.

[22]  Ewa Orlowska,et al.  Introduction: What You Always Wanted to Know about Rough Sets , 1998 .

[23]  Z. Pawlak.  Rough set elements , 1998 .

[24]  Marzena Kryszkiewicz,et al.  Rough Set Approach to Incomplete Information Systems , 1998, Inf. Sci..

[25]  M. Wajsberg.  Beiträge zum Metaaussagenkalkül I , 1935 .

[26]  Jaroslaw Stepaniuk,et al.  Approximation Spaces in Extensions of Rough Set Theory , 1998, Rough Sets and Current Trends in Computing.

[27]  Zdzislaw Pawlak,et al.  Information systems theoretical foundations , 1981, Inf. Syst..

[28]  L. Zadeh,et al.  Fuzzy Logic for the Management of Uncertainty , 1992 .

[29]  Brian F. Chellas.  Modal Logic: Normal systems of modal logic , 1980 .

[30]  Andrzej Skowron,et al.  Tolerance Approximation Spaces , 1996, Fundam. Informaticae.

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

[32]  Alexis Tsoukiàs,et al.  Valued Tolerance and Decision Rules , 2000, Rough Sets and Current Trends in Computing.

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

[34]  P. Pagliani.  Rough Set Theory and Logic-Algebraic Structures , 1998 .

[35]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

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