Some properties of equivalence soft set relations

The soft set theory is a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Babitha and Sunil [Soft set relations and function, Computers and Mathematics with Applications 60 (2010) 1840-1849] introduced the notion of soft set relations as a soft subset of the Cartesian product of soft sets and discussed many related concepts such as equivalence soft set relations, partitions and functions. In this paper, we further study the equivalence soft set relations and obtain soft analogues of many results concerning ordinary equivalence relations and partitions. Furthermore, we introduce and discuss the transitive closure of a soft set relation and prove that the poset of the equivalence soft set relations on a given soft set is a complete lattice with the least element and greatest element.

[1]  Sunil Jacob John,et al.  Soft set relations and functions , 2010, Comput. Math. Appl..

[2]  Yan Zou,et al.  Data analysis approaches of soft sets under incomplete information , 2008, Knowl. Based Syst..

[3]  H. Enderton.  Elements of Set Theory , 1977 .

[4]  Young Bae Jun,et al.  Soft semirings , 2008, Comput. Math. Appl..

[5]  Tsau Young Lin,et al.  Combination of interval-valued fuzzy set and soft set , 2009, Comput. Math. Appl..

[6]  Pinaki Majumdar,et al.  Generalised fuzzy soft sets , 2010, Comput. Math. Appl..

[7]  Muhammad Irfan Ali,et al.  Algebraic structures of soft sets associated with new operations , 2011, Comput. Math. Appl..

[8]  A. R. Roy,et al.  A fuzzy soft set theoretic approach to decision making problems , 2007 .

[9]  Aslihan Sezgin Sezer,et al.  On operations of soft sets , 2011, Comput. Math. Appl..

[10]  Shwu-Yeng T. Lin,et al.  Set theory: An intuitive approach , 1973 .

[11]  Janusz Zalewski,et al.  Rough sets: Theoretical aspects of reasoning about data , 1996 .

[12]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[13]  Daowu Pei,et al.  From soft sets to information systems , 2005, 2005 IEEE International Conference on Granular Computing.

[14]  E. Kerre.  A First View on the Alternatives of Fuzzy Set Theory. , 2001 .

[15]  Sunny Joseph Kalayathankal,et al.  A fuzzy soft flood alarm model , 2010, Math. Comput. Simul..

[16]  Pabitra Kumar Maji,et al.  FUZZY SOFT SETS , 2001 .

[17]  Xiaoyan Liu,et al.  On some new operations in soft set theory , 2009, Comput. Math. Appl..

[18]  Muhammad Shabir,et al.  On soft topological spaces , 2011, Comput. Math. Appl..

[19]  D. Dubois,et al.  Fuzzy Sets: History and Basic Notions , 2000 .

[20]  W.-L. Gau,et al.  Vague sets , 1993, IEEE Trans. Syst. Man Cybern..

[21]  Keyun Qin,et al.  On soft equality , 2010, J. Comput. Appl. Math..

[22]  Liqun Gao,et al.  Letter to the editor: Comment on A fuzzy soft set theoretic approach to decision making problems , 2009 .

[23]  Naim Çagman,et al.  Soft sets and soft groups , 2007, Inf. Sci..

[24]  Pabitra Kumar Maji,et al.  More on Intuitionistic Fuzzy Soft Sets , 2009, RSFDGrC.

[25]  Wieslaw A. Dudek,et al.  Soft MTL-algebras based on fuzzy sets , 2010 .

[26]  Jian Ma,et al.  Vague soft sets and their properties , 2010, Comput. Math. Appl..

[27]  Hai Liu,et al.  Interval-valued intuitionistic fuzzy soft sets and their properties , 2010, Comput. Math. Appl..

[28]  Athar Kharal,et al.  On Fuzzy Soft Sets , 2009, Adv. Fuzzy Syst..

[29]  A. R. Roy,et al.  An application of soft sets in a decision making problem , 2002 .

[30]  Young Bae Jun,et al.  Soft ordered semigroups , 2010, Math. Log. Q..

[31]  S. Varadhan,et al.  Probability Theory , 2001 .

[32]  Halis Aygün,et al.  Introduction to fuzzy soft groups , 2009, Comput. Math. Appl..

[33]  A. R. Roy,et al.  Soft set theory , 2003 .

[34]  M. Gorzałczany.  A method for inference in approximate reasoning based on interval-valued fuzzy sets , 1987 .

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

[36]  Zhi Xiao,et al.  A combined forecasting approach based on fuzzy soft sets , 2009 .

[37]  Hai-Long Yang,et al.  Computers and Mathematics with Applications Kernels and Closures of Soft Set Relations, and Soft Set Relation Mappings , 2022 .

[38]  Jianming Zhan,et al.  Soft p-ideals of soft BCI-algebras , 2009, Comput. Math. Appl..

[39]  K. Atanassov.  Operators over interval valued intuitionistic fuzzy sets , 1994 .

[40]  D. Molodtsov.  Soft set theory—First results , 1999 .

[41]  Young Bae Jun,et al.  Soft BCK/BCI-algebras , 2008, Comput. Math. Appl..

[42]  Young Bae Jun,et al.  Applications of soft sets in ideal theory of BCK/BCI-algebras , 2008, Inf. Sci..

[43]  Naim Çagman,et al.  Soft matrix theory and its decision making , 2010, Comput. Math. Appl..

[44]  Young Bae Jun,et al.  An adjustable approach to fuzzy soft set based decision making , 2010, J. Comput. Appl. Math..

[45]  Lotfi A. Zadeh,et al.  Is there a need for fuzzy logic? , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.