Categorical Properties of Soft Sets

The present study investigates some novel categorical properties of soft sets. By combining categorical theory with soft set theory, a categorical framework of soft set theory is established. It is proved that the category SFun of soft sets and soft functions has equalizers, finite products, pullbacks, and exponential properties. It is worth mentioning that we find that SFun is both a topological construct and Cartesian closed. The category SRel of soft sets and Z-soft set relations is also characterized, which shows the existence of the zero objects, biproducts, additive identities, injective objects, projective objects, injective hulls, and projective covers. Finally, by constructing proper adjoint situations, some intrinsic connections between SFun and SRel are established.

[1]  Shyamal Kumar Mondal,et al.  A balanced solution of a fuzzy soft set based decision making problem in medical science , 2012, Appl. Soft Comput..

[2]  Won Keun Min Similarity in soft set theory , 2012, Appl. Math. Lett..

[3]  Juan Lu,et al.  Categorical properties of M-indiscernibility spaces , 2011, Theor. Comput. Sci..

[4]  D. A. Molodtsov,et al.  Soft sets theory-based optimization , 2007 .

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

[6]  Tingquan Deng,et al.  An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets , 2013 .

[7]  Mustafa Mat Deris,et al.  A soft set approach for association rules mining , 2011, Knowl. Based Syst..

[8]  Rajab Ali Borzooei,et al.  The category of soft sets , 2015, J. Intell. Fuzzy Syst..

[9]  Alexander P. Sostak,et al.  Categories Related to Topology Viewed as Soft Sets , 2011, EUSFLAT Conf..

[10]  Jiri Mockor,et al.  Fuzzy sets and cut systems in a category of sets with similarity relations , 2012, Soft Comput..

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

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

[13]  Michael Barr,et al.  Category theory for computing science , 1995, Prentice Hall International Series in Computer Science.

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

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

[16]  Jin-Han Park,et al.  Some properties of equivalence soft set relations , 2012, Comput. Math. Appl..

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

[18]  Giuseppe Longo,et al.  Categories, types and structures - an introduction to category theory for the working computer scientist , 1991, Foundations of computing.

[19]  Naim Çagman,et al.  Soft set theory and uni-int decision making , 2010, Eur. J. Oper. Res..

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

[21]  Gunther Schmidt,et al.  Relational Mathematics , 2010, Encyclopedia of Mathematics and its Applications.

[22]  Elbert A. Walker,et al.  Categories with fuzzy sets and relations , 2014, Fuzzy Sets Syst..

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

[24]  Feng Feng,et al.  Soft subsets and soft product operations , 2013, Inf. Sci..

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

[26]  Muhammad Shabir,et al.  SOFT IDEALS AND GENERALIZED FUZZY IDEALS IN SEMIGROUPS , 2009 .

[27]  Ajoy Kumar Ray,et al.  Texture Classification Using a Novel, Soft-Set Theory Based Classification Algorithm , 2006, ACCV.

[28]  Ionel Bucur,et al.  Toposes, Algebraic Geometry and Logic , 1972 .

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

[30]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

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

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

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

[34]  Murat Diker,et al.  Categories of rough sets and textures , 2013, Theor. Comput. Sci..

[35]  Ju Wang,et al.  Extending soft sets with description logics , 2010, Comput. Math. Appl..

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

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

[38]  Sergey A. Solovyov,et al.  Lattice-valued soft algebras , 2013, Soft Comput..