Document Type: Research Paper


University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic


Powerset structures of extensional fuzzy sets in sets with similarity relations are investigated. It is proved that extensional fuzzy sets have powerset structures which are powerset theories in the category of sets with similarity relations, and some of these powerset theories are defined also by algebraic theories (monads). Between Zadeh's fuzzy powerset theory and the classical powerset theory there exists a strong relation, which can be represented as a homomorphism. Analogical results are also proved for new powerset theories of extensional fuzzy sets.


[1] C. De Mitri and C. Guido, Some remarks on fuzzy powerset operators, Fuzzy Sets and Systems
126 (2002), 241-251.
[2] G. File, and F. Ranzato, Improving Abstract Interpretations by Systematic Lifting to the
Powerset, Proceedings of the 1994 International Symposium on Logic programming, MIT
Press Cambridge, MA, USA (1994), 655{669.
[3] A. Frascella and C. Guido, Transporting many-valued sets along many-valued relations, Fuzzy
Sets and Systems 159(1) (2008), 1-22.
[4] A. Frascella and C. Guido, Structured lattices and ground categories of L-sets, Int. J. Math.
and Math. Sci. 17 (2005), 2783-2803.
[5] G. Georgescu and A. Popescu, Non-dual fuzzy connections, Archive Math. Logic 43 (2004),
[6] G. Gerla and L. Scarpati, Extension principles for fuzzy set theory, Journal of Information
Sciences 106 (1998), 49{69.
[7] H. Herrlich and C. G. Strecker, Category Theory, Sigma Ser. Pure Math. Vol. 1, Heldermann,
Berlin, 1979.
[8] U. Hohle, Fuzzy sets and sheaves. Part I, Basic concepts, Fuzzy Sets and Systems 158 (2007),
[9] E. G. Manes, Algebraic Theories, Springer-Verlag, Berlin, New York, 1976.
[10] E. G. Manes, A class of fuzzy theories, Journal of mathematical Analysis and Applications,
(85) (1982), 409{451.
[11] J. Mockor, Extensional subobjects in categories of
-fuzzy sets, Czech.Math.J. 57(132)
(2007), 631{645.
[12] J. Mockor, Morphisms in categories of sets with similarity relations, Proceedings of IFSA
Congress/EUSFLAT Conference. Lisabon (2009), 560{568.
[13] J. Mockor, Cut systems in sets with similarity relations, Fuzzy Sets and Systems, (161)
(2010), 3127{3140.
[14] J. Mockor, Fuzzy sets and cut systems in a category of sets with similarity relations, Soft
Computing 16 (2012), 101{107.
[15] H. T. Nguyen, A note on the extension principle for Fuzzy sets, J. Math. Anal. Appl., 64
(1978), 369{380.
[16] V. Novak and I. Perfi lijeva and J. Mockor, Mathematical Principles of Fuzzy Logic , Kluwer
Academic Publishers, Boston, Dordrecht, London, 1999.
[17] S. E. Rodabaugh, Powerset operator based foundation for point-set lattice theoretic (poslat)
fuzzy set theories and topologies, Quaestiones Mathematicae 20(3) (1997), 463{530.
[18] S. E. Rodabaugh, Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topo-
logical Theories for Lattice-Valued Mathematics, International Journal of Mathermatics and
Mathematical Sciences, (2007), 1{71.
[19] S. E. Rodabaugh, Relationship of algebraic theories to powersets over objects in Set and
Set  C, Fuzzy Sets and Systems, 161(3) (2010), 453{470.
[20] K. I. Rosenthal, Quantales and Their Applications , Pittman Res. Notes in Math. 234, Longman,
Burnt Mill, Harlow, 1990.
[21] S. A. Solovyov, Powerset operator foundations for catalc fuzzy set theories, Iranian Journal
of Fuzzy Systems 8(2) (2011), 1{46.
[22] R. R. Yager, A characterization of the extension principle, Fuzzy Sets and Systems, 18
(1996), 205{217.
[23] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.