# Fuzzy Function and the Generalized Extension Principle

### Irina Perfilieva, Alexandr Šostak

#### Abstract

The aim of this contribution is to develop a theory of such concepts as fuzzy point, fuzzy set and fuzzy function in a similar style as is common in classical mathematical analysis. We recall some known notions and propose new ones with the purpose to show that, similarly to the classical case, a (fuzzy) set is a collection of (fuzzy) points or singletons. We show a relationship between a fuzzy function and its ordinary “skeleton” that can be naturally associated with the original function. We show that any fuzzy function can be extended to the domain of fuzzy subsets and this extension is analogous to the Extension Principle of L. A. Zadeh.

#### References

- De Baets, B. and Mesiar, R. (1998). T-partitions. Fuzzy Sets Systems, 97:211-223.
- Demirci, M. (1999). Fuzzy functions and their fundamental properties. Fuzzy Sets and Systems, 106:239 - 246.
- Demirci, M. (2002). Fundamentals of m-vague algebra and m-vague arithmetic operations. Int. Journ. Uncertainty, Fuzziness and Knowledge-Based Systems, 10:25-75.
- Hájek, P. (1998). Metamathematics of Fuzzy Logic. Kluwer, Dordrecht.
- Höhle, U. (1998). Many-valued equalities, singletons and fuzzy partitions. Soft Computing, 2:134-140.
- Höhle, U., Porst, H., and S? ostak, A. (2000). Fuzzy functions: a fuzzy extension of the category set and some related categories. Applied General Topology, 1:115- 127.
- Klawonn, F. (2000). Fuzzy points, fuzzy relations and fuzzy functions. In Novák, V. and Perfilieva, I., editors, Discovering the World with Fuzzy Logic, pages 431-453. Springer, Berlin.
- Klawonn, F. and Castro, J. L. (1995). Similarity in fuzzy reasoning. Mathware Soft Comput., pages 197-228.
- Novák, V. (1989). Fuzzy Sets and Their Applications. Adam Hilger, Bristol.
- Perfilieva, I. (2004). Fuzzy function as an approximate solution to a system of fuzzy relation equations. Fuzzy Sets and Systems, 147:363-383.
- Perfilieva, I. (2011). Fuzzy function: Theoretical and practical point of view. In Proc. 7th conf. European Society for Fuzzy Logic and Technology, EUSFLAT 2011, AixLes-Bains, France, July 18-22, 2011, pages 480-486. Atlantis Press.
- Perfilieva, I., Dubois, D., Prade, H., Godo, L., Esteva, F., and Hod?áková, P. (2012). Interpolation of fuzzy data: analytical approach and overview. Fuzzy Sets and Systems, 192:134158.
- S?ostak, A. (2001). Fuzzy functions and an extension of the category l-top of chang-goguen l-topological spaces. In Proc. 9th Prague Topological Symp., pages 271- 294, Prague, Czech Republlic.
- Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning I, II, III. Information Sciences, 8-9:199-257, 301-357, 43-80.

#### Paper Citation

#### in Harvard Style

Perfilieva I. and Šostak A. (2014). **Fuzzy Function and the Generalized Extension Principle** . In *Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)* ISBN 978-989-758-053-6, pages 169-174. DOI: 10.5220/0005132701690174

#### in Bibtex Style

@conference{fcta14,

author={Irina Perfilieva and Alexandr Šostak},

title={Fuzzy Function and the Generalized Extension Principle},

booktitle={Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)},

year={2014},

pages={169-174},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005132701690174},

isbn={978-989-758-053-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)

TI - Fuzzy Function and the Generalized Extension Principle

SN - 978-989-758-053-6

AU - Perfilieva I.

AU - Šostak A.

PY - 2014

SP - 169

EP - 174

DO - 10.5220/0005132701690174