# Bijective Fuzzy Relations - A Graded Approach

### Martina Daňková

#### Abstract

The bijectivity is one of the crucial mathematical notions. In this paper, we will present a fuzzy bijective mapping as a fuzzy relation that has several special properties. These properties come with degrees and so the bijectivity is also graded property. We will focuse on properties of this type of relations and we show graded versions of theorems on fuzzy bijections that are known from the classical Fuzzy Set Theory.

#### References

- Batyrshin, I., Sudkamp, T., Schockaert, S., Cock, M. D., Cornelis, C., and Kerre, E. E. (2008). Special section: Perception based data mining and decision support systems fuzzy region connection calculus: An interpretation based on closeness. International Journal of Approximate Reasoning, 48(1):332 - 347.
- Be?hounek, L., Bodenhofer, U., and Cintula, P. (2008). Relations in Fuzzy Class Theory: Initial steps. Fuzzy Sets and Systems, 159(14):1729-1772.
- Be?hounek, L. and Cintula, P. (2005). Fuzzy class theory. Fuzzy Sets and Systems, 154(1):34-55.
- Be?hounek, L. and Novák, V. (2015). Towards fuzzy partial logic. In Proceedings of the IEEE 45th International Symposium on Multiple-Valued Logics (ISMVL 2015), pages 139-144. IEEE.
- Be?lohlávek, R. (2002). Fuzzy Relational Systems: Foundations and Principles, volume 20 of IFSR International Series on Systems Science and Engineering. Kluwer Academic/Plenum Press, New York.
- Dan?ková, M. (2007a). On approximate reasoning with graded rules. Fuzzy Sets and Systems, 158(6):652 - 673. The Logic of Soft ComputingThe Logic of Soft Computing {IV} and Fourth Workshop of the {ERCIM} working group on soft computing.
- Dan?ková, M. (2007b). On approximate reasoning with graded rules. Fuzzy Sets and Systems, 158:652-673.
- Dan?ková, M. (2010a). Approximation of extensional fuzzy relations over a residuated lattice. Fuzzy Sets and Systems, 161(14):1973 - 1991. Theme: Fuzzy and Uncertainty Logics.
- Dan?ková, M. (2010b). Representation theorem for fuzzy functions - graded form. In Abstracts of International Conference on Fuzzy Computation, pages 56- 64, SciTePress - Science and Technology Publications.
- Dan?ková, M. (2010c). Representation theorem for fuzzy functions - graded form. In INTERNATIONAL CONFERENCE ON FUZZY COMPUTATION 2010, pages 56 - 64, Portugal. SciTePress - Science and Technology Publications.
- Dan?ková, M. (2011). Generalized implicative model of a fuzzy rule base and its properties. In Cognitive 2011, pages 116 - 121. IARIA.
- Demirci, M. (1999). Fuzzy functions and their fundamental properties. Fuzzy Sets and Systems, 106(2):239 - 246.
- Demirci, M. (2000). Fuzzy functions and their applications. Journal of Mathematical Analysis and Applications, 252(1):495 - 517.
- Demirci, M. (2001). Gradation of being fuzzy function. Fuzzy Sets and Systems, 119(3):383 - 392.
- Demirci, M. and Recasens, J. (2004). Fuzzy groups, fuzzy functions and fuzzy equivalence relations. Fuzzy Sets and Systems, 144(3):441 - 458.
- Hájek, P. (1998). Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordercht.
- 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. Physica-Verlag, Heidelberg.
- Novák, V., Perfilieva, I., and Moc?ko?r, J. (1999). Mathematical Principles of Fuzzy Logic. Kluwer, Dordrecht.
- Perfilieva, I. (2001). Logical approximation. Soft Computing, 2:73-78.
- Perfiljeva I., S?ostak. A. (2014). Fuzzy function and the generalized extension principle. In FCTA 2014, pages 169 - 174, Lisboa, Portugal. SCITEPRESS.
- S?te?pnic?ka M., D. B. B. (2013). Implication-based models of monotone fuzzy rule bases. FUZZY SET SYST, 232(1):134 - 155.

#### Paper Citation

#### in Harvard Style

Daňková M. (2016). **Bijective Fuzzy Relations - A Graded Approach** . In *Proceedings of the 8th International Joint Conference on Computational Intelligence - Volume 2: FCTA, (IJCCI 2016)* ISBN 978-989-758-201-1, pages 42-50. DOI: 10.5220/0006053300420050

#### in Bibtex Style

@conference{fcta16,

author={Martina Daňková},

title={Bijective Fuzzy Relations - A Graded Approach},

booktitle={Proceedings of the 8th International Joint Conference on Computational Intelligence - Volume 2: FCTA, (IJCCI 2016)},

year={2016},

pages={42-50},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006053300420050},

isbn={978-989-758-201-1},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 8th International Joint Conference on Computational Intelligence - Volume 2: FCTA, (IJCCI 2016)

TI - Bijective Fuzzy Relations - A Graded Approach

SN - 978-989-758-201-1

AU - Daňková M.

PY - 2016

SP - 42

EP - 50

DO - 10.5220/0006053300420050