A Survey on Nullnorms on Bounded Lattices
Nullnorms are generalizations of triangular norms (t-norms) and triangular conorms (t-conorms) with a zero element to be an arbitrary point from an arbitrary bounded lattice. In this paper, we study nullnorms on bounded lattices. We examine some properties of nullnorms considering the concepts of idempotency, local internality, conjunctivity and disjunctivity on bounded lattices. We investigate relationships between such concepts for nullnorms on bounded lattices and some illustrative examples are added to clearly show connections between these. Moreover, we give two methods to obtain nullnorms on bounded lattices with a zero element by using the given nullnorm and t-norm (t-conorm) with some constraints.
The authors are very grateful to the anonymous reviewers and editors for their helpful comments and valuable suggestions.
- 3.Aşıcı, E.: An order induced by nullnorms and its properties. Fuzzy Sets Syst. doi: 10.1016/j.fss.2016.12.004
- 4.Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publishers, Providence (1967)Google Scholar
- 5.Calvo, T., Mesiar, R.: Distance operators. In: Proceedings of EUSFLAT 1999, Palma de Mallorca, pp. 363–366 (1999)Google Scholar
- 8.Çaylı, G.D., Ertuğrul, Ü., Köroğlu, T., Karaçal, F.: Notes on locally internal uninorms on bounded lattices. Kybernetika (Submitted)Google Scholar
- 9.Çaylı, G.D., KaraÇal, F.: Construction of uninorms on bounded lattices. Kybernetika 53, 394–417 (2017)Google Scholar
- 10.Çaylı, G.D.: On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets Syst. (in press) doi: 10.1016/j.fss.2017.07.015
- 13.Drygaś, P.: Isotonic operations with zero element in bounded lattices. In: Atanassov, K., Hryniewicz, O., Kacprzyk, J. (eds.) Soft Computing Foundations and Theoretical Aspect, EXIT Warszawa, pp. 181–190 (2004)Google Scholar
- 16.Ertuğrul, Ü., Kesicioğlu, M.N., Karaçal, F.: Ordering based on uninorms. Fuzzy Sets Syst. 330, 315–327 (2016)Google Scholar
- 18.Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation functions. Cambridge University Press, Cambridge (2009)Google Scholar
- 23.Mas, M., Mayor, G., Torrens, J.: t-Operators. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 7, 31–50 (1999)Google Scholar
- 26.Mesiar, R., Komorníková, M.: Classification of aggregation functions on bounded partially ordered sets. In: 2010 IEEE 8th International Symposium on Intelligent Systems and Informatics, SISY 2010, Subotica, Serbia, pp. 13–16 (2010)Google Scholar