Kybernetika 55 no. 6, 976-993, 2019

An extension method for t-norms on subintervals to t-norms on bounded lattices

Funda Karaçal, Ümit Ertuğrul and M. Nesibe KesicioğluDOI: 10.14736/kyb-2019-6-0976

Abstract:

In this paper, a construction method on a bounded lattice obtained from a given t-norm on a subinterval of the bounded lattice is presented. The supremum distributivity of the constructed t-norm by the mentioned method is investigated under some special conditions. It is shown by an example that the extended t-norm on $L$ from the t-norm on a subinterval of $L$ need not be a supremum-distributive t-norm. Moreover, some relationships between the mentioned construction method and the other construction methods in the literature are presented.

Keywords:

bounded lattice, T-norm, construction method, subinterval

Classification:

03E72, 03B52, 03G10, 18B35

paper.pdf

References:

  1. G. Birkhoff: Lattice Theory. Third edition. Providence 1967.   DOI:10.1090/coll/025
  2. G. D. Çaylı: On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets and Systems 332 (2018), 129-143.   DOI:10.1016/j.fss.2017.07.015
  3. P. Drygaś and B. Pekala: Properties of decomposable operations on same etension of the fuzzy set theory. In: Advences in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics (K. T. Atanassov, O. Hryniewicz, J. Kacprzyk, M. Krawczak, Z. Nahorski, E. Szmidt, S. Zadrozny, eds.), Vol. I, Foundations, EXIT, Warszawa 2008, pp. 115-118.   CrossRef
  4. Ü. Ertuğrul, F. Karaçal and R. Mesiar: Modified ordinal sums of triangular norms and triangular conorms on bounded lattices. Int. J. Intell. Systems 30 (2015), 807-817.   DOI:10.1002/int.21713
  5. M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap: Aggregation Functions. Cambridge University Press, 2009.   DOI:10.1017/cbo9781139644150
  6. F. Karaçal: On the direct decomposability of strong negations and S-implication operators on product lattices. Inform. Sci. 176 (2006), 3011-3025.   DOI:10.1016/j.ins.2005.12.010
  7. F. Karaçal and D. Khadjiev: $\vee$-Distributive and infinitely $\vee$-distributive t-norms on complete lattices. Fuzzy Sets Systems 151 (2005), 341-352.   DOI:10.1016/j.fss.2004.06.013
  8. F. Karaçal and M. N. Kesicioğlu: A T-partial order obtained from t-norms. Kybernetika 47 (2011), 300-314.   CrossRef
  9. E.. P Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  10. J. Martin, G. Mayor and J. Torrens: On locally internal monotonic operations. Fuzzy Sets Syst. 137 (2003), 27-42.   DOI:10.1016/s0165-0114(02)00430-x
  11. K.Menger: Statistical metrics. Proc. Natl. Acad. Sci. USA 8 (1942), 535-537.   DOI:10.1073/pnas.28.12.535
  12. S. Saminger: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets Systems 157 (2006), 10, 1403-1416.   DOI:10.1016/j.fss.2005.12.021
  13. B. Schweizer and A. Sklar: Espaces metriques aléatoires. C. R. Acad. Sci. Paris Ser. A 247 (1958), 2092-2094.   CrossRef
  14. B. Schweizer and A. Sklar: Statistical metric spaces. Pacific J. Math. 10 (1960), 313-334.   DOI:10.2140/pjm.1960.10.313
  15. A. K. Tsadiras and R. G. Margaritis: The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons. Fuzzy Sets Syst. 93 (1998), 263-274.   DOI:10.1016/s0165-0114(96)00185-6