On the T-partial order and properties

For a t-norm T on [0,1], a partial order @?"T was recently defined and studied. In this study, for any fixed [email protected]?(0,1), we define the set of incomparable elements according to @?"T and this set is deeply investigated. By means of the equivalence relation, defined in [14], it is shown the set [0,1]/~, denoting the set of all equivalence classes on t-norms on [0,1] is uncountably infinite. Finally, with the help of any t-norm T on [0,1], it is obtained that the family (T"@l)"@l"@?"("0","1") of t-norms on [0,1]. If T is a divisible t-norm, then it is obtained that ([0,1],@?"T"""@l) is a lattice. Thus, we give an answer to an open problem in [10].

[1]  S. Gottwald,et al.  Triangular norm-based mathematical fuzzy logics , 2005 .

[2]  Funda Karaçal,et al.  Pairwise comaximal elements and the classification of all ∨-distributive triangular norms of length 3 , 2012, Inf. Sci..

[3]  B. Schweizer,et al.  Statistical metric spaces. , 1960 .

[4]  Funda Karaçal,et al.  A t-partial order obtained from t-norms , 2011, Kybernetika.

[5]  Joan Torrens,et al.  Residual implications on the set of discrete fuzzy numbers , 2013, Inf. Sci..

[6]  Radko Mesiar,et al.  Triangular norms on product lattices , 1999, Fuzzy Sets Syst..

[7]  Vladimír Janis,et al.  Cuts of intuitionistic fuzzy sets respecting fuzzy connectives , 2013, Inf. Sci..

[8]  H. Mitsch,et al.  A natural partial order for semigroups , 1986 .

[9]  Jun-Lin Lin,et al.  On fuzzy relational equations and the covering problem , 2011, Inf. Sci..

[10]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

[11]  Joan Torrens,et al.  On a class of operators for expert systems , 1993, Int. J. Intell. Syst..

[12]  Funda Karaçal,et al.  On the direct decomposability of strong negations and S-implication operators on product lattices , 2006, Inf. Sci..

[13]  Józef Drewniak,et al.  Fuzzy equations max - * with conditionally cancellative operations , 2012, Inf. Sci..

[14]  L. Valverde,et al.  On Some Logical Connectives for Fuzzy Sets Theory , 1983 .

[15]  Radko Mesiar,et al.  Triangular norms on the real unit square , 1999, EUSFLAT-ESTYLF Joint Conf..

[16]  Funda Karaçal,et al.  Infinitely OR -distributive t-norms on complete lattices and pseudo-complements , 2009, Fuzzy Sets Syst..

[17]  Funda Karaçal,et al.  Distributive and infinitely-distributive t-norms on complete lattices , 2005, Fuzzy Sets Syst..

[18]  Zhudeng Wang.  TL-filters of integral residuated l-monoids , 2007, Inf. Sci..

[19]  Witold Pedrycz,et al.  Logic-based fuzzy networks: A study in system modeling with triangular norms and uninorms , 2009, Fuzzy Sets Syst..