The Archimedean property for t-norms in interval-valued fuzzy set theory

In this paper, the Archimedean property of t-norms on the lattice LI is introduced, where LI is the underlying lattice of interval-valued fuzzy set theory [R. Sambuc, Fonctions Φ-floues. Application à l’aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis, Université de Marseille, France, 1975.] and intuitionistic fuzzy set theory [K.T. Atanassov, Intuitionistic fuzzy sets, 1983, VII ITKR's Session, Sofia (deposed in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)]. Two additional variations on this property are also introduced: weakly and strong Archimedean property. We investigate the relationship with the (weak) limit property and obtain several necessary and sufficient conditions for a t-norm on LI to be Archimedean. We discuss the Archimedean property for special classes of t-norms on LI based on t-norms on the unit interval.