Arithmetic operators in interval-valued fuzzy set theory

We introduce the addition, subtraction, multiplication and division on LILI, where LILI is the underlying lattice of both 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)]. We investigate some algebraic properties of these operators. We show that using these operators the pseudo-t-representable extensions of the Łukasiewicz t-norm and the product t  -norm on the unit interval to LILI and some related operators can be written in a similar way as their counterparts on ([0,1],⩽)([0,1],⩽).