Studying On Balanced Triangular Norms And Balanced Fuzzy Relation Equations

Balanced triangular norm which is an extension of classical triangular norm was introduced by Wadyslaw Homenda aiming at resolving the problem of information aggregation. In this paper, basic properties of triangular norms are analyzed in detail. And a concept of balanced fuzzy relation equation with balanced triangular norm is introduced, as well as the problem of solving balanced fuzzy relation equation is studied primarily.The properties of balanced triangular norms were studied from two rather independent points of view, namely, the field of functional equations and the theory of semigroups. Continuity including continuity in one component and left-(right-) continuous is significant when balanced t-norms are binary function. Their notions are given and the relations between them which possess particularity due to the especial properties of balanced t-norms are discussed in this article. From an algebraic point of view, ([-1, 1], T,≤) is a totally ordered commutative semigroup with annihilator (zero element) 0 where T is a balanced t-norm. Firstly, idempotent and nilpotent elements and zero divisors of balanced t-norm are researched. Their investigative methods are similar to the ones which are used to investigate the properties of semigroup. Moreover, it is natural to consider additional algebraic properties a balanced t-norm may have such as monotone, Archimedean and limit property, a general construction method of balanced t-norms based on additive generators and the balanced t-conorms.The second keystone will concentrate on balanced fuzzy relation equation based on the composition of balanced fuzzy relation which was presented in this paper. As the classical fuzzy relation equation, the problem of solving balanced fuzzy relation equation is important. Tsukamoto method for solving sup-inf and sup-pro balanced fuzzy relation equation in finite field is showed.