THE DECOMPOSITION THEOREM AND THE REPRESENTATION THEOREM UNDER THE TRIANGULAR NORMS OPERATORS

摘要

The decomposition theorem of the fuzzy sets reveals the relation of the fuzzy sets and the classical sets; and the representation theorem shows how to use nested sets to construct a fuzzy set, but these results are all gotten under Zadeh operators. In this paper, at first, we discuss the decomposition theorem under the general triangular norms operators, and gain the necessary and sufficient condition of the theorem, namely: the "or" operator must be max, but the "and" operator may be an arbitrary T operator of the triangular norms; secondly, we discuss the representation theorem, and get the following result: the necessary and sufficient conditions of the two equations of the representation theorem under S norm are all that the "or" operator must be max.