On the convergence exponent of decomposable relations

This paper deals with an α-decomposable fuzzy relation on [0, 1]. The α-decomposition problem of a fuzzy relation can be stated as follows: given a fuzzy relation R, determine whether there exist two fuzzy sets A and B such that R=AαB, where the operator α is nothing but Godel implication. At first, two necessary and sufficient conditions that a fuzzy relation R is α-decomposable are given. Then the solution set of A and B with AαB=R is obtained. In the end, the convergence of the α-decomposable fuzzy relation is investigated and an algorithm is shown to calculate the convergence exponent for a given α-decomposable fuzzy relation.