A potential of fuzzy relations with a linear structure: the unbounded case

This paper is a sequel to Yoshida et al. (1993), in which the potential theory for linear fuzzy relations on the positive orthant R+n is considered in the class of fuzzy sets with a compact support under the contractive assumption. In this paper, potential treatment for unbounded fuzzy sets is developed without the assumption of contraction and compactness. The objective of this paper is to give the existence and the characterization of potentials for linear fuzzy relations under some reasonable conditions. Also, introducing a partial order in fuzzy sets, we prove Riesz decomposition theorem in the fuzzy potential theory. The proofs are shown by using only the linear structure and the monotonicity of fuzzy relations. In the one-dimensional case, the potential and its α-cuts are explicitly calculated. Numerical examples are given to comprehend further discussions.