Decomposition and construction of fuzzy sets and their applications to the arithmetic operations on fuzzy quantities

The membership function of a fuzzy set can be constructed from a family of subsets of a universal set via the form of resolution identity assuming that the family is nested. In this paper, we are going to consider the weak case of assuming that only a countable subfamily is nested. Using this generalized consideration, we can overcome the ambiguity of arithmetic operations on fuzzy quantities. Two well-known additions of fuzzy quantities in vector space have been adopted in the literature. One is based on the extension principle by directly considering the membership functions. Of course, this addition always exists. Another one uses the α -level sets without considering the membership functions. However, the existence of this addition cannot be guaranteed. In this paper, we are going to propose two other additions of fuzzy quantities in vector space. One generalizes the extension principle, and the other one also uses the α -level sets without considering the membership functions such that this generalized addition always exists. We can show that these four additions are equivalent.