Fuzzification using the extension principle and the expression in decomposition theorem

Fuzzifying the crisp functions is a well-known methodology to study the problems under fuzzy uncertainty. The traditional way for fuzzifying the crisp functions is based on the extension principle. In this paper, by referring to the expression in decomposition theorem, we present a new methodology to fuzzify the crisp functions for non-normal fuzzy sets. The main purpose of this paper is to establish the equivalence between the fuzzy functions obtained from the extension principle and the expression in decomposition theorem under the proposed concept of compatibility. The practical cases regarding the equivalences are also studied in order to be used for the practical problems.