The concept of inverse α-cuts in multi Q-fuzzy set

In various mathematical theories such as fuzzy sets, fuzzy multisets, fuzzy soft sets, the concept of α-Cuts were applied together with their inverses. However, we noticed that in multi Q-fuzzy sets only α-Cuts were studied without their inverses. In this paper the concept of inverse α-Cuts and their properties in multi Q-fuzzy sets were introduced. Some distinctive features of α-Cuts and inverse α-Cuts were demonstrated. It is shown that as both first and second decomposition theorems hold in the former, it actually fails in the latter. Moreover, unlike α-cuts, it was discovered that, a multi Q-fuzzy set cannot be uniquely represented as the family of all its weak inverse α-cuts. Thus, both α-cuts and inverse α-cuts attract applications in many mathematical fields.