Analysis of the  \(\varphi \) -Index of Inclusion Restricted to a Set of Indexes

The \(\varphi \) -index of inclusion has proven to be a suitable generalization of the inclusion in the fuzzy setting. In this paper, the properties of the \(\varphi \) -index of inclusion, when its definition is restricted to a subset of indexes, are analyzed. The theoretical results obtained in this work are necessary in order to develop fuzzy inference systems based on the \(\varphi \) -index of inclusion.