On the compactness of admissible transformations of fuzzy partitions in terms of T-equivalence relations

This paper analyzes constraints in terms of fuzzy-logical concepts for transforming families of fuzzy sets in order to guarantee linguistically interpretable solutions of fine-tuned or optimized fuzzy systems, and further, to make the whole process of fine tuning a well-posed problem. For this purpose, transformations for fuzzy sets are introduced which are constructed by means of the compositional rule of inference. It is investigated under which conditions on these transformations various characteristics of fuzzy partitions and fuzzy sets, like redundancy, convexity and topological aspects remain invariant. It is argued that further restrictions like the compactness of the set of admissible transformations are required in order to make the process of fine tuning well-posed. As main result it is demonstrated that compactness of this function space endowed with the uniform norm can be characterized in terms of purely fuzzy-logical concepts, that is, by means of fuzzy equivalence relations.