||This paper analyzes constraints in terms of fuzzy-logical concepts for transforming families of fuzzy sets in order to guarantee stable and linguistically interpretable solutions of fine-tuned or optimized fuzzy systems. 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. However, further restrictions like the compactness of the set of admissible transformations are required in order to guarantee stable solutions. It is pointed out that compactness in the uniform sense can be characterized in terms of purely fuzzy-logical concepts, that is, by means of fuzzy equivalence relations.