Algebraic Product Is the only “And-Like”-Operation for Which Normalized Intersection Is Associative: A Proof

For normalized fuzzy sets, intersection is, in general, not normalized. So, if we want to limit ourselves to normalized fuzzy sets, we need to normalize the intersection. It is known that for algebraic product, the normalized intersection is associative, and that for many other “and”-operations (t-norms), normalized intersection is not associative. In this paper, we prove that algebraic product is the only “and”-operation (and even the only “and-like” operation) for which normalized intersection is associative.