Algebraic Product Is the Only "And-like" Operation for Which Normalized Intersection Is Associative: A Proof

Thierry Denoeux (1, 2) , Vladik Kreinovitch (3)

1 Heudiasyc - Heuristique et Diagnostic des Systèmes Complexes [Compiègne]
2 IUF - Institut universitaire de France
3 UTEP - University of Texas [El Paso]

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 for which normalized intersection is associative. 1 Formulation of the Problem Fuzzy sets and normalized fuzzy sets: a brief reminder. A fuzzy set on a universal set X is a function µ(x) that assigns, to each element x ∈ X, a number from the interval [0, 1]; see, e.g., [1, 3, 4, 5, 6, 7]. Fuzzy sets were invented to describe imprecise ("fuzzy") natural-language properties such as "small"; the value µ(x) is then a degree to which, according to the user, the object x has the desired property (e.g., is small). • The degree 1 means that x definitely has the property. • The degree 0 means that x definitely does not have this property. • Intermediate value µ(x) correspond to x having the property "to some degree".

Domains: Artificial Intelligence [cs.AI]

Thierry Denoeux, Vladik Kreinovitch. Algebraic Product Is the Only "And-like" Operation for Which Normalized Intersection Is Associative: A Proof. Fifth International Conference on Artificial Intelligence and Computational Intelligence (AICI 2024), Jan 2024, Hanoi (Vietnam), Vietnam. ⟨hal-04436177⟩