TNorms

T-norms, or triangular norms, are binary operations T: [0,1]2 → [0,1] used as a generalization of conjunction in fuzzy logic and as an intersection operator in fuzzy set theory. They provide a mathematical means to combine degrees of truth or membership and form the basis for many aggregation frameworks in fuzzy systems.

A t-norm must satisfy four axioms: commutativity (T(a,b) = T(b,a)), associativity (T(T(a,b),c) = T(a,T(b,c))), monotonicity (if a ≤ a′ and b ≤ b′ then T(a,b) ≤ T(a′,b′)), and the neutral element 1 (T(a,1) = a for all a in [0,1]). In many common t-norms, 0 acts as an absorbing element (T(0,a) = 0 for all a).

Common examples include: the minimum t-norm T(a,b) = min(a,b), often called the Gödel t-norm; the product t-norm T(a,b) = a·b; and the Łukasiewicz t-norm T(a,b) = max(a + b − 1, 0). The drastic t-norm, defined by T(a,b) = 0 if a < 1 and b < 1, and T(a,b) = min(a,b) otherwise, is a non-continuous boundary case. Several parameterized families exist, such as the Schweizer–Sklar and Hamacher t-norms, which interpolate between these extremes.

T-norms have a dual relationship with t-conorms (S-norms): S(a,b) = 1 − T(1 − a, 1 − b). Together, t-norms and t-conorms form the core operators for conjunction and disjunction in fuzzy reasoning.

Applications include fuzzy inference systems, fuzzy aggregation, clustering, and probabilistic reasoning with fuzzy events.