tnorm

A t-norm, short for triangular norm, is a binary operation used in fuzzy logic and related fields to model the conjunction of fuzzy propositions. It is defined on the unit interval [0,1] and maps two truth values to another truth value between 0 and 1.

A t-norm T is required to satisfy four standard properties: commutativity (T(a,b) = T(b,a)), associativity (T(a,T(b,c)) = T(T(a,b),c)), monotonicity (if a1 ≤ a2 and b1 ≤ b2 then T(a1,b1) ≤ T(a2,b2)), and the neutral element condition T(a,1) = a for all a in [0,1]. Many formulations also adopt a boundary condition such as T(a,0) = 0. These properties ensure the operation behaves like a logical conjunction in a fuzzy setting.

Common examples of t-norms include:

- Minimum t-norm: T_min(a,b) = min(a,b); interprets conjunction as taking the smaller truth value.

- Product t-norm: T_prod(a,b) = a·b; corresponds to probabilistic interpretation of independent events.

- Lukasiewicz t-norm: T_L(a,b) = max(0, a + b − 1); provides a linear, continuous alternative.

- Drastic t-norm: T_D(a,b) = if a = 1 then b, else if b = 1 then a, else 0; a extreme form for conjunction.

T-norms have a dual concept, t-conorms, which model disjunction in fuzzy logic. They underpin many fuzzy inference systems, aggregation operators, and decision-making frameworks, offering a principled way to combine partial truths into a single value. The study of t-norms encompasses a family of operations with various continuity, algebraic, and interpretive properties suitable for different applications.