[0, 1] → [0, +∞] be an additive generator of a t-norm T. Then: T is an Archimedean t-norm. T is continuous if and only if f is continuous. T is strictly monotone if and only if f(0) = +∞. Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞. The multiple of f by a positive constant is also an additive generator of T. T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.) Multiplicative generators The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive generator of a t-norm T, then the function h: [0, 1] → [0, 1] defined as h(x) = e−f (x) is a multiplicative generator of T, that is, a function h such that h is strictly increasing h(1) = 1 h(x) · h(y) is in the range of h or equal to 0 or h(0+) for all x, y in [0, 1] h is right-continuous in 0 T(x, y) = h (−1)(h(x) · h(y)). Vice versa, if h is a multiplicative generator of T, then f: [0, 1] → [0, +∞] defined by f(x) = −log(h(x)) is an additive generator of T. Parametric classes of t-norms Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list: A family of t-norms Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(x, y) for all x, y in [0, 1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing). A family of t-norms Tp is continuous with respect to the parameter p if for all values p0 of the parameter. Schweizer–Sklar t-norms The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition A Schweizer–Sklar t-norm is Archimedean if and only if p > −∞ Continuous if and only if p < +∞ Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product) Nilpotent if and only if 0 < p < +∞ (for p = 1 it