Construction of t-norms
In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1].
Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms.
This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic.
The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms.
Relevant background can be found in the article on t-norms.
The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm.
In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed: The construction of t-norms by additive generators is based on the following theorem: Alternatively, one may avoid using the notion of pseudo-inverse function by having
If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T. Examples: Basic properties of additive generators are summarized by the following theorem: 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 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. 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: 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 The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞].
for −∞ < p < +∞ is The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ p ≤ +∞: The t-norm
The family is strictly decreasing and continuous with respect to p. An additive generator of
The family is strictly decreasing and continuous with respect to p. An additive generator for
The family is strictly increasing and continuous with respect to p. The Yager t-norm
for 0 < p < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of
for 0 < p < +∞ is The family of Aczél–Alsina t-norms, introduced in the early 1980s by János Aczél and Claudi Alsina, is given for 0 ≤ p ≤ +∞ by The Aczél–Alsina t-norm
The family is strictly increasing and continuous with respect to p. The Aczél–Alsina t-norm
for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of
The family is strictly increasing and continuous with respect to p. The Dombi t-norm
for 0 < p < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of p. An additive generator of
for 0 < p < +∞ is The family of Sugeno–Weber t-norms was introduced in the early 1980s by Siegfried Weber; the dual t-conorms were defined already in the early 1970s by Michio Sugeno.
The family is strictly increasing and continuous with respect to p. An additive generator of
for 0 < p < +∞ [sic] is The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval [0, 1] and completing the t-norm by using the minimum on the rest of the unit square.
It is based on the following theorem: The resulting t-norm is called the ordinal sum of the summands (Ti, ai, bi) for i in I, denoted by or
Ordinal sums of t-norms enjoy the following properties: If
By the Mostert–Shields theorem, every continuous t-norm is expressible as the ordinal sum of Archimedean continuous t-norms.
Important examples of ordinal sums of continuous t-norms are the following ones: The construction of t-norms by rotation was introduced by Sándor Jenei (2000).
It is based on the following theorem: Geometrically, the construction can be described as first shrinking the t-norm T to the interval [0.5, 1] and then rotating it by the angle 2π/3 in both directions around the line connecting the points (0, 0, 1) and (1, 1, 0).
The theorem can be generalized by taking for N any strong negation, that is, an involutive strictly decreasing continuous function on [0, 1], and for t taking the unique fixed point of N. The resulting t-norm enjoys the following rotation invariance property with respect to N: The negation induced by Trot is the function N, that is, N(x) = Rrot(x, 0) for all x, where Rrot is the residuum of Trot.
