Construction of t-norms


In mathematics, t-norms are a special kind of binary operations on the real unit interval . 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.

Generators of t-norms

The method of constructing t-norms by generators consists in using a unary function to transform some known binary function 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:

Additive generators

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. The corresponding residuum can then be expressed as. And the biresiduum as.
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:

Multiplicative generators

The isomorphism between addition on and multiplication on 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: → defined as h = e−f 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: → defined by f = −log 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:
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 . An additive generator for for −∞ < p < +∞ is

Hamacher t-norms

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 is called the Hamacher product.
Hamacher t-norms are the only t-norms which are rational functions.
The Hamacher t-norm is strict if and only if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator of for p < +∞ is

Frank t-norms

The family of Frank t-norms, introduced by M.J. Frank in the late 1970s, is given by the parametric definition for 0 ≤ p ≤ +∞ as follows:
The Frank t-norm is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for is

Yager t-norms

The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by
The Yager t-norm is nilpotent if and only if 0 < p < +∞. 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

Aczél–Alsina t-norms

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 is strict if and only if 0 < p < +∞. 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 for 0 < p < +∞ is

Dombi t-norms

The family of Dombi t-norms, introduced by József Dombi, is given for 0 ≤ p ≤ +∞ by
The Dombi t-norm is strict if and only if 0 < p < +∞. 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

Sugeno–Weber t-norms

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. It is given for −1 ≤ p ≤ +∞ by
The Sugeno–Weber t-norm is nilpotent if and only if −1 < p < +∞. The family is strictly increasing and continuous with respect to p. An additive generator of for 0 < p < +∞ is

Ordinal sums

The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval and completing the t-norm by using the minimum on the rest of the unit square. It is based on the following theorem:
Image:OrdSum-Luk-prod-graph-contours.png|thumb|270px|Ordinal sum of the Łukasiewicz t-norm on the interval and the product t-norm on the interval
The resulting t-norm is called the ordinal sum of the summands for i in I, denoted by
or if I is finite.
Ordinal sums of t-norms enjoy the following properties:
If is a left-continuous t-norm, then its residuum R is given as follows:
where Ri is the residuum of Ti, for each i in I.

Ordinal sums of continuous t-norms

The ordinal sum of a family of continuous t-norms is a continuous t-norm. By the Mostert–Shields theorem, every continuous t-norm is expressible as the ordinal sum of Archimedean continuous t-norms. Since the latter are either nilpotent or strict, each continuous t-norm is isomorphic to the ordinal sum of Łukasiewicz and product 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. It is based on the following theorem:
Geometrically, the construction can be described as first shrinking the t-norm T to the interval and then rotating it by the angle 2π/3 in both directions around the line connecting the points and.
The theorem can be generalized by taking for N any strong negation, that is, an involutive strictly decreasing continuous function on , 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 = Rrot for all x, where Rrot is the residuum of Trot.