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. Generators of 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: 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 formula_2. The corresponding residuum can then be  expressed as formula_3. And the biresiduum as formula_4. 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 [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 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". 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: 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 formula_7 is The family is strictly decreasing for "p" ≥ 0 and continuous with respect to "p" in [−∞, +∞]. An additive generator for formula_7 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 formula_11 is called the "Hamacher  product." Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm formula_12 is strict if and only if "p" < +∞ (for "p" = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to "p". An additive generator of formula_12 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 formula_16 is strict if  p < +∞. The family is strictly decreasing and continuous with respect to "p". An additive generator for formula_16 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 formula_20 is nilpotent if and only if 0 < "p" < +∞ (for "p" = 1 it is the Łukasiewicz t-norm). The family is strictly increasing and continuous with respect to "p". The Yager t-norm formula_20 for 0 < "p" < +∞ arises  from the Łukasiewicz t-norm by raising its additive generator to the power of "p". An additive generator of formula_20 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 formula_25 is strict if and only if 0 < "p" < +∞ (for "p" = 1 it is the product t-norm). The family is strictly increasing and continuous with respect to "p". The  Aczél–Alsina t-norm formula_25 for 0 < "p" < +∞ arises from the product t-norm by raising its additive generator to the power of "p". An additive generator of formula_25 for 0 < "p" < +∞ is Dombi t-norms. The family of "Dombi t-norms", introduced by József Dombi (1982), is given for 0 ≤ "p" ≤ +∞ by The Dombi t-norm formula_30 is strict if and only if 0 < "p" < +∞ (for "p" = 1 it is the Hamacher product). The family is strictly increasing and continuous with respect  to "p". The Dombi t-norm formula_30 for 0 < "p" < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of "p". An additive generator of formula_30 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 formula_35 is nilpotent if and only  if −1 < "p" < +∞ (for "p" = 0 it is the Łukasiewicz t-norm). The family is strictly increasing and continuous with respect to "p". An additive generator of formula_35 for 0 < "p" < +∞ [sic] 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 [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 (T", "a", "b") for "i" in "I", denoted by or formula_40 if "I" is finite. Ordinal sums of t-norms enjoy the following properties: If formula_41 is a left-continuous t-norm, then its residuum "R" is given as follows: where "R" is the residuum of "T", 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 (and then isomorphic to the Łukasiewicz t-norm) or strict (then isomorphic to the product t-norm), 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: Rotations. 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 "T" is the function "N", that is, "N"("x") = "R"("x", 0) for all "x", where "R" is the residuum of "T".