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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:

Let f: [a,&nbsp;b] → [c,&nbsp;d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f&nbsp;<sup>(−1)</sup>: [c,&nbsp;d] → [a,&nbsp;b] defined as
:

Additive generators

The construction of t-norms by additive generators is based on the following theorem:

Let f: [0,&nbsp;1] → [0,&nbsp;+&infin;] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or in [f(0<sup>+</sup>),&nbsp;+&infin;] for all x, y in [0,&nbsp;1]. Then the function T: [0,&nbsp;1]<sup>2</sup> → [0,&nbsp;1] defined as
:T(x, y) = f&nbsp;<sup>(-1)</sup>(f(x) + f(y))
is a t-norm.

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:

  • The function f(x) = 1 – x for x in [0,&nbsp;1] is an additive generator of the Łukasiewicz t-norm.
  • The function f defined as f(x) = –log(x) if 0 &lt; x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm.
  • The function f defined as f(x) = 2 – x if 0 ≤ x &lt; 1 and f(1) = 0 is an additive generator of the drastic t-norm.

Basic properties of additive generators are summarized by the following theorem:

Let f: [0,&nbsp;1] → [0,&nbsp;+&infin;] 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) = +&infin;.
* Each element of (0,&nbsp;1) is a nilpotent element of T if and only if f(0) &lt; +&infin;.
* 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,&nbsp;+∞] and multiplication on [0,&nbsp;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,&nbsp;1] → [0,&nbsp;1] defined as h(x) = e<sup>−f&nbsp;(x)</sup> 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,&nbsp;1]
  • h is right-continuous in 0
  • T(x, y) = h&nbsp;<sup>(−1)</sup>(h(x) · h(y)).

Vice versa, if h is a multiplicative generator of T, then f: [0,&nbsp;1] → [0,&nbsp;+∞] 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 T<sub>p</sub> parameterized by p is increasing if T<sub>p</sub>(x, y) ≤ T<sub>q</sub>(x, y) for all x, y in [0,&nbsp;1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing).
  • A family of t-norms T<sub>p</sub> is continuous with respect to the parameter p if
:
for all values p<sub>0</sub> 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 &gt; −∞
  • Continuous if and only if p &lt; +∞
  • Strict if and only if −∞ &lt; p ≤ 0 (for p = −1 it is the Hamacher product)
  • Nilpotent if and only if 0 &lt; p &lt; +∞ (for p = 1 it is the Łukasiewicz t-norm).

The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞,&nbsp;+∞]. An additive generator for for −∞ &lt; p &lt; +∞ 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 &lt; +∞ (for p = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to p. An additive generator of for p &lt; +∞ 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 &lt; +∞. 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 &lt; p &lt; +∞ (for p = 1 it is the Łukasiewicz t-norm). The family is strictly increasing and continuous with respect to p. The Yager t-norm for 0 &lt; p &lt; +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of for 0 &lt; p &lt; +∞ 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 &lt; p &lt; +∞ (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 for 0 &lt; p &lt; +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of for 0 &lt; p &lt; +∞ 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 is strict if and only if 0 &lt; p &lt; +∞ (for p = 1 it is the Hamacher product). The family is strictly increasing and continuous with respect to p. The Dombi t-norm for 0 &lt; p &lt; +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of p. An additive generator of for 0 &lt; p &lt; +∞ 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 &lt; p &lt; +∞ (for p = 0 it is the Łukasiewicz t-norm). The family is strictly increasing and continuous with respect to p. An additive generator of for 0 &lt; p &lt; +∞ [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,&nbsp;1] and completing the t-norm by using the minimum on the rest of the unit square. It is based on the following theorem:

Let T<sub>i</sub> for i in an index set I be a family of t-norms and (a<sub>i</sub>,&nbsp;b<sub>i</sub>) a family of pairwise disjoint (non-empty) open subintervals of [0,&nbsp;1]. Then the function T: [0,&nbsp;1]<sup>2</sup> → [0,&nbsp;1] defined as
:
is a t-norm.

] The resulting t-norm is called the ordinal sum of the summands (T<sub>i</sub>, a<sub>i</sub>, b<sub>i</sub>) for i in I, denoted by

or if I is finite.

Ordinal sums of t-norms enjoy the following properties:

  • Each t-norm is a trivial ordinal sum of itself on the whole interval [0,&nbsp;1].
  • The empty ordinal sum (for the empty index set) yields the minimum t-norm T<sub>min</sub>. Summands with the minimum t-norm can arbitrarily be added or omitted without changing the resulting t-norm.
  • It can be assumed without loss of generality that the index set is countable, since the real line can only contain at most countably many disjoint subintervals.
  • An ordinal sum of t-norm is continuous if and only if each summand is a continuous t-norm. (Analogously for left-continuity.)
  • An ordinal sum is Archimedean if and only if it is a trivial sum of one Archimedean t-norm on the whole unit interval.
  • An ordinal sum has zero divisors if and only if for some index i, a<sub>i</sub> = 0 and T<sub>i</sub> has zero divisors. (Analogously for nilpotent elements.)

If is a left-continuous t-norm, then its residuum R is given as follows:

where R<sub>i</sub> is the residuum of T<sub>i</sub>, 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:

  • Dubois–Prade t-norms, introduced by Didier Dubois and Henri Prade in the early 1980s, are the ordinal sums of the product t-norm on [0,&nbsp;p] for a parameter p in [0,&nbsp;1] and the (default) minimum t-norm on the rest of the unit interval. The family of Dubois–Prade t-norms is decreasing and continuous with respect to p..
  • Mayor–Torrens t-norms, introduced by Gaspar Mayor and Joan Torrens in the early 1990s, are the ordinal sums of the Łukasiewicz t-norm on [0,&nbsp;p] for a parameter p in [0,&nbsp;1] and the (default) minimum t-norm on the rest of the unit interval. The family of Mayor–Torrens t-norms is decreasing and continuous with respect to p..

Rotations

The construction of t-norms by rotation was introduced by Sándor Jenei (2000). It is based on the following theorem:

Let T be a left-continuous t-norm without zero divisors, N: [0,&nbsp;1] → [0,&nbsp;1] the function that assigns 1 − x to x and t = 0.5. Let T<sub>1</sub> be the linear transformation of T into [t,&nbsp;1] and Then the function
:
is a left-continuous t-norm, called the rotation of the t-norm T.

Geometrically, the construction can be described as first shrinking the t-norm T to the interval [0.5,&nbsp;1] and then rotating it by the angle 2π/3 in both directions around the line connecting the points (0,&nbsp;0,&nbsp;1) and (1,&nbsp;1,&nbsp;0).

The theorem can be generalized by taking for N any strong negation, that is, an involutive strictly decreasing continuous function on [0,&nbsp;1], and for t taking the unique fixed point of&nbsp;N.

The resulting t-norm enjoys the following rotation invariance property with respect to&nbsp;N:

T(x, y) &le; z if and only if T(y, N(z)) &le; N(x) for all x, y, z in [0,&nbsp;1].

The negation induced by T<sub>rot</sub> is the function N, that is, N(x) = R<sub>rot</sub>(x, 0) for all x, where R<sub>rot</sub> is the residuum of&nbsp;T<sub>rot</sub>.

See also

References