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, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f (−1): [c, d] → [a, b] defined as
 * $$f^{(-1)}(y) = \begin{cases}

\sup \{ x\in[a,b] \mid f(x) < y \} & \text{for } f \text{ non-decreasing} \\ \sup \{ x\in[a,b] \mid f(x) > y \} & \text{for } f \text{ non-increasing.} \end{cases}$$

Additive generators
The construction of t-norms by additive generators is based on the following theorem:
 * Let f: [0, 1] → [0, +&infin;] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or equal to f(0+) or +&infin; for all x, y in [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as
 * T(x, y) = f (-1)(f(x) + f(y))
 * is a t-norm.

Alternatively, one may avoid using the notion of pseudo-inverse function by having $$T(x,y)=f^{-1}\left(\min\left(f(0^+),f(x)+f(y)\right)\right)$$. The corresponding residuum can then be expressed as $$(x \Rightarrow y) = f^{-1}\left(\max\left(0,f(y)-f(x)\right)\right)$$. And the biresiduum as $$(x \Leftrightarrow y) = f^{-1}\left(\left|f(x)-f(y)\right|\right)$$.

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, 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, 1] → [0, +&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, 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, +∞] 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.
 * 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)).

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
 * $$\lim_{p\to p_0} T_p = T_{p_0}$$
 * 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
 * $$T^{\mathrm{SS}}_p(x,y) = \begin{cases}

T_{\min}(x,y)         & \text{if } p = -\infty \\ (x^p + y^p - 1)^{1/p}         & \text{if } -\infty < p < 0 \\ T_{\mathrm{prod}}(x,y)        & \text{if } p = 0 \\ (\max(0, x^p + y^p - 1))^{1/p} & \text{if } 0 < p < +\infty \\ T_{\mathrm{D}}(x,y)           & \text{if } p = +\infty. \end{cases}$$

A Schweizer–Sklar t-norm $$T^{\mathrm{SS}}_p$$ is The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for $$T^{\mathrm{SS}}_p$$ for −∞ &lt; p &lt; +∞ 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).
 * $$f^{\mathrm{SS}}_p (x) = \begin{cases}

-\log x          & \text{if } p = 0 \\ \frac{1 - x^p}{p} & \text{otherwise.} \end{cases}$$

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 ≤ +∞:
 * $$T^{\mathrm{H}}_p (x,y) = \begin{cases}

T_{\mathrm{D}}(x,y)               & \text{if } p = +\infty \\ 0                                 & \text{if } p = x = y = 0 \\ \frac{xy}{p + (1 - p)(x + y - xy)} & \text{otherwise.} \end{cases}$$ The t-norm $$T^{\mathrm{H}}_0$$ is called the Hamacher product.

Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm $$T^{\mathrm{H}}_p$$ 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 $$T^{\mathrm{H}}_p$$ for p &lt; +∞ is
 * $$f^{\mathrm{H}}_p(x) = \begin{cases}

\frac{1 - x}{x}           & \text{if } p = 0 \\ \log\frac{p + (1 - p)x}{x} & \text{otherwise.} \end{cases}$$

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:
 * $$T^{\mathrm{F}}_p(x,y) = \begin{cases}

T_{\mathrm{min}}(x,y) & \text{if } p = 0 \\ T_{\mathrm{prod}}(x,y) & \text{if } p = 1 \\ T_{\mathrm{Luk}}(x,y) & \text{if } p = +\infty \\ \log_p\left(1 + \frac{(p^x - 1)(p^y - 1)}{p - 1}\right) & \text{otherwise.} \end{cases}$$

The Frank t-norm $$T^{\mathrm{F}}_p$$ is strict if p &lt; +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for $$T^{\mathrm{F}}_p$$ is
 * $$f^{\mathrm{F}}_p(x) = \begin{cases}

-\log x                  & \text{if } p = 1 \\ 1 - x                    & \text{if } p = +\infty \\ \log\frac{p - 1}{p^x - 1} & \text{otherwise.} \end{cases} $$

Yager t-norms
The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by
 * $$T^{\mathrm{Y}}_p (x,y) = \begin{cases}

T_{\mathrm{D}}(x,y)  & \text{if } p = 0 \\ \max\left(0, 1 - ((1 - x)^p + (1 - y)^p)^{1/p}\right) & \text{if } 0 < p < +\infty \\ T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \end{cases} $$

The Yager t-norm $$T^{\mathrm{Y}}_p$$ 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 $$T^{\mathrm{Y}}_p$$ 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 $$T^{\mathrm{Y}}_p$$ for 0 &lt; p &lt; +∞ is
 * $$f^{\mathrm{Y}}_p(x) = (1 - x)^p.$$

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
 * $$T^{\mathrm{AA}}_p (x,y) = \begin{cases}

T_{\mathrm{D}}(x,y)  & \text{if } p = 0 \\ e^{-\left(|-\log x|^p + |-\log y|^p\right)^{1/p}} & \text{if } 0 < p < +\infty \\ T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \end{cases}$$

The Aczél–Alsina t-norm $$T^{\mathrm{AA}}_p$$ 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 $$T^{\mathrm{AA}}_p$$ 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 $$T^{\mathrm{AA}}_p$$ for 0 &lt; p &lt; +∞ is
 * $$f^{\mathrm{AA}}_p(x) = (-\log x)^p.$$

Dombi t-norms
The family of Dombi t-norms, introduced by József Dombi (1982), is given for 0 ≤ p ≤ +∞ by
 * $$T^{\mathrm{D}}_p (x,y) = \begin{cases}

0                    & \text{if } x = 0 \text{ or } y = 0 \\ T_{\mathrm{D}}(x,y)  & \text{if } p = 0 \\ T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \\ \frac{1}{1 + \left(   \left(\frac{1 - x}{x}\right)^p + \left(\frac{1 - y}{y}\right)^p  \right)^{1/p}} & \text{otherwise.} \\ \end{cases} $$

The Dombi t-norm $$T^{\mathrm{D}}_p$$ 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 $$T^{\mathrm{D}}_p$$ 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 $$T^{\mathrm{D}}_p$$ for 0 &lt; p &lt; +∞ is
 * $$f^{\mathrm{D}}_p(x) = \left(\frac{1-x}{x}\right)^p.$$

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
 * $$T^{\mathrm{SW}}_p (x,y) = \begin{cases}

T_{\mathrm{D}}(x,y)   & \text{if } p = -1 \\ \max\left(0, \frac{x + y - 1 + pxy}{1 + p}\right) & \text{if } -1 < p < +\infty \\ T_{\mathrm{prod}}(x,y) & \text{if } p = +\infty \end{cases} $$

The Sugeno–Weber t-norm $$T^{\mathrm{SW}}_p$$ 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 $$T^{\mathrm{SW}}_p$$ for 0 &lt; p &lt; +∞ [sic] is
 * $$f^{\mathrm{SW}}_p(x) = \begin{cases}

1 - x  & \text{if } p = 0 \\ 1 - \log_{1 + p}(1 + px) & \text{otherwise.} \end{cases}$$

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:


 * Let Ti for i in an index set I be a family of t-norms and (ai, bi) a family of pairwise disjoint (non-empty) open subintervals of [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as
 * $$T(x, y) = \begin{cases}

a_i + (b_i - a_i) \cdot T_i\left(\frac{x - a_i}{b_i - a_i}, \frac{y - a_i}{b_i - a_i}\right) & \text{if } x, y \in [a_i, b_i]^2 \\ \min(x, y) & \text{otherwise} \end{cases}$$
 * is a t-norm.

The resulting t-norm is called the ordinal sum of the summands (Ti, ai, bi) for i in I, denoted by
 * $$T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i),$$

or $$(T_1, a_1, b_1) \oplus \dots \oplus (T_n, a_n, b_n)$$ 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, 1].
 * The empty ordinal sum (for the empty index set) yields the minimum t-norm Tmin. 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, ai = 0 and Ti has zero divisors. (Analogously for nilpotent elements.)

If $$T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i)$$ is a left-continuous t-norm, then its residuum R is given as follows:
 * $$R(x, y) = \begin{cases}

1 & \text{if } x \le y \\ a_i + (b_i - a_i) \cdot R_i\left(\frac{x - a_i}{b_i - a_i}, \frac{y - a_i}{b_i - a_i}\right) & \text{if } a_i < y < x \le b_i \\ y & \text{otherwise.} \end{cases}$$ 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 (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, p] for a parameter p in [0, 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, p] for a parameter p in [0, 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, 1] → [0, 1] the function that assigns 1 − x to x and t = 0.5. Let T1 be the linear transformation of T into [t, 1] and $$R_{T_1}(x,y) = \sup\{z \mid T_1(z,x)\le y\}.$$ Then the function
 * $$T_{\mathrm{rot}} = \begin{cases}

T_1(x, y) & \text{if } x, y \in (t, 1] \\ N(R_{T_1}(x, N(y))) & \text{if } x \in (t, 1] \text{ and } y \in [0, t] \\ N(R_{T_1}(y, N(x))) & \text{if } x \in [0, t] \text{ and } y \in (t, 1] \\ 0 & \text{if } x, y \in [0, t] \end{cases}$$
 * 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, 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:
 * T(x, y) &le; z if and only if T(y, N(z)) &le; N(x) for all x, y, z in [0, 1].

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.