Construction of t-norms

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

Let f: [0, 1] → [0, +∞] 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 +∞ 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 ${\displaystyle T(x,y)=f^{-1}\left(\min \left(f(0^{+}),f(x)+f(y)\right)\right)}$. The corresponding residuum can then be expressed as ${\displaystyle (x\Rightarrow y)=f^{-1}\left(\max \left(0,f(y)-f(x)\right)\right)}$. And the biresiduum as ${\displaystyle (x\Leftrightarrow y)=f^{-1}\left(\left|f(x)-f(y)\right|\right)}$.

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 < 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 < 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, +∞] 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) = +∞.
• Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞.
• 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.)

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

• 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)).

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.

Graph (3D and contours) of the Schweizer–Sklar t-norm with p = 2

The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition

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

A Schweizer–Sklar t-norm ${\displaystyle T_{p}^{\mathrm {SS} }}$ is

• Archimedean if and only if p > −∞
• Continuous if and only if p < +∞
• Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product)
• Nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm).

The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for ${\displaystyle T_{p}^{\mathrm {SS} }}$ for −∞ < p < +∞ is

${\displaystyle f_{p}^{\mathrm {SS} }(x)={\begin{cases}-\log x&{\text{if }}p=0\\{\frac {1-x^{p}}{p}}&{\text{otherwise.}}\end{cases}}}$

The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ p ≤ +∞:

${\displaystyle T_{p}^{\mathrm {H} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=+\infty \\0&{\text{if }}p=x=y=0\\{\frac {xy}{p+(1-p)(x+y-xy)}}&{\text{otherwise.}}\end{cases}}}$

The t-norm ${\displaystyle T_{0}^{\mathrm {H} }}$ is called the Hamacher product.

Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm ${\displaystyle T_{p}^{\mathrm {H} }}$ 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 ${\displaystyle T_{p}^{\mathrm {H} }}$ for p < +∞ is

${\displaystyle f_{p}^{\mathrm {H} }(x)={\begin{cases}{\frac {1-x}{x}}&{\text{if }}p=0\\\log {\frac {p+(1-p)x}{x}}&{\text{otherwise.}}\end{cases}}}$

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:

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

The Frank t-norm ${\displaystyle T_{p}^{\mathrm {F} }}$ is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for ${\displaystyle T_{p}^{\mathrm {F} }}$ is

${\displaystyle f_{p}^{\mathrm {F} }(x)={\begin{cases}-\log x&{\text{if }}p=1\\1-x&{\text{if }}p=+\infty \\\log {\frac {p-1}{p^{x}-1}}&{\text{otherwise.}}\end{cases}}}$

Graph of the Yager t-norm with p = 2

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

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

The Yager t-norm ${\displaystyle T_{p}^{\mathrm {Y} }}$ 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 ${\displaystyle T_{p}^{\mathrm {Y} }}$ for 0 < p < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of ${\displaystyle T_{p}^{\mathrm {Y} }}$ for 0 < p < +∞ is

${\displaystyle f_{p}^{\mathrm {Y} }(x)=(1-x)^{p}.}$

The family of Aczél–Alsina t-norms, introduced in the early 1980s by János Aczél and Claudi Alsina, is given for 0 ≤ p ≤ +∞ by

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

The Aczél–Alsina t-norm ${\displaystyle T_{p}^{\mathrm {AA} }}$ 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 ${\displaystyle T_{p}^{\mathrm {AA} }}$ for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of ${\displaystyle T_{p}^{\mathrm {AA} }}$ for 0 < p < +∞ is

${\displaystyle f_{p}^{\mathrm {AA} }(x)=(-\log x)^{p}.}$

The family of Dombi t-norms, introduced by József Dombi (1982), is given for 0 ≤ p ≤ +∞ by

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

The Dombi t-norm ${\displaystyle T_{p}^{\mathrm {D} }}$ 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 ${\displaystyle T_{p}^{\mathrm {D} }}$ for 0 < p < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of p. An additive generator of ${\displaystyle T_{p}^{\mathrm {D} }}$ for 0 < p < +∞ is

${\displaystyle f_{p}^{\mathrm {D} }(x)=\left({\frac {1-x}{x}}\right)^{p}.}$

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

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

The Sugeno–Weber t-norm ${\displaystyle T_{p}^{\mathrm {SW} }}$ 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 ${\displaystyle T_{p}^{\mathrm {SW} }}$ for 0 < p < +∞ [sic] is

${\displaystyle f_{p}^{\mathrm {SW} }(x)={\begin{cases}1-x&{\text{if }}p=0\\1-\log _{1+p}(1+px)&{\text{otherwise.}}\end{cases}}}$

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..