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| In mathematics, [[t-norm]]s 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-example]]s 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.
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| Relevant background can be found in the article on [[t-norm]]s.
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| == Generators of t-norms ==
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| 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.
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| In order to allow using non-bijective generators, which do not have the [[inverse function]], the following notion of ''pseudo-inverse function'' is employed:
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| :Let ''f'': [''a'', ''b''] → [''c'', ''d''] be a monotone function between two closed subintervals of [[affinely extended real number system|extended real line]]. The ''pseudo-inverse function'' to ''f'' is the function ''f'' <sup>(−1)</sup>: [''c'', ''d''] → [''a'', ''b''] defined as
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| ::<math>f^{(-1)}(y) = \begin{cases}
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| \sup \{ x\in[a,b] \mid f(x) < y \} & \text{for } f \text{ non-decreasing} \\
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| \sup \{ x\in[a,b] \mid f(x) > y \} & \text{for } f \text{ non-increasing.}
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| \end{cases}</math>
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| === Additive generators ===
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| The construction of t-norms by additive generators is based on the following theorem:
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| : 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<sup>+</sup>) or +∞ for all ''x'', ''y'' in [0, 1]. Then the function ''T'': [0, 1]<sup>2</sup> → [0, 1] defined as
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| ::''T''(''x'', ''y'') = ''f'' <sup>(-1)</sup>(''f''(''x'') + ''f''(''y''))
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| : is a t-norm.
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| 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''.
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| Examples:
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| * The function ''f''(''x'') = 1 – ''x'' for ''x'' in [0, 1] is an additive generator of the Łukasiewicz t-norm.
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| * 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.
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| * 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.
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| Basic properties of additive generators are summarized by the following theorem:
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| :Let ''f'': [0, 1] → [0, +∞] be an additive generator of a t-norm ''T''. Then:
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| :* ''T'' is an Archimedean t-norm.
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| :* ''T'' is continuous if and only if ''f'' is continuous.
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| :* ''T'' is strictly monotone if and only if ''f''(0) = +∞.
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| :* Each element of (0, 1) is a nilpotent element of ''T'' if and only if f(0) < +∞.
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| :* The multiple of ''f'' by a positive constant is also an additive generator of ''T''.
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| :* ''T'' has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.)
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| === Multiplicative generators ===
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| 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<sup>−''f'' (''x'')</sup> is a ''multiplicative generator'' of ''T'', that is, a function ''h'' such that
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| * ''h'' is strictly increasing
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| * ''h''(1) = 1
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| * ''h''(''x'') · ''h''(''y'') is in the range of ''h'' or equal to 0 or ''h''(0+) for all ''x'', ''y'' in [0, 1]
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| * ''h'' is right-continuous in 0
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| * ''T''(''x'', ''y'') = ''h'' <sup>(−1)</sup>(''h''(''x'') · ''h''(''y'')).
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| 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''.
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| == Parametric classes of t-norms ==
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| 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:
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| * 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, 1] whenever ''p'' ≤ ''q'' (similarly for ''decreasing'' and ''strictly'' increasing or decreasing).
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| * A family of t-norms ''T''<sub>''p''</sub> is ''continuous'' with respect to the parameter ''p'' if
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| ::<math>\lim_{p\to p_0} T_p = T_{p_0}</math>
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| :for all values ''p''<sub>0</sub> of the parameter.
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| === Schweizer–Sklar t-norms ===
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| [[Image:Schweizer-Sklar-2-Tnorm-graph-contour.png|thumb|270px|Graph (3D and contours) of the Schweizer–Sklar t-norm with ''p'' = 2]]
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| The family of ''Schweizer–Sklar t-norms'', introduced by Berthold Schweizer and [[Abe Sklar]] in the early 1960s, is given by the parametric definition
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| :<math>T^{\mathrm{SS}}_p(x,y) = \begin{cases}
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| T_\min(x,y) & \text{if } p = -\infty \\
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| (x^p + y^p - 1)^{1/p} & \text{if } -\infty < p < 0 \\
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| T_{\mathrm{prod}}(x,y) & \text{if } p = 0 \\
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| (\max(0, x^p + y^p - 1))^{1/p} & \text{if } 0 < p < +\infty \\
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| T_{\mathrm{D}}(x,y) & \text{if } p = +\infty.
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| \end{cases}</math>
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| A Schweizer–Sklar t-norm <math>T^{\mathrm{SS}}_p</math> is
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| * Archimedean if and only if ''p'' > −∞
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| * Continuous if and only if ''p'' < +∞
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| * Strict if and only if −∞ < ''p'' ≤ 0 (for ''p'' = −1 it is the Hamacher product)
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| * Nilpotent if and only if 0 < ''p'' < +∞ (for ''p'' = 1 it is the Łukasiewicz t-norm).
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| The family is strictly decreasing for ''p'' ≥ 0 and continuous with respect to ''p'' in [−∞, +∞]. An additive generator for <math>T^{\mathrm{SS}}_p</math> for −∞ < ''p'' < +∞ is
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| :<math>f^{\mathrm{SS}}_p (x,y) = \begin{cases}
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| -\log x & \text{if } p = 0 \\
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| \frac{1 - x^p}{p} & \text{otherwise.}
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| \end{cases}</math>
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| === Hamacher t-norms ===
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| The family of ''Hamacher t-norms'', introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ ''p'' ≤ +∞:
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| :<math>T^{\mathrm{H}}_p (x,y) = \begin{cases}
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| T_{\mathrm{D}}(x,y) & \text{if } p = +\infty \\
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| 0 & \text{if } p = x = y = 0 \\
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| \frac{xy}{p + (1 - p)(x + y - xy)} & \text{otherwise.}
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| \end{cases}</math>
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| The t-norm <math>T^{\mathrm{H}}_0</math> is called the ''Hamacher product.''
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| Hamacher t-norms are the only t-norms which are rational functions.
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| The Hamacher t-norm <math>T^{\mathrm{H}}_p</math> 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 <math>T^{\mathrm{H}}_p</math> for ''p'' < +∞ is
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| :<math>f^{\mathrm{H}}_p(x) = \begin{cases}
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| \frac{1 - x}{x} & \text{if } p = 0 \\
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| \log\frac{p + (1 - p)x}{x} & \text{otherwise.}
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| \end{cases}</math>
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| === Frank t-norms ===
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| 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:
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| :<math>T^{\mathrm{F}}_p(x,y) = \begin{cases}
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| T_{\mathrm{min}}(x,y) & \text{if } p = 0 \\
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| T_{\mathrm{prod}}(x,y) & \text{if } p = 1 \\
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| T_{\mathrm{Luk}}(x,y) & \text{if } p = +\infty \\
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| \log_p\left(1 + \frac{(p^x - 1)(p^y - 1)}{p - 1}\right) & \text{otherwise.}
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| \end{cases}</math>
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| The Frank t-norm <math>T^{\mathrm{F}}_p</math> is strict if ''p'' < +∞. The family is strictly decreasing and continuous with respect to ''p''. An additive generator for <math>T^{\mathrm{F}}_p</math> is
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| :<math>f^{\mathrm{F}}_p(x) = \begin{cases}
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| -\log x & \text{if } p = 1 \\
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| 1 - x & \text{if } p = +\infty \\
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| \log\frac{p - 1}{p^x - 1} & \text{otherwise.}
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| \end{cases}
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| </math>
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| === Yager t-norms ===
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| [[Image:Yager-2-Tnorm-graph-contours.png|thumb|270px|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
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| :<math>T^{\mathrm{Y}}_p (x,y) = \begin{cases}
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| T_{\mathrm{D}}(x,y) & \text{if } p = 0 \\
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| \max\left(0, 1 - ((1 - x)^p + (1 - y)^p)^{1/p}\right) & \text{if } 0 < p < +\infty \\
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| T_{\mathrm{min}}(x,y) & \text{if } p = +\infty
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| \end{cases}
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| </math>
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| The Yager t-norm <math>T^{\mathrm{Y}}_p</math> 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 <math>T^{\mathrm{Y}}_p</math> for 0 < ''p'' < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of ''p''. An additive generator of <math>T^{\mathrm{Y}}_p</math> for 0 < ''p'' < +∞ is
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| :<math>f^{\mathrm{Y}}_p(x) = (1 - x)^p.</math>
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| === Aczél–Alsina t-norms ===
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| The family of ''Aczél–Alsina t-norms'', introduced in the early 1980s by János Aczél and Claudi Alsina, is given for 0 ≤ ''p'' ≤ +∞ by
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| :<math>T^{\mathrm{AA}}_p (x,y) = \begin{cases}
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| T_{\mathrm{D}}(x,y) & \text{if } p = 0 \\
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| e^{-\left(|\log x|^p + |\log y|^p\right)^{1/p}} & \text{if } 0 < p < +\infty \\
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| T_{\mathrm{min}}(x,y) & \text{if } p = +\infty
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| \end{cases}</math>
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| The Aczél–Alsina t-norm <math>T^{\mathrm{AA}}_p</math> 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 <math>T^{\mathrm{AA}}_p</math> for 0 < ''p'' < +∞ arises from the product t-norm by raising its additive generator to the power of ''p''. An additive generator of <math>T^{\mathrm{AA}}_p</math> for 0 < ''p'' < +∞ is
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| :<math>f^{\mathrm{AA}}_p(x) = (-\log x)^p.</math>
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| === Dombi t-norms ===
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| The family of ''Dombi t-norms'', introduced by József Dombi (1982), is given for 0 ≤ ''p'' ≤ +∞ by
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| :<math>T^{\mathrm{D}}_p (x,y) = \begin{cases}
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| 0 & \text{if } x = 0 \text{ or } y = 0 \\
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| T_{\mathrm{D}}(x,y) & \text{if } p = 0 \\
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| T_{\mathrm{min}}(x,y) & \text{if } p = +\infty \\
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| \frac{1}{1 + \left(
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| \left(\frac{1 - x}{x}\right)^p + \left(\frac{1 - y}{y}\right)^p
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| \right)^{1/p}} & \text{otherwise.} \\
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| \end{cases}
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| </math>
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| The Dombi t-norm <math>T^{\mathrm{D}}_p</math> 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 <math>T^{\mathrm{D}}_p</math> for 0 < ''p'' < +∞ arises from the Hamacher product t-norm by raising its additive generator to the power of ''p''. An additive generator of <math>T^{\mathrm{D}}_p</math> for 0 < ''p'' < +∞ is
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| :<math>f^{\mathrm{D}}_p(x) = \left(\frac{1-x}{x}\right)^p.</math>
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| === Sugeno–Weber t-norms ===
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| The family of ''Sugeno–Weber t-norms'' was introduced in the early 1980s by Siegfried Weber; the dual [[t-conorm]]s were defined already in the early 1970s by Michio Sugeno. It is given for −1 ≤ ''p'' ≤ +∞ by
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| :<math>T^{\mathrm{SW}}_p (x,y) = \begin{cases}
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| T_{\mathrm{D}}(x,y) & \text{if } p = -1 \\
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| \max\left(0, \frac{x + y - 1 + pxy}{1 + p}\right) & \text{if } -1 < p < +\infty \\
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| T_{\mathrm{prod}}(x,y) & \text{if } p = +\infty
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| \end{cases}
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| </math>
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| The Sugeno–Weber t-norm <math>T^{\mathrm{SW}}_p</math> 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 <math>T^{\mathrm{SW}}_p</math> for 0 < ''p'' < +∞ [sic] is
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| :<math>f^{\mathrm{SW}}_p(x) = \begin{cases}
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| 1 - x & \text{if } p = 0 \\
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| 1 - \log_{1 + p}(1 + px) & \text{otherwise.}
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| \end{cases}</math>
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| == Ordinal sums ==
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| 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:
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| :Let ''T''<sub>''i''</sub> for ''i'' in an index set ''I'' be a family of t-norms and (''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>) a family of pairwise disjoint (non-empty) open subintervals of [0, 1]. Then the function ''T'': [0, 1]<sup>2</sup> → [0, 1] defined as
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| ::<math>T(x, y) = \begin{cases}
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| 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)
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| & \text{if } x, y \in [a_i, b_i]^2 \\
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| \min(x, y) & \text{otherwise}
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| \end{cases}</math>
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| :is a t-norm.
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| [[Image:OrdSum-Luk-prod-graph-contours.png|thumb|270px|Ordinal sum of the Łukasiewicz t-norm on the interval [0.05, 0.45] and the product t-norm on the interval [0.55, 0.95]]]
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| 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
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| :<math>T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i),</math>
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| or <math>(T_1, a_1, b_1) \oplus \dots \oplus (T_n, a_n, b_n)</math> if ''I'' is finite.
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| Ordinal sums of t-norms enjoy the following properties:
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| * Each t-norm is a trivial ordinal sum of itself on the whole interval [0, 1].
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| * 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.
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| * 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.
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| * An ordinal sum of t-norm is continuous if and only if each summand is a continuous t-norm. (Analogously for left-continuity.)
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| * An ordinal sum is Archimedean if and only if it is a trivial sum of one Archimedean t-norm on the whole unit interval.
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| * 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.)
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| If <math>T = \bigoplus\nolimits_{i\in I} (T_i, a_i, b_i)</math> is a left-continuous t-norm, then its residuum ''R'' is given as follows:
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| :<math>R(x, y) = \begin{cases}
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| 1 & \text{if } x \le y \\
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| 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)
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| & \text{if } a_i < y < x \le b_i \\
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| y & \text{otherwise.}
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| \end{cases}</math>
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| where ''R''<sub>i</sub> is the residuum of ''T''<sub>i</sub>, for each ''i'' in ''I''.
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| === Ordinal sums of continuous t-norms ===
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| 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.
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| Important examples of ordinal sums of continuous t-norms are the following ones:
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| * '''Dubois–Prade t-norms''', introduced by [[Didier Dubois (mathematician)|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''..
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| * '''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''..
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| == Rotations ==
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| The construction of t-norms by rotation was introduced by Sándor Jenei (2000). It is based on the following theorem:
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| :Let ''T'' be a left-continuous t-norm without [[zero divisor]]s, ''N'': [0, 1] → [0, 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'', 1] and <math>R_{T_1}(x,y) = \sup\{z \mid T_1(z,x)\le y\}.</math> Then the function
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| ::<math>T_{\mathrm{rot}} = \begin{cases}
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| T_1(x, y) & \text{if } x, y \in (t, 1] \\
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| N(R_{T_1}(x, N(y))) & \text{if } x \in (t, 1] \text{ and } y \in [0, t] \\
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| N(R_{T_1}(y, N(x))) & \text{if } x \in [0, t] \text{ and } y \in (t, 1] \\
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| 0 & \text{if } x, y \in [0, t]
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| \end{cases}</math>
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| :is a left-continuous t-norm, called the ''rotation'' of the t-norm ''T''.
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| [[Image:NilpotentMinimum-as-rotation.png|thumb|120px|right|The [[T-norm#Nilpotent-minimum t-norm|nilpotent minimum]] as a rotation of the [[T-norm#Minimum t-norm|minimum]] t-norm]]
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| 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).
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| [[Image:Rotation-Luk-prod-nM-drast-Tnorm-graphs.png|thumb|200px|left|Rotations of the [[T-norm#Lukasiewicz t-norm|Łukasiewicz]], [[T-norm#Product t-norm|product]], [[T-norm#Nilpotent-minimum t-norm|nilpotent minimum]], and [[T-norm#Drastic t-norm|drastic]] t-norm]]
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| The theorem can be generalized by taking for ''N'' any ''strong negation'', that is, an [[Involution (mathematics)|involutive]] strictly decreasing continuous function on [0, 1], and for ''t'' taking the unique [[Fixed point (mathematics)|fixed point]] of ''N''.
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| The resulting t-norm enjoys the following ''rotation invariance'' property with respect to ''N'':
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| :''T''(''x'', ''y'') ≤ ''z'' if and only if ''T''(''y'', ''N''(''z'')) ≤ ''N''(''x'') for all ''x'', ''y'', ''z'' in [0, 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 ''T''<sub>rot</sub>.
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| == See also ==
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| * [[T-norm]]
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| * [[T-norm fuzzy logics]]
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| == References ==
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| * Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), ''Triangular Norms''. Dordrecht: Kluwer. ISBN 0-7923-6416-3.
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| * Fodor, János (2004), [http://www.uni-obuda.hu/journal/Fodor_2.pdf "Left-continuous t-norms in fuzzy logic: An overview"]. ''Acta Polytechnica Hungarica'' '''1'''(2), ISSN 1785-8860 [http://www.uni-obuda.hu/journal/]
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| * Dombi, József (1982), [http://www.dopti.com/~dombi/publications/1982-J.Dombi---A_general_class_of_fuzzy_operators.pdf "A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators"]. ''[[Fuzzy Sets and Systems]]'' '''8''', 149–163.
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| * Jenei, Sándor (2000), "Structure of left-continuous t-norms with strong induced negations. (I) Rotation construction". ''[[Journal of Applied Non-Classical Logics]]'' '''10''', 83–92.
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| * Mirko Navara (2007), [http://www.scholarpedia.org/article/Triangular_Norms_and_Conorms "Triangular norms and conorms"], Scholarpedia [http://www.scholarpedia.org/].
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| [[Category:Fuzzy logic]]
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