Construction of 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 ) = { sup { x ∈ [ a , b ] ∣ f ( x ) < y } for  f  non-decreasing sup { x ∈ [ a , b ] ∣ f ( x ) > y } for  f  non-increasing.

Additive generators

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 T ( x , y ) = f − 1 ( min ( f ( 0 + ) , f ( x ) + f ( y ) ) ) . The corresponding residuum can then be expressed as ( x ⇒ y ) = f − 1 ( max ( 0 , f ( y ) − f ( x ) ) ) . And the biresiduum as ( x ⇔ y ) = f − 1 ( | f ( x ) − f ( y ) | ) .

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.)

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

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.

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_p parameterized by p is increasing if T_p(x, y) ≤ T_q(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 T_p is continuous with respect to the parameter p if

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 p S S ( x , y ) = { T min ( x , y ) if  p = − ∞ ( x p + y p − 1 ) 1 / p if  − ∞ < p < 0 T p r o d ( x , y ) if  p = 0 ( max ( 0 , x p + y p − 1 ) ) 1 / p if  0 < p < + ∞ T D ( x , y ) if  p = + ∞ .

A Schweizer–Sklar t-norm T p S S 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 T p S S for −∞ < p < +∞ is

f p S S ( x ) = { − log ⁡ x if  p = 0 1 − x p p otherwise.

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 p H ( x , y ) = { T D ( x , y ) if  p = + ∞ 0 if  p = x = y = 0 x y p + ( 1 − p ) ( x + y − x y ) otherwise.

The t-norm T 0 H is called the Hamacher product.

Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm T p 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 T p H for p < +∞ is

f p H ( x ) = { 1 − x x if  p = 0 log ⁡ p + ( 1 − p ) x x otherwise.

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 p F ( x , y ) = { T m i n ( x , y ) if  p = 0 T p r o d ( x , y ) if  p = 1 T L u k ( x , y ) if  p = + ∞ log p ⁡ ( 1 + ( p x − 1 ) ( p y − 1 ) p − 1 ) otherwise.

The Frank t-norm T p F is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for T p F is

f p F ( x ) = { − log ⁡ x if  p = 1 1 − x if  p = + ∞ log ⁡ p − 1 p x − 1 otherwise.

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 p Y ( x , y ) = { T D ( x , y ) if  p = 0 max ( 0 , 1 − ( ( 1 − x ) p + ( 1 − y ) p ) 1 / p ) if  0 < p < + ∞ T m i n ( x , y ) if  p = + ∞

The Yager t-norm T p 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 T p Y for 0 < p < +∞ arises from the Łukasiewicz t-norm by raising its additive generator to the power of p. An additive generator of T p Y for 0 < p < +∞ is

f p Y ( 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 p A A ( x , y ) = { T D ( x , y ) if  p = 0 e − ( | log ⁡ x | p + | log ⁡ y | p ) 1 / p if  0 < p < + ∞ T m i n ( x , y ) if  p = + ∞

The Aczél–Alsina t-norm T p A A 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 T p A A for 0 < p < +∞ arises from the product t-norm by raising its additive generator to the power of p. An additive generator of T p A A for 0 < p < +∞ is

f p A A ( 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 p D ( x , y ) = { 0 if  x = 0  or  y = 0 T D ( x , y ) if  p = 0 T m i n ( x , y ) if  p = + ∞ 1 1 + ( ( 1 − x x ) p + ( 1 − y y ) p ) 1 / p otherwise.

The Dombi t-norm T p 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 T p 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 T p D for 0 < p < +∞ is

f p D ( x ) = ( 1 − x x ) 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 p S W ( x , y ) = { T D ( x , y ) if  p = − 1 max ( 0 , x + y − 1 + p x y 1 + p ) if  − 1 < p < + ∞ T p r o d ( x , y ) if  p = + ∞

The Sugeno–Weber t-norm T p S W 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 T p S W for 0 < p < +∞ [sic] is

f p S W ( x ) = { 1 − x if  p = 0 1 − log 1 + p ⁡ ( 1 + p x ) otherwise.

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 T_i for i in an index set I be a family of t-norms and (a_i, b_i) 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 ) = { a i + ( b i − a i ) ⋅ T i ( x − a i b i − a i , y − a i b i − a i ) if  x , y ∈ [ a i , b i ] 2 min ( x , y ) otherwise is a t-norm.

The resulting t-norm is called the ordinal sum of the summands (T_i, a_i, b_i) for i in I, denoted by

T = ⨁ i ∈ I ( T i , a i , b i ) ,

or ( T 1 , a 1 , b 1 ) ⊕ ⋯ ⊕ ( 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 T_min. 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_i = 0 and T_i has zero divisors. (Analogously for nilpotent elements.)

If T = ⨁ i ∈ I ( T i , a i , b i ) is a left-continuous t-norm, then its residuum R is given as follows:

R ( x , y ) = { 1 if  x ≤ y a i + ( b i − a i ) ⋅ R i ( x − a i b i − a i , y − a i b i − a i ) if  a i < y < x ≤ b i y otherwise.

where R_i is the residuum of T_i, 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 ∣ T 1 ( z , x ) ≤ y } . Then the function T r o t = { T 1 ( x , y ) if  x , y ∈ ( t , 1 ] N ( R T 1 ( x , N ( y ) ) ) if  x ∈ ( t , 1 ]  and  y ∈ [ 0 , t ] N ( R T 1 ( y , N ( x ) ) ) if  x ∈ [ 0 , t ]  and  y ∈ ( t , 1 ] 0 if  x , y ∈ [ 0 , t ] 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) ≤ z if and only if T(y, N(z)) ≤ N(x) for all x, y, z in [0, 1].

The negation induced by T_rot is the function N, that is, N(x) = R_rot(x, 0) for all x, where R_rot is the residuum of T_rot.