T-norm

In

.

Definition

A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following properties:

• Commutativity
  : T(a, b) = T(b, a)
• Monotonicity
  : T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d
• Associativity
  : T(a, T(b, c)) = T(T(a, b), c)
• The number 1 acts as identity element: T(a, 1) = a

Since a t-norm is a binary algebraic operation on the interval [0, 1], infix algebraic notation is also common, with the t-norm usually denoted by ${\displaystyle *}$.

The defining conditions of the t-norm are exactly those of a

.) The monoidal operation of any partially ordered abelian monoid L is therefore by some authors called a triangular norm on L.

Classification of t-norms

A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]². (Similarly for left- and right-continuity.)

A t-norm is called strict if it is continuous and strictly monotone.

A t-norm is called nilpotent if it is continuous and each x in the open interval (0, 1) is nilpotent, that is, there is a natural number n such that x ${\displaystyle *}$ ... ${\displaystyle *}$ x (n times) equals 0.

A t-norm ${\displaystyle *}$ is called Archimedean if it has the Archimedean property, that is, if for each x, y in the open interval (0, 1) there is a natural number n such that x ${\displaystyle *}$ ... ${\displaystyle *}$ x (n times) is less than or equal to y.

The usual partial ordering of t-norms is pointwise, that is,

T₁ ≤ T₂   if   T₁(a, b) ≤ T₂(a, b) for all a, b in [0, 1].

As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of fuzzy logic, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents.

Prominent examples

• Minimum t-norm ${\displaystyle \top _{\mathrm {min} }(a,b)=\min\{a,b\},}$ also called the Gödel t-norm, as it is the standard semantics for conjunction in
  Gödel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the properties of t-norms
  below).

• Product t-norm ${\displaystyle \top _{\mathrm {prod} }(a,b)=a\cdot b}$ (the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in
  product fuzzy logic
  . It is a strict Archimedean t-norm.

• Łukasiewicz t-norm ${\displaystyle \top _{\mathrm {Luk} }(a,b)=\max\{0,a+b-1\}.}$ The name comes from the fact that the t-norm is the standard semantics for strong conjunction in
  Łukasiewicz fuzzy logic
  . It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm.

• Drastic t-norm

${\displaystyle \top _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=1\\a&{\mbox{if }}b=1\\0&{\mbox{otherwise.}}\end{cases}}}$

The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the properties of t-norms below). It is a right-continuous Archimedean t-norm.

• Nilpotent minimum

${\displaystyle \top _{\mathrm {nM} }(a,b)={\begin{cases}\min(a,b)&{\mbox{if }}a+b>1\\0&{\mbox{otherwise}}\end{cases}}}$

is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm.

• Hamacher product

${\displaystyle \top _{\mathrm {H} _{0}}(a,b)={\begin{cases}0&{\mbox{if }}a=b=0\\{\frac {ab}{a+b-ab}}&{\mbox{otherwise}}\end{cases}}}$

is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer–Sklar t-norms.

Properties of t-norms

The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm:

${\displaystyle \top _{\mathrm {D} }(a,b)\leq \top (a,b)\leq \mathrm {\top _{min}} (a,b),}$ for any t-norm ${\displaystyle \top }$ and all a, b in [0, 1].

For every t-norm T, the number 0 acts as null element: T(a, 0) = 0 for all a in [0, 1].

A t-norm T has

elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval [0, a] or [0, a), for some a in [0, 1].

Properties of continuous t-norms

Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]², this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions f_y(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left- and right-continuity of a t-norm.

A continuous t-norm is Archimedean if and only if 0 and 1 are its only

.

A continuous Archimedean t-norm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that

${\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).}$

If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function p(x, y) = max(0.25, x · y) on [0.25, 1]².

For each continuous t-norm, the set of its idempotents is a closed subset of [0, 1]. Its complement—the set of all elements that are not idempotent—is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows:

A t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm.

A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the construction of t-norms have been found.

Residuum

For any left-continuous t-norm ${\displaystyle \top }$, there is a unique binary operation ${\displaystyle \Rightarrow }$ on [0, 1] such that

${\displaystyle \top (z,x)\leq y}$ if and only if ${\displaystyle z\leq (x\Rightarrow y)}$

for all x, y, z in [0, 1]. This operation is called the residuum of the t-norm. In prefix notation, the residuum of a t-norm ${\displaystyle \top }$ is often denoted by ${\displaystyle {\vec {\top }}}$ or by the letter R.

The interval [0, 1] equipped with a t-norm and its residuum forms a residuated lattice. The relation between a t-norm T and its residuum R is an instance of adjunction (specifically, a Galois connection): the residuum forms a right adjoint R(x, –) to the functor T(–, x) for each x in the lattice [0, 1] taken as a poset category.

In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often called R-implication).

Basic properties of residua

If ${\displaystyle \Rightarrow }$ is the residuum of a left-continuous t-norm ${\displaystyle \top }$, then

${\displaystyle (x\Rightarrow y)=\sup\{z\mid \top (z,x)\leq y\}.}$

Consequently, for all x, y in the unit interval,

${\displaystyle (x\Rightarrow y)=1}$ if and only if ${\displaystyle x\leq y}$

and

${\displaystyle (1\Rightarrow y)=y.}$

If ${\displaystyle *}$ is a left-continuous t-norm and ${\displaystyle \Rightarrow }$ its residuum, then

${\displaystyle {\begin{array}{rcl}\min(x,y)&\geq &x*(x\Rightarrow y)\\\max(x,y)&=&\min((x\Rightarrow y)\Rightarrow y,(y\Rightarrow x)\Rightarrow x).\end{array}}}$

If ${\displaystyle *}$ is continuous, then equality holds in the former.

Residua of common left-continuous t-norms

If x ≤ y, then R(x, y) = 1 for any residuum R. The following table therefore gives the values of prominent residua only for x > y.

Residuum of the	Name	Value for x > y	Graph
Minimum t-norm	Standard Gödel implication	y
Product t-norm	Goguen implication	y / x
Łukasiewicz t-norm	Standard Łukasiewicz implication	1 – x + y
Nilpotent minimum	Fodor implication	max(1 – x, y)

T-conorms

T-conorms (also called S-norms) are dual to t-norms under the order-reversing operation that assigns 1 – x to x on [0, 1]. Given a t-norm ${\displaystyle \top }$, the complementary conorm is defined by

${\displaystyle \bot (a,b)=1-\top (1-a,1-b).}$

This generalizes De Morgan's laws.

It follows that a t-conorm satisfies the following conditions, which can be used for an equivalent axiomatic definition of t-conorms independently of t-norms:

• Commutativity: ⊥(a, b) = ⊥(b, a)
• Monotonicity: ⊥(a, b) ≤ ⊥(c, d) if a ≤ c and b ≤ d
• Associativity: ⊥(a, ⊥(b, c)) = ⊥(⊥(a, b), c)
• Identity element: ⊥(a, 0) = a

T-conorms are used to represent

.

Examples of t-conorms

Important t-conorms are those dual to prominent t-norms:

• Maximum t-conorm ${\displaystyle \bot _{\mathrm {max} }(a,b)=\max\{a,b\}}$, dual to the minimum t-norm, is the smallest t-conorm (see the
  Gödel fuzzy logic
  and for weak disjunction in all t-norm based fuzzy logics.

• Probabilistic sum ${\displaystyle \bot _{\mathrm {sum} }(a,b)=a+b-a\cdot b=1-(1-a)\cdot (1-b)}$ is dual to the product t-norm. In
  product fuzzy logic
  in which it is definable (e.g., those containing involutive negation).

• Bounded sum ${\displaystyle \bot _{\mathrm {Luk} }(a,b)=\min\{a+b,1\}}$ is dual to the Łukasiewicz t-norm. It is the standard semantics for strong disjunction in
  Łukasiewicz fuzzy logic
  .

• Drastic t-conorm

${\displaystyle \bot _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=0\\a&{\mbox{if }}b=0\\1&{\mbox{otherwise,}}\end{cases}}}$

dual to the drastic t-norm, is the largest t-conorm (see the properties of t-conorms below).

• Nilpotent maximum, dual to the nilpotent minimum:

${\displaystyle \bot _{\mathrm {nM} }(a,b)={\begin{cases}\max(a,b)&{\mbox{if }}a+b<1\\1&{\mbox{otherwise.}}\end{cases}}}$

• Einstein sum (compare the velocity-addition formula under special relativity)

${\displaystyle \bot _{\mathrm {H} _{2}}(a,b)={\frac {a+b}{1+ab}}}$

is a dual to one of the Hamacher t-norms.

Properties of t-conorms

Many properties of t-conorms can be obtained by dualizing the properties of t-norms, for example:

• For any t-conorm ⊥, the number 1 is an annihilating element: ⊥(a, 1) = 1, for any a in [0, 1].
• Dually to t-norms, all t-conorms are bounded by the maximum and the drastic t-conorm:

${\displaystyle \mathrm {\bot _{max}} (a,b)\leq \bot (a,b)\leq \bot _{\mathrm {D} }(a,b)}$, for any t-conorm ${\displaystyle \bot }$ and all a, b in [0, 1].

Further properties result from the relationships between t-norms and t-conorms or their interplay with other operators, e.g.:

• A t-norm T
  distributes
  over a t-conorm ⊥, i.e.,

T(x, ⊥(y, z)) = ⊥(T(x, y), T(x, z)) for all x, y, z in [0, 1],

if and only if ⊥ is the maximum t-conorm. Dually, any t-conorm distributes over the minimum, but not over any other t-norm.

Non-standard negators

A negator ${\displaystyle n\colon [0,1]\to [0,1]}$ is a monotonically decreasing mapping such that ${\displaystyle n(0)=1}$ and ${\displaystyle n(1)=0}$. A negator n is called

• strict in case of strict monotonocity, and
• strong if it is strict and involutive, that is, ${\displaystyle n(n(x))=x}$ for all ${\displaystyle x}$ in [0, 1].

The standard (canonical) negator is ${\displaystyle n(x)=1-x,\ x\in [0,1]}$, which is both strict and strong. As the standard negator is used in the above definition of a t-norm/t-conorm pair, this can be generalized as follows:

A De Morgan triplet is a triple (T,⊥,n) such that^[1]

1. T is a t-norm
2. ⊥ is a t-conorm according to the axiomatic definition of t-conorms as mentioned above
3. n is a strong negator
4. ${\displaystyle \forall a,b\in [0,1]\colon \ n({\perp }(a,b))=\top (n(a),n(b))}$.

See also

• Construction of t-norms
• T-norm fuzzy logics

References

• Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer.
  ISBN 0-7923-6416-3
  .
• ISBN 0-7923-5238-6
• Cignoli, Roberto L.O.;
  ISBN 0-7923-6009-5
• Fodor, János (2004), "Left-continuous t-norms in fuzzy logic: An overview". Acta Polytechnica Hungarica 1(2),
  ISSN 1785-8860 [1]