# Triangular norms and conorms

Triangular norms and conorms are operations which generalize the logical conjunction and logical disjunction to fuzzy logic. They are a natural interpretation of the conjunction and disjunction in the semantics of mathematical fuzzy logics [Hájek (1998)] and they are used to combine criteria in multi-criteria decision making.

## Triangular Norms

### Definition

A **triangular norm** (abbreviation **t-norm**) is a binary operation \(T\) on the interval [0,1] satisfying the following conditions:

- \(T(x,y)=T(y,x)\) (commutativity)
- \(T(x,T(y,z))=T(T(x,y),z)\) (associativity)
- \(y\le z\Longrightarrow T(x,y)\le T(x,z)\) (monotonicity)
- \(T(x,1)=x\) (neutral element 1)

### Examples

- \(T_M(x,y)=\min(x,y) \) (
**minimum or Gödel t-norm**) - \(T_P(x,y)=x\cdot y\) (
**product t-norm**) - \(T_L(x,y)=\max(x+y-1,0)\) (
**Lukasiewicz t-norm**)

No t-norm can attain greater values than \(T_M\ .\)
There are many parametrized families of t-norms [Klement et al. (2000)].
The **Frank t-norms** are defined for all
\(r>0, r \ne 1\) by
\[T_{F_{r}}(x,y)=
\log_r\left(1+\frac{(r^x-1)(r^y-1)}{r-1}\right)\,.\]
The limit elements of this family are the above t-norms\[T_{F_0}=T_M\ ,\]
\(T_{F_1}=T_P\ ,\) and \(T_{F_{\infty}}=T_L\ .\)
The only t-norms which are rational functions are the **Hamacher t-norms** defined for all
\(r>0\) by
\[T_{H_r}(x,y) =
\frac{xy}{r + (1-r)(x+y-xy)}\]
and for \(r=0\) by
\[T_{H_0}(x,y)= \frac{x y}{x+y-xy}\]
(\(T_{H_0}(0,0)=0\)).

### Classification and representations

The **idempotents of a t-norm** \(T\) are those \(x\) satisfying
\(T(x,x)=x\ .\) The bounds 0 and 1 are trivial idempotents.
A t-norm is called **Archimedean** if each sequence \(x_n, n\in\mathbb{N},\)
where \(x_1<1\) and
\(x_{n+1}=T(x_n,x_n),\)
converges to 0.
A continuous t-norm is Archimedean iff it has no idempotents between 0 and 1.
A continuous Archimedean t-norm is called **strict** if
\(T(x,x)>0\) for all \(x>0\ .\)
Continuous Archimedean t-norms which are not strict are called **nilpotent**.
The product t-norm is strict, the Lukasiewicz t-norm is nilpotent.

If \(T^*\) is a t-norm and \(h\colon [0,1]\to[0,1]\) is an increasing bijection, then \[\tag{1} T(x,y)=h^{-1}(T^*(h(x),h(y)))\]

is a t-norm. This way,
all strict t-norms can be obtained from the product t-norm and
all nilpotent t-norms from the Lukasiewicz t-norm.
(These t-norms serve as *universal examples* of these classes.)

More generally, each continuous Archimedean t-norm can be obtained from the product t-norm using the formula
\[T(x,y)=
\left\{\begin{array}{ll}
\theta^{-1}(\theta(x)\cdot\theta(y)) & \mbox{if }\theta(x)\cdot\theta(y)\ge b,
\\0 & \mbox{otherwise.}
\end{array}\right.\]
where \(\theta\colon [0,1]\to[b,1]\ (b\in\left[0,1\right[)\) is an increasing bijection called a **multiplicative generator** of \(T\ .\) (It is not uniquely determined by \(T\ .\))
Each continuous Archimedean t-norm has also a (non-unique) **additive generator**, which is a decreasing bijection
\(t\colon [0,1]\to[0,B]\ (B\in\left]0,\infty\right])\) such that
\[T(x,y)=
\left\{\begin{array}{ll}
t^{-1}(t(x)+t(y)) & \mbox{if }t(x)+t(y)\le B,
\\0 & \mbox{otherwise.}
\end{array}\right.\]

### Generalizations

More generally, triangular norms can be defined (exactly the same way) on any ordered set with an upper bound (serving as the neutral element). They can be also restricted to (possibly finite) subsets of the unit interval. The term *triangular norm* is usually used for these operations, too. In particular, a t-norm \(T\) on an interval [a,b] can be defined by (1), where \(h\colon [a,b]\to[0,1]\) is an increasing bijection and
\(T^*\) is a t-norm on [0,1].

For a family of disjoint subintervals
\(\left]a_j,b_j\right[\subseteq[0,1],\ j\in J,\)
we may define a t-norm \(T\) called an **ordinal sum**:
\[T(x,y)=
\left\{\begin{array}{ll}
h_j^{-1}(T_j(h_j(x),h_j(y))) & \mbox{if } x,y \in \left]a_j,b_j\right[\mbox{ for some } j\in J,
\\\min(x,y) & \mbox{otherwise,}
\end{array}\right.\]
where \(h_j\colon [a_j,b_j]\to[0,1]\) are increasing bijections and \(T_j\) are t-norms on \([0,1],\,
j\in J.\)
All continuous t-norms are ordinal sums of Archimedean t-norms, we may choose
\(T_j\in \{T_L,T_P\}.\)

There are t-norms which are not continuous or even not measurable.

## Triangular conorms

### Definition

The dual notion to a triangular norm is a **triangular conorm** (abbreviation **t-conorm**, also **s-norm**), \(S\ .\) Its neutral element is 0 instead of 1, all other conditions remain unchanged:

- \(S(x,y)=S(y,x)\) (commutativity)
- \(S(x,S(y,z))=S(S(x,y),z)\) (associativity)
- \(y\le z\Longrightarrow S(x,y)\le S(x,z)\) (monotonicity)
- \(S(x,0)=x\) (neutral element 0)

### Examples of t-conorms

- \(S_M(x,y)=\max(x,y)\) (
**maximum or Gödel t-conorm**) - \(S_P(x,y)=x+y-x\cdot y\) (
**product t-conorm**,**probabilistic sum**) - \(S_L(x,y)=\min(x+y,1)\) (
**Lukasiewicz t-conorm**,**bounded sum**))

No t-conorm can attain smaller values than \(S_M\ .\)

If \(T\) is a t-norm, then
\(S(x,y)=1-T(1-x,1-y)\)
is a t-conorm, and vice versa. We obtain a *dual pair* \((T,S)\) of a t-norm and a t-conorm.
(Instead of the *standard fuzzy negation*,
\(x\mapsto 1-x\ ,\) another strong fuzzy negation can be used in the duality formula.)

The classification and representations of t-conorms are dual to those of t-norms. Each continuous Archimedean t-conorm \(S\) has a (non-unique) additive generator, which is an increasing bijection \(s\colon [0,1]\to[0,B]\ (B\in\left]0,\infty\right])\) such that \[S(x,y)= \left\{\begin{array}{ll} s^{-1}(s(x)+s(y)) & \mbox{if }s(x)+s(y)\le B, \\1 & \mbox{otherwise.} \end{array}\right.\]

## Derived operations

### Fuzzy intersections and unions

If \(A,B\) are fuzzy sets and
\(\mu_A,\mu_B\) their *membership functions*, then the fuzzy intersection \(C\) of \(A\) and \(B\) has the membership function
\(\mu_C(u)=T(\mu_A(u),\mu_B(u))\ .\) Thus a t-norm is sometimes called a *fuzzy intersection*.
Depending on the choice of a t-norm, we obtain different fuzzy intersections.
Dually, a t-conorm corresponds to a *fuzzy union*.

### Residua (fuzzy implications)

The residuum of a left-continuous t-norm is defined by \(I(x,y)=\max\{z\in [0,1]\mid T(x,z)\le y\}.\) It is usually used as a fuzzy implication [Nguyen and Walker (2000)].

Originally t-norms appeared in the context of probabilistic metric spaces [Schweizer and Sklar (1983)].
Then they were used as a natural interpretation of the conjunction in the semantics of mathematical fuzzy logics [Hájek (1998)] and they are used to combine criteria in multi-criteria decision making.
T-norms and t-conorms allow to evaluate the truth degrees of compound formulas.
They are applied in fuzzy control to formulate assumptions of rules as conjunctions (fuzzy intersections) of fuzzy sets called *antecedents* or *premises*. (In such applications, the minimum or product t-norm are usually used because of a lack of motivation for other t-norms [Driankov et al. (1993)].)
The Lukasiewicz t-conorm is closely related to the basic binary operation of MV-algebras.
T-norms and t-conorms form also examples of aggregation operators.
They play a crucial role in the axiomatic definition of the concept of triangular
norm based measure and, in particular, of a concept of *probability of fuzzy events*; the Frank family of t-norms and t-conorms plays a particular role here [Butnariu and Klement (1993)].

T-norms overlap with copulas [Nelsen (1999), Alsina et al. (2006)]: commutative associative copulas are t-norms;
t-norms which satisfy the 1-*Lipschitz condition* are copulas.
Some families of t-norms are known as families of copulas under different names.

## References

- Alsina, Claudi; Frank, Maurice J.; and Schweizer, Berthold:
*Associative Functions: Triangular Norms and Copulas.*World Scientific, 2006. ISBN 981-256-671-6. doi:10.1142/6036. - Butnariu, Dan and Klement, Erich Peter:
*Triangular Norm-Based Measures and Games wih Fuzzy Coalitions.*Kluwer, Dordrecht, Netherlands, 1993. ISBN 0-7923-2369-6. doi:10.1007/978-94-017-3602-2_4. - Driankov, Dimiter; Hellendoorn, Hans; and Reinfrank, Michael:
*An Introduction to Fuzzy Control.*Springer, Berlin/Heidelberg, 1993. ISBN 3-540-56362-8. doi:10.1007/978-3-662-11131-4_1. - Hájek, Petr:
*Metamathematics of Fuzzy Logic.*Kluwer, Dordrecht, 1998. ISBN 0-7923-5238-6 - Klement, Erich Peter; Mesiar, Radko; and Pap, Endre:
*Triangular Norms.*Kluwer, Dordrecht, 2000. ISBN 0-7923-6416-3. doi:10.1007/978-94-015-9540-7. - Nelsen, Roger B.:
*An Introduction to Copulas.*Lecture Notes in Statistics 139, Springer, New York, 1999. ISBN 0-387-98623-5. doi:10.1007/978-1-4757-3076-0. - Nguyen, Hung T. and Walker, Elbert A.:
*A First Course in Fuzzy Logic.*2nd ed., Chapman & Hall/CRC, Boca Raton/London/New York/Washington, 2000. ISBN 0-8493-1659-6. - Schweizer, Berthold and Sklar, Abe:
*Probabilistic Metric Spaces.*North-Holland, New York, 1983. ISBN 0-444-00666-4.

**Internal references**

- Milan Mares (2006) Fuzzy sets. Scholarpedia, 1(10):2031. doi:10.4249/scholarpedia.2031.

## External links

## See also

Copulas, Fuzzy Control, Fuzzy Implications, Fuzzy Negations, Fuzzy Logic, Fuzzy Sets