T-norm


In mathematics, a t-norm is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces.

Definition

A t-norm is a function T: × → which satisfies the following properties:
Since a t-norm is a binary algebraic operation on the interval , infix algebraic notation is also common, with the t-norm usually denoted by .
The defining conditions of the t-norm are exactly those of the partially ordered Abelian monoid on the real unit interval . The monoidal operation of any partially ordered Abelian monoid L is therefore by some authors called a triangular norm on L.

Motivations and applications

T-norms are a generalization of the usual two-valued logical conjunction, studied by classical logic, for fuzzy logics. Indeed, the classical Boolean conjunction is both commutative and associative. The monotonicity property ensures that the degree of truth of conjunction does not decrease if the truth values of conjuncts increase. The requirement that 1 be an identity element corresponds to the interpretation of 1 as true. Continuity, which is often required from fuzzy conjunction as well, expresses the idea that, roughly speaking, very small changes in truth values of conjuncts should not macroscopically affect the truth value of their conjunction.
T-norms are also used to construct the intersection of fuzzy sets or as a basis for aggregation operators. In probabilistic metric spaces, t-norms are used to generalize triangle inequality of ordinary metric spaces. Individual t-norms may of course frequently occur in further disciplines of mathematics, since the class contains many familiar functions.

Classification of t-norms

A t-norm is called continuous if it is continuous as a function, in the usual interval topology on 2.
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 is its nilpotent element, i.e., there is a natural number n such that x ... x equals 0.
A t-norm is called Archimedean if it has the Archimedean property, i.e., if for each x, y in the open interval there is a natural number n such that x ... x is less than or equal to y.
The usual partial ordering of t-norms is pointwise, i.e.,
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 conjunction it represents.

Prominent examples

The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm:
For every t-norm T, the number 0 acts as null element: T = 0 for all a in .
A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval or .

Properties of continuous t-norms

Although real functions of two variables can be continuous in each variable without being continuous on 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 fy = T are continuous for each y in . 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 idempotents.
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 are nilpotent. If it is the case that all elements in are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that
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 = max on 2.
For each continuous t-norm, the set of its idempotents is a closed subset of . Its complement — the set of all elements which 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 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 similar characterization theorem for non-continuous t-norms is not known, only some non-exhaustive methods for the construction of t-norms have been found.

Residuum

For any left-continuous t-norm, there is a unique binary operation on such that
for all x, y, z in . This operation is called the residuum of the t-norm. In prefix notation, the residuum to a t-norm is often denoted by or by the letter R.
The interval 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 : the residuum forms a right adjoint R to the functor T for each x in the lattice 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.

Basic properties of residua

If is the residuum of a left-continuous t-norm, then
Consequently, for all x, y in the unit interval,
and
If is a left-continuous t-norm and its residuum, then
If is continuous, then equality holds in the former.

Residua of prominent left-continuous t-norms

If x ≤ y, then R = 1 for any residuum R. The following table therefore gives the values of prominent residua only for x > y.
Residuum of theNameValue for x > yGraph
Minimum t-normStandard Gōdel implicationy
Product t-normGoguen implicationy / x
Łukasiewicz t-normStandard Łukasiewicz implication1 – x + y
Nilpotent minimummax

T-conorms

T-conorms are dual to t-norms under the order-reversing operation which assigns 1 – x to x on . Given a t-norm, the complementary conorm is defined by
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:
T-conorms are used to represent logical disjunction in fuzzy logic and union in fuzzy set theory.

Examples of t-conorms

Important t-conorms are those dual to prominent t-norms:
Many [|properties of t-conorms] can be obtained by dualizing the [|properties of t-norms], for example:
Further properties result from the relationships between t-norms and t-conorms or their interplay with other operators, e.g.:
A negator is a monotonous falling, i. e. order-reversion mapping with and .
A negator n is called
The standard negator is
,
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 iff
  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. .