In mathematics, a t-norm (also T-norm or, unabbreviated, triangular 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 the triangle inequality of ordinary metric spaces.

Definition

A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following properties: 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 *. The defining conditions of the t-norm are exactly those of a partially ordered abelian monoid on the real unit interval [0, 1]. (Cf. ordered group.) 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 * ... * x (n times) equals 0. A t-norm * 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 * ... * x (n times) is less than or equal to y. The usual partial ordering of t-norms is pointwise, that is, 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

Properties of t-norms

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(a, 0) = 0 for all a in [0, 1]. 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 [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 fy(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 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 (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 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 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 \top, there is a unique binary operation \Rightarrow on [0, 1] such that 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 \top is often denoted by \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 \Rightarrow is the residuum of a left-continuous t-norm \top, then Consequently, for all x, y in the unit interval, and If * is a left-continuous t-norm and \Rightarrow its residuum, then If * 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.

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 \top, 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:

Properties of t-conorms

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.:

Non-standard negators

A negator is a monotonically decreasing mapping such that n(0) = 1 and n(1) = 0. A negator n is called The standard (canonical) 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 (T,⊥,n) such that