T-norm
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.
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].
The monoidal operation of any partially ordered abelian monoid L is therefore by some authors called a triangular norm on L. A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]2.
In the semantics of fuzzy logic, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents.
Analogous theorems hold for left- and right-continuity of a t-norm.
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.
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.
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.
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).
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 (also called S-norms) are dual to t-norms under the order-reversing operation that assigns 1 – x to x on [0, 1].
, 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.
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
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]
