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 triangle inequality of ordinary metric spaces.
Definition.
A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] which 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 formula_1.
The defining conditions of the t-norm are exactly those of the 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".
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" (and consequently 0 as "false"). 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 (see fuzzy set operations). 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 [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 its nilpotent element, i.e., there is a natural number "n" such that "x" formula_1 ... formula_1 "x" ("n" times) equals 0.
A t-norm formula_1 is called "Archimedean" if it has the Archimedean property, i.e., if for each "x", "y" in the open interval (0, 1) there is a natural number "n" such that "x" formula_1 ... formula_1 "x" ("n" times) 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" (in terms of logical strength) conjunction it represents.
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 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 (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 formula_14, there is a unique binary operation formula_17 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 to a t-norm formula_14 is often denoted by formula_21 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 formula_17 is the residuum of a left-continuous t-norm formula_14, then
Consequently, for all "x", "y" in the unit interval,
and
If formula_1 is a left-continuous t-norm and formula_17 its residuum, then
If formula_1 is continuous, then equality holds in the former.
Residua of prominent 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 which assigns 1 – "x" to "x" on [0, 1]. Given a t-norm formula_14, 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 formula_42 is a monotonous falling, i.e., order-reversion mapping with formula_43 and formula_44 (in other notation: formula_45 and formula_46). 
A negator "n" is called
The standard (canonical) negator is 
formula_49, 
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