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: Commutativity: T(a, b) = T(b, a) Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d Associativity: T(a, T(b, c)) = T(T(a, b), c) The number 1 acts as identity element: T(a, 1) = a 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 ∗ {\displaystyle } . 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 ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) equals 0. A t-norm ∗ {\displaystyle } 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 ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) is less than or equal to y. The usual partial ordering of t-norms is pointwise, that is, T1 ≤ T2 if T1(a, b) ≤ T2(a, b) for all a, b in [0, 1]. As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of t-norm fuzzy logics, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents. == Prominent examples == Minimum t-norm ⊤ m i n ( a , b ) = min { a , b } , {\displaystyle \top _{\mathrm {min} }(a,b)=\min\{a,b\},} also called the Gödel t-norm, as it is the standard semantics for conjunction in Gödel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the properties of t-norms below). Product t-norm ⊤ p r o d ( a , b ) = a ⋅ b {\displaystyle \top _{\mathrm {prod} }(a,b)=a\cdot b} (the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm. Łukasiewicz t-norm ⊤ L u k ( a , b ) = max { 0 , a + b − 1 } . {\displaystyle \top _{\mathrm {Luk} }(a,b)=\max\{0,a+b-1\}.} The name comes from the fact that the t-norm is the standard semantics for strong conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm. Drastic t-norm ⊤ D ( a , b ) = { b if a = 1 a if b = 1 0 otherwise. {\displaystyle \top _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=1\\a&{\mbox{if }}b=1\\0&{\mbox{otherwise.}}\end{cases}}} The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the properties of t-norms below). It is a right-continuous Archimedean t-norm. Nilpotent minimum ⊤ n M ( a , b ) = { min ( a , b ) if a + b > 1 0 otherwise {\displaystyle \top _{\mathrm {nM} }(a,b)={\begin{cases}\min(a,b)&{\mbox{if }}a+b>1\\0&{\mbox{otherwise}}\end{cases}}} is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm. Hamacher product ⊤ H 0 ( a , b ) = { 0 if a = b = 0 a b a + b − a b otherwise {\displaystyle \top _{\mathrm {H} _{0}}(a,b)={\begin{cases}0&{\mbox{if }}a=b=0\\{\frac {ab}{a+b-ab}}&{\mbox{otherwise}}\end{cases}}} is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer–Sklar t-norms. == Properties of t-norms == The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm: ⊤ D ( a , b ) ≤ ⊤ ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top (a,b)\leq \mathrm {\top _{min}} (a,b),} for any t-norm ⊤ {\displaystyle \top } and all a, b in [0, 1]. In particular, we have that: ⊤ D ( a , b ) ≤ ⊤ L u k ( a , b ) ≤ ⊤ p r o d ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top _{\mathrm {Luk} }(a,b)\leq \top _{\mathrm {prod} }(a,b)\leq \mathrm {\top _{min}} (a,b),} for all a, b in [0, 1]. 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 ⊤ ( x , y ) = f − 1 ( ⊤ L u k ( f ( x ) , f ( y ) ) ) . {\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).} 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 t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm. 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 ⊤ {\displaystyle \top } , there is a unique binary operation ⇒ {\displaystyle \Rightarrow } on [0, 1] such that ⊤ ( z , x ) ≤ y {\displaystyle \top (z,x)\leq y} if and only if z ≤ ( x ⇒ y ) {\displaystyle z\leq (x\Rightarrow y)} 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 ⊤ {\displaystyle \top } is often denoted by ⊤ → {\displaystyle {\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
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