In mathematics, a tnorm (also Tnorm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multivalued logic, specifically in fuzzy logic. A tnorm 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 tnorms are used to generalize triangle inequality of ordinary metric spaces.
Definition
A tnorm is a function T: [0, 1] × [0, 1] [0, 1] which satisfies the following properties:
Since a tnorm is a binary algebraic operation on the interval [0, 1], infix algebraic notation is also common, with the tnorm usually denoted by *.
The defining conditions of the tnorm 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
Tnorms are a generalization of the usual twovalued 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.
Tnorms 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, tnorms are used to generalize triangle inequality of ordinary metric spaces. Individual tnorms may of course frequently occur in further disciplines of mathematics, since the class contains many familiar functions.
Classification of tnorms
A tnorm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]^{2}. (Similarly for left and rightcontinuity.)
A tnorm is called strict if it is continuous and strictly monotone.
A tnorm 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 * ... * x (n times) equals 0.
A tnorm * 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 * ... * x (n times) is less than or equal to y.
The usual partial ordering of tnorms is pointwise, i.e.,
 T_{1} T_{2} if T_{1}(a, b) T_{2}(a, b) for all a, b in [0, 1].
As functions, pointwise larger tnorms are sometimes called stronger than those pointwise smaller. In the semantics of fuzzy logic, however, the larger a tnorm, the weaker (in terms of logical strength) conjunction it represents.
Prominent examples
Graph of the minimum tnorm (3D and contours)

Minimum tnorm \top_{\mathrm{min}}(a, b) = \min \{a, b\}, also called the G del tnorm, as it is the standard semantics for conjunction in G del fuzzy logic. Besides that, it occurs in most tnorm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest tnorm (see the properties of tnorms below).
Graph of the product tnorm

Product tnorm \top_{\mathrm{prod}}(a, b) = a \cdot b (the ordinary product of real numbers). Besides other uses, the product tnorm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean tnorm.
Graph of the ukasiewicz tnorm

ukasiewicz tnorm \top_{\mathrm{Luk}}(a, b) = \max \{0, a+b1\}. The name comes from the fact that the tnorm is the standard semantics for strong conjunction in ukasiewicz fuzzy logic. It is a nilpotent Archimedean tnorm, pointwise smaller than the product tnorm.
Graph of the drastic tnorm. The function is discontinuous at the lines 0 < x = 1 and 0 < y = 1.

 \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 tnorm is the pointwise smallest tnorm (see the properties of tnorms below). It is a rightcontinuous Archimedean tnorm.
Graph of the nilpotent minimum. The function is discontinuous at the line 0 < x = 1 y < 1.

 \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 tnorm which is leftcontinuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent tnorm.
Graph of the Hamacher product

 \top_{\mathrm{H}_0}(a, b) = \begin{cases} 0 & \mbox{if } a=b=0 \\ \frac{ab}{a+bab} & \mbox{otherwise} \end{cases}
 is a strict Archimedean tnorm, and an important representative of the parametric classes of Hamacher tnorms and Schweizer Sklar tnorms.
Properties of tnorms
The drastic tnorm is the pointwise smallest tnorm and the minimum is the pointwise largest tnorm:

\top_{\mathrm{D}}(a, b) \le \top(a, b) \le \mathrm{\top_{min}}(a, b), for any tnorm \top and all a, b in [0, 1].
For every tnorm T, the number 0 acts as null element: T(a, 0) = 0 for all a in [0, 1].
A tnorm 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 tnorms
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 tnorms: a tnorm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions f_{y}(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left and rightcontinuity of a tnorm.
A continuous tnorm is Archimedean if and only if 0 and 1 are its only idempotents.
A continuous Archimedean tnorm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean tnorm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean tnorm 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 tnorm is isomorphic to the ukasiewicz tnorm; i.e., there is a strictly increasing function f such that
 \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 tnorm is isomorphic to the product tnorm. In other words, all nilpotent tnorms are isomorphic, the ukasiewicz tnorm being their prototypical representative; and all strict tnorms are isomorphic, with the product tnorm as their prototypical example. The ukasiewicz tnorm is itself isomorphic to the product tnorm 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 tnorm, 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 nonoverlapping open intervals. The restriction of the tnorm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the ukasiewicz tnorm or the product tnorm. For such x, y that do not fall into the same open interval of nonidempotents, the tnorm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous tnorms, called the Mostert Shields theorem, since every continuous tnorm can in this way be decomposed, and the described construction always yields a continuous tnorm. The theorem can also be formulated as follows:
 A tnorm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, ukasiewicz, and product tnorm.
A similar characterization theorem for noncontinuous tnorms is not known (not even for leftcontinuous ones), only some nonexhaustive methods for the construction of tnorms have been found.
Residuum
For any leftcontinuous tnorm \top, there is a unique binary operation \Rightarrow on [0, 1] such that

\top(z, x) \le y if and only if z \le (x \Rightarrow y)
for all x, y, z in [0, 1]. This operation is called the residuum of the tnorm. In prefix notation, the residuum to a tnorm \top is often denoted by \vec{\top} or by the letter R.
The interval [0, 1] equipped with a tnorm and its residuum forms a residuated lattice. The relation between a tnorm T and its residuum R is an instance of adjunction: 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 tnorm based fuzzy logics, where conjunction is interpreted by a tnorm, the residuum plays the role of implication (often called Rimplication).
Basic properties of residua
If \Rightarrow is the residuum of a leftcontinuous tnorm \top, then
 (x \Rightarrow y) = \sup\{z\mid\top(z,x) \le y\}.
Consequently, for all x, y in the unit interval,

(x \Rightarrow y) = 1 if and only if x \le y
and
 (1 \Rightarrow y) = y.
If * is a leftcontinuous tnorm and \Rightarrow its residuum, then
 \begin{array}{rcl} \min(x,y) & \ge & x * (x \Rightarrow y) \\ \max(x, y) & = & \min((x \Rightarrow y)\Rightarrow y, (y \Rightarrow x)\Rightarrow x). \end{array}
If * is continuous, then equality holds in the former.
Residua of prominent leftcontinuous tnorms
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.
Residuum of the 
Name 
Value for x > y

Graph 
Minimum tnorm 
Standard G del implication 
y 
Standard G del implication. The function is discontinuous at the line y = x < 1.

Product tnorm 
Goguen implication 
y / x

Goguen implication. The function is discontinuous at the point x = y = 0.

ukasiewicz tnorm 
Standard ukasiewicz implication 
1 x + y

Standard ukasiewicz implication.

Nilpotent minimum 

max(1 x, y) 
Residuum of the nilpotent minimum. The function is discontinuous at the line 0 < y = x < 1.

Tconorms
Tconorms (also called Snorms) are dual to tnorms under the orderreversing operation which assigns 1 x to x on [0, 1]. Given a tnorm, the complementary conorm is defined by
 \bot(a,b) = 1\top(1a, 1b).
This generalizes De Morgan's laws.
It follows that a tconorm satisfies the following conditions, which can be used for an equivalent axiomatic definition of tconorms independently of tnorms:
 Commutativity: (a, b) = (b, a)
 Monotonicity: (a, b) (c, d) if a c and b d
 Associativity: (a, (b, c)) = ( (a, b), c)
 Identity element: (a, 0) = a
Tconorms are used to represent logical disjunction in fuzzy logic and union in fuzzy set theory.
Examples of tconorms
Important tconorms are those dual to prominent tnorms:
Graph of the maximum tconorm (3D and contours)

Maximum tconorm \bot_{\mathrm{max}}(a, b) = \max \{a, b\}, dual to the minimum tnorm, is the smallest tconorm (see the properties of tconorms below). It is the standard semantics for disjunction in G del fuzzy logic and for weak disjunction in all tnorm based fuzzy logics.
Graph of the probabilistic sum

Probabilistic sum \bot_{\mathrm{sum}}(a, b) = a + b  a \cdot b is dual to the product tnorm. In probability theory it expresses the probability of the union of independent events. It is also the standard semantics for strong disjunction in such extensions of product fuzzy logic in which it is definable (e.g., those containing involutive negation).
Graph of the bounded sum tconorm

Bounded sum \bot_{\mathrm{Luk}}(a, b) = \min \{a+b, 1\} is dual to the ukasiewicz tnorm. It is the standard semantics for strong disjunction in ukasiewicz fuzzy logic.
Graph of the drastic tconorm. The function is discontinuous at the lines 1 > x = 0 and 1 > y = 0.

 \bot_{\mathrm{D}}(a, b) = \begin{cases} b & \mbox{if }a=0 \\ a & \mbox{if }b=0 \\ 1 & \mbox{otherwise,} \end{cases}
 dual to the drastic tnorm, is the largest tconorm (see the properties of tconorms below).
Graph of the nilpotent maximum. The function is discontinuous at the line 0 < x = 1 y < 1.

Nilpotent maximum, dual to the nilpotent minimum:

 \bot_{\mathrm{nM}}(a, b) = \begin{cases} \max(a,b) & \mbox{if }a+b < 1 \\ 1 & \mbox{otherwise.} \end{cases}
Graph of the Einstein sum

 \bot_{\mathrm{H}_2}(a, b) = \frac{a+b}{1+ab}
 is a dual to one of the Hamacher tnorms.
Properties of tconorms
Many properties of tconorms can be obtained by dualizing the properties of tnorms, for example:
 For any tconorm , the number 1 is an annihilating element: (a, 1) = 1, for any a in [0, 1].
 Dually to tnorms, all tconorms are bounded by the maximum and the drastic tconorm:


\mathrm{\bot_{max}}(a, b) \le \bot(a, b) \le \bot_{\mathrm{D}}(a, b), for any tconorm \bot and all a, b in [0, 1].
Further properties result from the relationships between tnorms and tconorms or their interplay with other operators, e.g.:

 T(x, S(y, z)) = S(T(x, y), T(x, z)) for all x, y, z in [0, 1],
 if and only if S is the maximum tconorm. Dually, any tconorm distributes over the minimum, but not over any other tnorm.
See also
References
 Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer. ISBN 0792364163.
 H jek, Petr (1998), Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. ISBN 0792352386
 Cignoli, Roberto L.O.; D'Ottaviano, Itala M.L.; and Mundici, Daniele (2000), Algebraic Foundations of Manyvalued Reasoning. Dordrecht: Kluwer. ISBN 0792360095
 Fodor, J nos (2004), "Leftcontinuous tnorms in fuzzy logic: An overview". Acta Polytechnica Hungarica 1(2), ISSN 17858860 http://www.bmf.hu/journal/
de:TNorm ru:T
