From Wikipedia, the free encyclopedia T-norm
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviat- Classification of t-norms
norm)
ed, triangular norm is a kind of binary operation used
A t-norm is called continuous if it is continuous as a func-
in the framework of probabilistic metric spaces and in
tion, in the usual interval topology on [0, 1]2. (Similarly
multi-valued logic, specifically in fuzzy logic. A t-norm
for left- and right-continuity.)
generalizes intersection in a lattice and conjunction in
A t-norm is called strict if it is continuous and strictly
logic. The name triangular norm refers to the fact that in
monotone.
the framework of probabilistic metric spaces t-norms are
A t-norm is called nilpotent if it is continuous and each
used to generalize triangle inequality of ordinary metric
x in the open interval (0, 1) is its nilpotent element, i.e.,
spaces.
there is a natural number n such that x * ... * x (n times)
equals 0.
Definition A t-norm * is called Archimedean if it has the
Archimedean property, i.e., if for each x, y in the open in-
A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] which sat-
terval (0, 1) there is a natural number n such that x * ... *
isfies the following properties:
x (n times) is less than or equal to y.
• Commutativity: T(a, b) = T(b, a)
The usual partial ordering of t-norms is pointwise,
• Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d
i.e.,
• Associativity: T(a, T(b, c)) = T(T(a, b), c)
T1 ≤ T2 if T1(a, b) ≤ T2(a, b) for all a, b in [0, 1].
• The number 1 acts as identity element: T(a, 1) = a
Since a t-norm is a binary algebraic operation on the in- As functions, pointwise larger t-norms are sometimes
terval [0, 1], infix algebraic notation is also common, with called stronger than those pointwise smaller. In the se-
the t-norm usually denoted by * . mantics of fuzzy logic, however, the larger a t-norm, the
The defining conditions of the t-norm are exactly weaker (in terms of logical strength) conjunction it repre-
those of the partially ordered Abelian monoid on the real sents.
unit interval [0, 1]. (Cf. ordered group.) The monoidal oper-
ation of any partially ordered Abelian monoid L is there-
fore by some authors called a triangular norm on L. Prominent examples
Motivations and applications
T-norms are a generalization of the usual two-valued log-
ical conjunction, studied by classical logic, for fuzzy log-
ics. Indeed, the classical Boolean conjunction is both
commutative and associative. The monotonicity proper-
ty 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 corre-
sponds to the interpretation of 1 as true (and consequent- Graph of the minimum t-norm (3D and contours)
ly 0 as false). Continuity, which is often required from
fuzzy conjunction as well, expresses the idea that, rough-
ly speaking, very small changes in truth values of con- • also called the
juncts should not macroscopically affect the truth value t-norm,
Gōdel t-norm as it is the standard semantics for
of their conjunction. conjunction in Gōdel fuzzy logic. Besides that, it
T-norms are also used to construct the intersection occurs in most t-norm based fuzzy logics as the
of fuzzy sets or as a basis for aggregation operators (see standard semantics for weak conjunction. It is the
fuzzy set operations). In probabilistic metric spaces, t- pointwise largest t-norm (see the properties of t-
norms are used to generalize triangle inequality of or- norms below).
dinary metric spaces. Individual t-norms may of course • (the ordinary product of
frequently occur in further disciplines of mathematics, real numbers). Besides other uses, the product t-
since the class contains many familiar functions. norm is the standard semantics for strong
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From Wikipedia, the free encyclopedia T-norm
Graph of the product t-norm Graph of the nilpotent minimum. The function is discontinuous
at the line 0 y.
construction always yields a continuous t-norm. The the-
orem can also be formulated as follows:
A t-norm is continuous if and only if it is T-conorms
isomorphic to an ordinal sum of the minimum,
S-norms)
T-conorms (also called S-norms are dual to t-norms un-
Łukasiewicz, and product t-norm.
der the order-reversing operation which assigns 1 – x to
A similar characterization theorem for non-continuous t- x on [0, 1]. Given a t-norm, the complementary conorm is
norms is not known (not even for left-continuous ones), defined by
only some non-exhaustive methods for the construction
of t-norms have been found.
This generalizes De Morgan’s laws.
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From Wikipedia, the free encyclopedia T-norm
Residuum of the Name Value for x > y Graph
Minimum t-norm Standard Gōdel implication y
Standard
Gödel im-
plication.
The func-
tion is dis-
continu-
ous at the
line y = x x = 0 and 1 > y = 0. , for any t-conorm and all a, b in [0, 1].
•
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From Wikipedia, the free encyclopedia T-norm
Further properties result from the relationships between
t-norms and t-conorms or their interplay with other op-
References
erators, e.g.: • Klement, Erich Peter; Mesiar, Radko; and Pap, Endre
• A t-norm T distributes over a t-conorm S, i.e., (2000), Triangular Norms. Dordrecht: Kluwer. ISBN
T(x, S(y, z)) = S(T(x, y), T(x, z)) for all x, y, z in 0-7923-6416-3.
[0, 1], • Hájek, Petr (1998), Metamathematics of Fuzzy Logic.
Dordrecht: Kluwer. ISBN 0-7923-5238-6
if and only if S is the maximum t-conorm. Dually, • Cignoli, Roberto L.O.; D’Ottaviano, Itala M.L.; and
any t-conorm distributes over the minimum, but Mundici, Daniele (2000), Algebraic Foundations of
not over any other t-norm. Many-valued Reasoning. Dordrecht: Kluwer. ISBN
0-7923-6009-5
See also • Fodor, János (2004), "Left-continuous t-norms in
fuzzy logic: An overview". Acta Polytechnica Hungarica
• Construction of t-norms 1(2), ISSN 1785-8860 [1]
Retrieved from "http://en.wikipedia.org/w/index.php?title=T-norm&oldid=450488463"
Categories:
• Fuzzy logic
• Norms (mathematics)
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