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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.

: T(a, b) = T(b, a)

Commutativity

: T(a, b) ≤ T(c, d) if ac and bd

Monotonicity

: T(a, T(b, c)) = T(T(a, b), c)

Associativity

The number 1 acts as : T(a, 1) = a

identity element

Minimum t-norm also called the Gödel t-norm, as it is the standard semantics for conjunction in . 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).

Gödel fuzzy logic

Commutativity: ⊥(a, b) = ⊥(b, a)

Monotonicity: ⊥(a, b) ≤ ⊥(c, d) if ac and bd

Associativity: ⊥(a, ⊥(b, c)) = ⊥(⊥(a, b), c)

Identity element: ⊥(a, 0) = a

Construction of t-norms

T-norm fuzzy logics

Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer.  0-7923-6416-3.

ISBN

(1998), Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. ISBN 0-7923-5238-6

Hájek, Petr

Cignoli, Roberto L.O.; ; and Mundici, Daniele (2000), Algebraic Foundations of Many-valued Reasoning. Dordrecht: Kluwer. ISBN 0-7923-6009-5

D'Ottaviano, Itala M.L.

Fodor, János (2004), Acta Polytechnica Hungarica 1(2), ISSN 1785-8860 [1]

"Left-continuous t-norms in fuzzy logic: An overview".