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.

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 ${\displaystyle *}$.

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.

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.

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