nLab t-norm

Definition

A t-norm is a semicartesian commutative monoidal structure on the unit interval $[0,1]$ as a poset.

That means it is a commutative monoid structure on the set $[0,1]$ that is order-preserving in each argument and with $1$ as the unit element.

Related concepts

T-norms are used in fuzzy logic.

Examples

• $T(x,y) = \min(x,y)$. This is the cartesian monoidal structure, also known as the minimum t-norm or the Godel t-norm.
• $T(x,y) = x y$, the product t-norm.
• $T(x,y) = \max(x+y-1,0)$, the Lukasiewicz t-norm (see Lukasiewicz logic?). This monoidal structure is moreover star-autonomous with the obvious involution $x^\ast = 1-x$.