The triple rotation method for constructing t-norms

Abstract

Given an involutive negator N and a left-continuous t-norm T that either has no zero divisors or is rotation invariant, we build a rotation-invariant t-norm from a rescaled version of T and its left, right and front rotation. Depending on the involutive negator N and the set of zero divisors of T, some reshaping of the rescaled version of T may occur during the rotation process. The rescaled version of T itself can be understood as the β -zoom of the newly constructed rotation-invariant t-norm, with β the unique fixpoint of N. Starting with a rotation-invariant t-norm T there is, however, one important restriction. The triple rotation method based on the involutive negator N will yield a t-norm if and only if the companion Q of T is commutative on [ 0 , 1 [ 2 . When Q is not commutative on [ 0 , 1 [ 2 , there even does not exist a rotation-invariant t-norm with β -zoom equal to T.