The triple rotation method for constructing t-norms

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 resealed version of T may occur during the rotation process. The resealed version of T itself can be understood as the beta-zoom of the newly constructed rotation-invariant t-nonn, with beta 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 beta-zoom equal to T. (c) 2007 Elsevier B.V. All rights reserved.