The triple rotation method for constructing t-norms

Authors:
Koen C. Maes;Bernard De Baets
Affiliations:
Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium;Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, B-9000 Gent, Belgium
Venue:
Fuzzy Sets and Systems
Year:
2007

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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 @b-zoom of the newly constructed rotation-invariant t-norm, with @b 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 @b-zoom equal to T.