Rotation-invariant t-norm solutions of a system of functional equations

Authors:
Koen 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:
2006

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Abstract

The generalization of the disjunctive and conjunctive Boolean normal forms to fuzzy set theory, obtained by interpreting @? as a t-norm, @? as a t-conorm and ^' as a negator, often provides a kind of standard fuzzification procedure. A system of functional equations turns up if some functional independence of the difference between both generalizations is demanded. In this paper, we explore which De Morgan triplets, based on a left-continuous t-norm T, solve this system. Imposing some extra continuity conditions on T, these t-norm solutions can be obtained by the rotation construction of Jenei. The Cauchy equation plays a key role in the reasoning process.