FUZZY LOGICAL OPERATORS ON FINITE CHAINS

Abstract The extension of classical binary logical operators to multi-valued operators is a very interesting topic, both from a theoretical and a practical point of view. In this paper we pay special attention to fuzzy implication operators on the unit interval and on finite chains. We discuss some properties and define several special classes, in particular the class of Lukasiewicz-like implicators. On the unit interval we prove a nice characterization for this class, while we have to replace the continuity condition on [0,1] by a smoothness condition on finite chains to obtain a similar characterization. In contrast to implicators on [0,1], there is no direct way to construct implicators on a finite chain. Therefore, we discuss several algorithms that generate different types of implicators. In this way we also obtain the number of implicators on a finite chain, which gives us an idea of the restrictivity of the involved properties.