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On rational fuzzy implication functions

The proposal of new classes of fuzzy implication functions must take into account the final expression of the operator to be potentially used in a concrete application. In particular, fuzzy implication functions with simple expressions have low computational cost and they reduce the spreading of numerical errors. Following this line of research, in this work, the class of fuzzy rational implications is introduced as those fuzzy implication functions whose expression is given by the quotient of two polynomials of two variables. Since not all quotients of polynomials are adequate to generate a fuzzy implication function, some necessary properties related to the values of the coefficients of the polynomials are given to ensure obtaining a fuzzy implication function. In addition, we characterize all fuzzy rational implications constructed from polynomials of some fixed degrees. Finally, several construction methods of these implications and relationships with other families of fuzzy implication functions are given.