Fuzzy implication functions with a specific expression: The polynomial case

In the last decades, more than a hundred families of fuzzy implication functions have been proposed. These families are generated by adequately combining other logical connectives or univariate functions. However, no attention has been given to the final expression of the fuzzy implication function, a crucial feature for any application. In this paper, fuzzy polynomial implications are introduced as those fuzzy implication functions whose expression is given by a polynomial of two variables. Polynomials present advantages with respect to other types of functions making them interesting for applications. Several results are proved for polynomials of any degree and the characterisation of all fuzzy polynomial implications of degree less or equal to 4 is achieved.