A construction method of semicopulas from fuzzy negations

In this paper a method of defining commutative semicopulas from fuzzy negations is introduced. Some properties are investigated that lead to understand these semicopulas as non-associative generalizations of the Łukasiewicz t-norm. In particular, it is proved that some well known examples of copulas and t-norms can be obtained by this method. Moreover, any commutative semicopula constructed by this method can be always obtained from a negation N which is symmetric with respect to the diagonal. Then, those symmetric fuzzy negations N for which the corresponding semicopula is a copula are characterized. Also, several examples of symmetric negations N are given such that the corresponding semicopula is a t-norm.