Fuzzy Bi-implications Generated by t-norms and Fuzzy Negations

Abstract

In the literature, there are several forms of extensions of the classical bi-implication for the fuzzy logic, as for example, the axiomatization proposed by Fodor and Roubens [1]. Another way to obtain a generalization is to provide a definition based on the classical equivalence (Formula Presented), in which the classical operators of conjunction and implication are replaced, respectively, by a t-norm (T) and a fuzzy implication (I). In this paper, we investigate a particular class of fuzzy bi-implications (Formula Presented), in which I is a fuzzy (T, N)-implication introduced by Bedregal [2]. We study several properties satisfied by (T, N)-bi-implications, such as the sufficient conditions that they must satisfy in order to be a f-bi-implication.