Generating methods of some classes of fuzzy implications obtained by unary functions and algebraic structures

Abstract

The existing generating methods of fuzzy implications are closed in the set of all fuzzy implications, but not when applied to some families of these operators. In this paper, some binary operations are defined on some well-established families of fuzzy implications. Namely, the families of $(S,N)$-implications with continuous t-conorms, R-implications obtained from continuous t-norms, Yager's f- and g-generated implications, h- and generalized $(h,e)$-implications and k-implications are considered and it is proved that with these operations, they are lattices. Moreover, some other lattice substructures are defined in some of the families when some restrictions on the underlying generators of the implications are imposed. Furthermore, it is emphasized that these generating methods preserve important properties like the exchange principle and the law of importation.