On Extension of Fuzzy Connectives
lattice-valued Fuzzy connectives, extensions, retractions, e-operators.
There are several operators related to fuzzy logic which in general are generalization of classical ones. For instance, triangular norms and conorms (t-norms and t-conorms for short) generalize the operators ``and" and ``or" from classical logic. Considering that most of these fuzzy operators are in particular functions a natural question arises: Without loss of generality if we consider a t-norm $T$ defined on a sublattice $M$ of bounded lattice $L$ which satisfies a property $P$ is it possible (1) to extend $T$ to a function $T^E$ on $L$ in such way that $T^E$ is also a t-norm and (2) that property $P$ is preserved?\
In this framework we present here two different methods for extending t-norms, t-conorms, fuzzy negations and implications (namely, extension via retractions and extension via e-operators) from a sublattice to a lattice considering a generalized notion of sub lattices. For both methods goal (1) is
completely achieved, however extension via retractions fails in preserving some properties of these fuzzy connectives. Moreover, we prove some results involving extension and automorphisms,
De Morgan triples and t-subnorms. Finally, we present a study of extension of restricted equivalence functions.