Extension of fuzzy logic operators defined on bounded lattices via retractions

The primary goal of this paper is to present a method of extending tt-norms, tt-conorms and fuzzy negations from a sublattice MM to the bounded lattice LL by considering a more general version of the idea of the sublattice. In general terms, we consider MM as a sublattice of the bounded lattice LL, if MM has the same lattice structure of the LL equipped with the restriction of operations of LL and is a subset of LL. However, this latter condition may be relaxed without losing the essence of the usual definition of the sublattice. This is done through the use of retractions. Furthermore, the same idea is employed to extend tt-subnorms and present some results related to extension and automorphism. Additionally, a formalization of a relaxed notion of De Morgan triple and its extension is provided.