On retractions and extension of quasi-overlap and quasi-grouping functions defined on bounded lattices

Saminger-Platz, Klement, and Mesiar (2008) extended t-norms from a complete sublattice to its respective lattice using the conventional definition of sublattice. In contrast, Palmeira and Bedregal (2012) introduced a more inclusive sublattice definition, via retractions. They expanded various important mathematical operators, including t-norms, t-conorms, fuzzy negations, and automorphisms. They also introduced De Morgan triples (semi-triples) for these operators and provided their extensions in their groundbreaking work. In this paper, we propose a method of extending quasi-overlap functions and quasi-grouping functions defined on bounded sublattices (in a broad sense) to a bounded superlattice. To achieve that, we use the technique proposed by Palmeira and Bedregal. We also define: quasi-overlap (resp. quasi-grouping) functions generated from quasi-grouping (resp. quasi-overlap) functions and frontier fuzzy negations, De Morgan (semi)triples for the classes of quasi-overlap functions, quasi-grouping functions and fuzzy negations, as well as its respective extensions. Finally we study properties of all extensions defined.