Extension principles for closure operators on fuzzy sets and cuts

We investigate closure operators defined on various partly ordered sets of fuzzy objects, including sets of all fuzzy sets with values in a complete residuated lattice and sets of all cuts systems, defined in crisp sets and sets with similarity relations, respectively. We proved several extension theorems, under which a closure operator defined on one universe can be extended to a closure operator defined on another universe. We also investigate relationships between continuity of pairs of maps with respect to different closure operators.