Leibniz-linked Pairs of Deductive Systems

A pair of deductive systems (S,S??) is Leibniz-linked when S?? is an extension of S and on every algebra there is a map sending each filter of S to a filter of S?? with the same Leibniz congruence. We study this generalization to arbitrary deductive systems of the notion of the strong version of a protoalgebraic deductive system, studied in earlier papers, and of some results recently found for particular non-protoalgebraic deductive systems. The necessary examples and counterexamples found in the literature are described.