Extending functions to natural extensions

We investigate the problem of extending maps between algebras of a finitely generated prevariety to their natural extensions. As for canonical extension of lattice-based algebras, a new topology has to be introduced in order to be able to define an algebra inside its natural extension. Under the assumption that there is a structure that yields a logarithmic duality for the prevariety, this topology is used to define the natural extension of a map. This extension turns out to be a multivalued map and we investigate its properties related to continuity, composition and smoothness. We also prove that our approach completely subsume the lattice-based one. In the meanwhile, we characterize the natural extension of Boolean products.