On the expansion map of lattices

It is well known that the closure operator on a lattice is an extensive, isotone, and idempotent map. In this paper, we extend this concept by introducing the notion of an expansion map on lattices, which serves as a generalization of closure operators. The focus is to explore the properties of the collection of all expansion maps on a lattice, which forms a lattice. We delve into the discussion of their covering relation and present a comprehensive characterization of the atoms and dual atoms within the lattice obtained from these expansion maps.