Integrations on lattices

In this paper, we introduce the notion of integration with respect to a given derivation on a lattice. More precisely, we give the definitions of integrable elements of a lattice and their integral sets. We investigate several characterizations and properties of integrations on a lattice. Also, we give a lattice structure to the family of integral sets with respect to a given integration. Further, we provide a representation theorem for the lattice of fixed points of an isotone derivation based on the family of integral sets. As an application of this notion of integration, we use the integrable elements of a Boolean lattice to determine the necessary and sufficient conditions under which a linear differential equation on this Boolean lattice has a solution.