On The Characterizations Of Complete Distributive Lattices By Up-Sets

This paper first describes a characterization of a lattice L which can be represented as the collection of all up-sets of a poset. It then obtains a representation of a complete distributive lattice L-0 which can be embedded into the lattice L such that all infima, suprema, the top and bottom elements are preserved under the embedding by defining a monotonic operator on a poset. This paper finally studies the algebraic characterization of a finite distributive.