The Implication on a Partial Ordered Set

The concept of a basic implication algebra and an implication algebra on a partial ordered set are obtained by studying the conditions that the implicative operator in a logic system should be satisfied.The basic properties with different conditions and the characterizes of the implication algebra are given. The iff conditions for a basic implication algebra to be a implication algebra are discussed. The relations between the basic implication algebra and the implication algebra are gained. From the view of lattices, some lattice properties of the implication algebra and the conditions under which a implication algebra is a lattice are found.