Some Theoretical Study Of Irreducible Filter

In this paper,some new elements on posets are introduced and examined.For example: irreducible filters,locally maximal filters and weakly maximal filters,some good properties of these elements and conditions of equivalence among them are studied.The structure of this thesis is organized as follows:In the first part,we briefly introduce the history of the current research situation on irreducible and filters.Through citing and analyzing numerous works in this filed,we also give some preliminaries that related to continuous lattices definition and theory which we will be used throughout this paper.In the second part.From the point of maximal,the definition of the irreducible filters is obtained and some properties in poset continuous semilattice are studied,the relativity between irreducible filters and prime filters are discussed.At the end of this part,Some relevant example are obtained,some properties of prime filters in distributive lattice are also obtained.In the third part,locally maximal filters on poset and weakly maximal filters relative to idea are defined,the relations between locally maximal filters and maximal filters,maximal filters irreducible filters are obtained;the relations between weakly prime filters and weakly maximal filters relative to idea,weakly irreducible filters are showed by some examples and theorem.Inspired by this,the follow theorem is obtained every filters on posets can be decomposed as intersection of some locally maximal filters.And an intrinsic characterizations of a kind of distributive posets is obtained.