Study Of The Property Of Join-irreducible Element And Coprime On Poset

This dissertation, we mainly introduce some new elements in poset, e.g. join-irreducible element coprime,pseudo-join-prime,=>-prime element. We give the conditions of equivalence among them. The thesis comprises three chapters:In the first chapter, we introduce the history and current situation of irre-ducible element,prime,pseudoprime and<=prime element in post. And a brief introduction to the launching of this thesis workIn second chapter, First we give the concept of join-irreducible and coprime on poset. An element p in a poset is called join-irreducible iff p is minimal or↓p\{p} is an ideal. An element p in a poset L is called coprime iff p=0 or L\↓pis an ideal. The set of coprime is denoted by COPRIME L. we give the concept of the open ideal, prove that if the Domain L satisfy certain conditions, the set of non-zero irreducebile elements will have some special character, by using the open ideal. we obtain the results that join-irreducible and coprime are equivalent. Then in To topological spaces, we give join-irreducible and coprime. V is join-irreducible and coprime of X, if V is also a compact saturated subset in X(((?)v∈V)V= sat{v}), C is open set in X, then ((?)x∈X)C= X\{x}-is join-irreducible and coprime of Q(X).And the pseudoprime element is dual a pseudo-join-prime elements, We show that pseudo-join-prime is a special form of coprime, but they are equivalent under certain conditions.In the third chapter, we presented the definition of=>-prime elements. And divestigats the relations among the=>- prime elements, coprime elements and pseudo-join-prime elements. We also give the equivalent conditions of=> - prime element, pseudo-join-prime and coprime.