Study On The Properties Of Direct Product And Direct Sum In Riesz Module Category

In this paper,we study the properties of direct product and direct sum of Riesz module category.The content is arranged as follows:In Chapter 1,it introduces the research background of Riesz module and the basic concepts and lemmas related to this article.In Chapter 2,it provides the concept of direct product of two objects in the Riesz module category;Defined the projection of the direct product onto the Riesz module(Mi,+,≤);The necessary and sufficient condition for proving the uniqueness of the element decomposition formula of M1+M2 is(M1,+,≤)∩(M2,+,≤)=0,Provide the concept of the inner direct sum of two objects in the Riesz category along the way;Proved that when(M1,+,≤)∩(M2,+,≤)=0,the direct product is isomorphic to the inner direct sum(hereinafter collectively referred to as the direct sum);Defined the embedding of Riesz module(Mi,+,≤)to the direct sum;Some other properties of direct product and direct sum are given.In Chapter 3,it presents a Cartesian product of a family of Riesz module,where(?)is a Riesz module and the Cartesian product is projected onto(M;,+,≤);At the category level,it has been proven that the Cartesian product is a direct product of Riesz module families;Proved that subset(?)of the direct product is a sub Riesz module of the direct product;Embedding from(Mj,+,≤)to(?)is provided;At the category level,it has been proven that(?)is the(outer)direct sum of Riesz module clusters;Analogizing the inner direct sum of two Riesz modules gives the inner direct sum of the Riesz module family,and proves that when the Riesz module family is a sub Riesz module of the inner direct sum,the inner direct sum is isomorphic to the outer direct sum;Some other properties of direct product and direct sum are given.In Chapter 4,it compares the isomorphic properties of two Riesz module and provides three isomorphic properties of Riesz module families.