Some Studies Of Homology Theory Over Semimodule Complex

Semi-ring is one of the most natural algebraic structure which people feel,because we add the usual addition and multiplication to the natural numbers can forma semi-ring.The Semi-module is a generalization of the mold, but this special algebrastructure which also has semi-ring structure and the mode structure, has not beenadequately studied.Currently, the semi-ring and the semi-modules on the semi-ring have beenapplied in mathematics and the oretical computer science as an important tool, and itis an effective way to use semi-modules to characterize semi-ring, so the research ofsemi-modules is natural and necessary.This paper describes the proper-exactsequences of semi-modules and its related properties, such as “the nine lemma ofsemi-modules”, special cases which can make the semi-modules to be splited inthese special cases, homology theory of semi-modules complex, puresubsemimodules and their related properties.This paper is divided into four chapters：The first chapter is the preliminaries, We introduce some correspondingdefinitions and conclusions which are be used frequently.In the second chapter, we first give the concept of the short exact sequence thecongruence defined and on this basis, we generalize some properties of short exactsequence in the category of modules, such as Nine Lemma, to semimodulecategory.And we use the short exact sequences of semimodules to derive thecharacterizations of split semi-modules.The last result gives a very usefulcharacterization of single semimodule.In the third chapter, we first give the concept of complex and chain mappingwhich belong to the semi-module category, prove some related properties andresults.And we generalize connecting homomorphism theorem to the category ofsemimodules in these special cases。In addition,we introduce the notion of homotopy,decribe the relationship of homotopy and synthesis, homotopy and Additive functor.In the fourth chapter, we study the characterizations of pure subsemimodules inthe category of semimodules and and generalizes some results.