S-systems Theory

This thesis consists of three chapters.In chapter 1, we study chain conditions on S-acts. A PS-act is defined for the first time. An S-act M is called a PS-act if for any subact N of M, there exists a summand K of M, such that SocK N K, where SocK is the union of all simple subacts of K. The structure of a PS-act is characterized. We show-that an 5-act M is PS if and only if M is a 0-direct union of MI and M2, where M1 is a CS-subact of M, SocM1 an essential subact of M1, and M2 a subact of M with SocM2=0. We also get that a finitely generated PS-act is Xoetherian (Artinian) if it has ACC (DCC) on essential subacts.In chatper 2, we obtain seven equivalent conditions for which a modular lattice is semisimple. It is shown that if L is a modular lattice with finite length, and with 0, 1 the least and the largest elements in L respectively, then 1 is the union of some minimal elements in L if and only if 0 is the intersection of some maximal elements in L, if and only if each non-zero element in L is the union of some minimal elements in L. By using these results, we give the necessary and sufficient conditions of an algebra module, module, ring, group and semigroup, respectively, to be semisimple. These are the generalizations of related ones in algebra module, module, ring, and group.In chapter 3. we study the exact sequences on sets and S-acts. By using the technique of diagram chasing, we get some characterizations of exact sequences on sets, which generalize the results of commuting diagrams about modules (S-acts) in the module theory (S-act theory). Some characterizations of exact sequences on S-acts, especially on semisimple S-acts are also obtained.