The Convexity Of Finitely Generated Module Over Max-plus Algebra

Max-plus algebra provides an important algebraic method to solve the problems of discretemathematics. Since max-plus algebra has been proposed, it has been widely used in manyproblems of computer, communication networks, machinery manufacturing, automatic devices,as well as graph theory and petri nets, etc. The special structure of max-plus algebra givesthese problems some features of linear algebra, and then the nonlinear model is transformedinto linear model, which is more convenient to solve practical problems.Many important mathematical concepts can be reintroduced in the sense of max-plus al-gebra, and then discuss their properties. For example, there have been many scholars do theresearch of problems such as determinant, eigenvalue and linear independent in the sense ofmax-plus algebra. But because the max-plus algebra does not satisfy the characteristics of lin-ear properties, many problems are difficult to solve. So max-plus algebra theory developedslowly. Generated module is an important mathematical concept, however, there are not manyresearches of module in max-plus algebra. Baccelli F et al pointed out that the module in max-plus algebra is moduloid. Chen W. gave the concept of finitely generated module over max-plusalgebra, and discussed the geometric shape of finitely generated module over max-plus algebra.The main work of this thesis is further discussing the convexity of finitely generated mod-ules over max-plus algebra on the basis of the research of predecessors. This thesis is dividedinto six sections.In the first section, we introduce the related background and status of finitely generatedmodule over max-plus algebra.In the second section, basic concepts and theorems are introduced, such as semi-field,quasi-field, max-plus algebra and the determination theorem of convex set, etc. We emphati-cally explains the concept of finitely generated module over max-plus algebra. The scalar op-erations and the operations between matrices on max-plus algebra are introduced by examples.These concepts and theorems provide a theoretical support for the later sections.In the third section, the geometric shape of finitely generated module over max-plus al-gebra have been further studied. According to the limit that K is a quasi-base, the geometricshape of finitely generated module over max-plus algebra when n=3, m≥3is renewed. M isalso prismoid which formed by the set of three-dimensional equal add lines, but the geometricshape of the section of four extremal point in x1=0is reduced from ten to three. This willreduce the complexity of further analysis of the geometric shape. The fourth section is the core of this thesis. By applying the geometric shape of finitelygenerated module and using algebraic method combined with geometric method, the convexityof finitely generated module with dimension n≤3and the number of generating vector m≥1is analyzed. It is proved that the finitely generated module with n=1,2is a convex set. Forn=3, the necessary and sufficient condition for the convexity with m=2and the sufficientcondition for the convexity with m≥3are obtained, respectively.In the fifth section, we study the geometric shape and convexity of the finitely generatedmodule over another quasi-field. Firstly, the concept of nonnegative real number algebra quasi-field is given. Secondly, we analysis the geometric shape and the convexity of finitely generatedmodules over nonnegative real number algebra. Finally, we compare them with that of finitelygenerated module over max-plus algebra on the similarities and differences in the convexity.The characteristic of nonlinearity in max-plus algebra is illustrated.In the sixth section, we summarize the main conclusions of this thesis, and raise the issuewhich needs further study.