Some Studies Of Matrices Over Semirings

The algebra theories of matrices over semirings have broad applications in optimization theory, models of discrete event networks and graph theory etc. They not only have occupied an important position in mathematics, fuzzy mathematics, theoretical computer science and information science, but also have bright background, rich theory and many open questions. This dissertation mainly studies several kinds of matrices over semirings which have distinctive background.The dissertation consists of five chapters.The first chapter is the Introduction, mainly introduce the research back-ground and present situation of the semiring matrix algebra theory. Also, it briefly introduces some main results obtained in this dissertation.The second chapter will study the n×n matrices over a finite distributive lattice. We firstly give some concrete ways to decompose the finite distributive lattice into a subdircct product of some chains. And then, we show that a square matrix over a finite distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the corresponding results in [1] and [2]. As some applications, we present a method to calculate the indices and periods of the matrices over a finite distributive lattice, and discuss the Green's relations on the multiplicative semigroup of semiring of matrices over a finite distributive lattice.The third chapter will generalizes the main results in Chapter2to more general cases. In this chapter, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of ma-trices over finite distributive lattices, chain semirings and fuzzy semirings etc (including the corresponding results in [1],[2] and [3]. Also, as some applica-tions, we present a method to calculate the indices and periods of the matrices over a distributive lattice, and characterize the structures of idempotent and nilpotent matrices over it, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings and fuzzy semirings etc.The forth chapter mainly investigates some linear preserver problems of the matrices over general Boolean algebra. The linear preserver problem is one of the most active and fertile subjects in matrix theory during the past100years. In this chapter, we give complete characterizations of the linear operators that strongly preserve regular (invcrtiblc, respectively) matrices over general Boolean algebra. And then, note that a general Boolean algebra is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertiblc, respectively) matrices over general Boolean algebra from another point of view. Consequently, our main results enrich the contents of the linear operators that strongly preserve regular (invcrtiblc, respectively) matrices, and also widened the research scopes of the linear preserver problem of the matrices over semirings.The five chapter discusses and studies upper triangular nonnegative ma-trices. Upper triangular nonncgative matrices is a class of important matrices which are well-studied. In this chapter, we first explore some properties and characteristics of the upper triangular nonncgative matrices over the set of natural numbers, and made some answers of the open problem4and open problem2in [4]. And then, we also study some properties and characteristics of the upper triangular nonnegative matrices over the positive integer set, and partly answer the open problem3and open problem1in [4].