On Some Semi-ring Properties

Algebraic theory of semiring is still one of the most active fields of algebra. The goal of this dissertation is to study this field. It mainly achieved in the following aspects:1. The semirings for which additive reducts are semilattices and multiplica-tive reducts are left normal orthogroups are studied. That semiring (S,+,·) is embedded into the endomorphism semiring of the semilattice (S,+) is proved. Further, we construct the partial order relation on semiring S and obtain the sufficient and necessary condition for the natural order on multiplicative reduct of semiring S is equivalent to the constructed partial order on semiring S.2. The semirings of n x n matrices over a chain are studied. We give out the decomposition theorem of n×n matrices over a chain. Further, the index and periodicity of a matrix over a chain are studied. Finally, the Green's relations on the multiplicative semigroup of semiring of matrices over a chain are discussed.3. The semirings of n×n matrices over a semiring are studied. The decom-position theorem of matrices over a semiring is given. We give out the concrete ways to find decompositions of a finite distributive lattice. The decomposition theorem of matrices over a finite distributive lattice is proved. As an applica-tion, we give the way to calculate indices and periods of the matrices over a finite distributive lattice.4. Partial inductive*-semirings and partial weak inductive*-semirings are studied. We proved that every partial inductive*-semiring and partial weak inductive*-semiring are partial Conway semiring. Moreover, the Kleene theorem of them are obtained.