Several Types Of Semi-ring And Half Ring Semimodule Study

Derived from theoretical computer science and information science, several semirings are attached great importance and concern.This paper precisely studies these semirings and semiring-semimodule pairs which are corresponding to the several semirings above. The main results are as follows:1. The matrix semiring and the formal power series semiring of weak in-ductive *-semiring are studied.We obtain its upper (lower) triangular matrix semiring is also a weak inductive *-semiring. By analyzing the structural of such semiring, we give another way to prove "the formal power series semiring of weak inductive *-semiring is still a weak inductive *-semiring "2. The matrix semiring ofμ-semiring is studied.We prove its upper (lower) triangular matrix semiring is also aμ-semiring.3. The formal power series of semiring-semimodule pairs is studied.We give the definition of formal power series on semiring-semimodule pairs and constructωoperation on it. Moreover, we prove that the formal power series of bi-inductive semiring-semimodule pairs is still a bi-inductive semiring-semimodule pairs.4. Several semiring-semimodule pairs are studied.We give the definition of several semiring-semimodule pairs which are corresponding to some semir-ings,discuss the relationship between them, prove upper (lower) triangular ma-trix of bi-μ-semiring-semimodule pairs is also a bi-μ-semiring-semimodule pairs,and get the matrix of bi-*-μ-semiring-semimodule pairs is also itself. Fur-ther, we use the relationship between these semiring-semimodule pairs to discuss their formal power series and get some well results.5. The bi-weak inductive semiring-semimodule pairs is studied. We give the definition of bi-weak inductive semiring-semimodule pairs and discuss its matrix and formal power series.Then, the result that its upper (lower) triangular matrix is also itself is proved. Moreover, we get a necessary and sufficient condition under which its formal power series is still its own.