On Some Properties Of The Semiring And Semimodule

In this paper, we study several classes of *-semirings and the semimodule over a *-semiring. The main results are as follows:1. The UF-inductive *-semiring is studied. Some properties of the UF-inductive *-semiring are given. For any non-negative integer n, we prove that the n×n matrix semiring and the formal power series semiring over a UF-inductive *-semiring are also UF-inductive *-semirings. Then the results are the extension of the known theorems of the inductive *-semiring.2. The UF-weak inductive *-semiring is studied. The relationship between the UF-weak inductive *-semiring and the UF-inductive *-semiring is discussed. It is proved that the n×n matrix semiring over a UF-weak inductive *-semiring is also a UF-weak inductive *-semiring. Then we give a partial answer of the open problem whether the n×n matrix semiring over a weak inductive *-semiring is also a weak inductive *-semiring.3. The strong continuous *-ω-semiring is introduced and studied. Through the research of the greatest and the least fixed points of the linear map of the strong continuous *-ω-semiring, we obtain the relationship between the greatest and the least fixed points. Finally, a sufficient condition for the uniqueness of the fixed point of the linear map is given.4. The UF-inductive semimodule over a *-semiring is studied. We give a sufficient and necessary condition that a UF-inductive semimodule is a Kleene module. At the same time, we give some properties of the UF-inductive semi-module. We prove that the vector module over a UF-inductive semimodule is also a UF-inductive semimodule.