Categories ~ * - Half Ring And Its Dual Semi-ring

The dissertation mainly studies the fixed point theory of the*-semiring. The results are as follows:1. The strong inductive*-semiring is studied. Some properties of the set of additively idempotents of a strong inductive*-semiring are given. And a sufficient condition for a strong inductive*-semiring (inductive*-semiring) to be a symmetric strong inductive*-semiring (symmetric inductive*-semiring) is gotten. Finally, it is proved that the semiring of formal power series over a strong inductive*-semiring is also a strong inductive*-semiring.2. The*-λ-semiring satisfying ascending chain condition (ACC) is stud-ied. The result is gotten that a*-λ-semiring satisfying ACC is a continuous *-semiring and symmetric*-λ-semiring. Ascending, it is proved that all matrix semirings over a weak inductive*-semiring satisfying ACC are weak inductive *-semirings. Thus the open problem whether all matrix semirings over a weak inductive*-semiring are weak inductive*-semirings is partially answered.3. The index stable*-semiring is introduced and studied. Some properties of an index stable*-semiring are given. It is showed that S is an index stable *-semiring, then for any n∈N,n>1, the matrix semiring Sn×n js not an index stable*-semiring. Meanwhile, the result that the semiring of formal power series over an index stable semiring is also an index stable semiring is gotten.