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Inductive semimodules and the vector modules over them
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DOI：10.1007/s00500-008-0384-y.png)
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Abstract
En 中文
The notion of an inductive semimodule over an ordered *-semiring is introduced and some related properties are investigated. Inductive semimodules are extensions of several important algebraic structures such as Kleene modules, Kleene algebras and inductive *-semirings. We prove that an inductive semimodule over an ordered *-semiring K is a Kleene module if and only if K is a Kleene algebra. Moreover, we establish that the vector module of an inductive semimodule over an ordered Conway semiring is again an inductive semimodule over the matrix semiring. Consequently, in an inductive semimodule over an ordered Conway semiring, least solutions to linear inequation systems can be denoted by linear expressions, avoiding the least fixed point operator. In addition, we also introduce a related notion called weak inductive semimodules, and propose several open problems on them.
Keywords:
Semiring
Semimodule
Inductive *-semiring
Kleene algebra
Kleene module
Inductive semimodule
Linear inequation system