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Inductive semimodules and the vector modules over them

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Abstract

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.

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Notes

  1. Note that a more restrictive notion of *-semirings is discussed in Lehmann (1977) and is also known as Lehmann semirings (Golan 2003).

  2. Note that a more restrictive notion of Kleene (left-) modules is given by Ehm et al. (2004).

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Acknowledgments

The authors are highly grateful to the reviewers for their valuable comments and suggestions which greatly helped in improving the paper. This work was supported by a grant (no. 08JK432) from the Education Department of Shaanxi Province of China.

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Correspondence to Feng Feng.

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Feng, F., Jun, Y.B. Inductive semimodules and the vector modules over them. Soft Comput 13, 1113–1121 (2009). https://doi.org/10.1007/s00500-008-0384-y

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