Abstract
Let R be a partially-ordered semiring. A left R-semimodule M is partially ordered if and only if there exists a partial-order relation ≤ defined on M satisfying the following conditions:
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(1)
If m ≤ m’ in M and if m“ ∈ M then m + m” ≤ m’ + m“,
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(2)
If m ≤ m’ in M and if 0 ≤ a in R then am ≤ am’;
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(3)
If a ≤ b in R and 0 M ≤ m in M then am ≤ bm.
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© 2003 Springer Science+Business Media Dordrecht
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Golan, J.S. (2003). Partially-ordered semimodules. In: Semirings and Affine Equations over Them: Theory and Applications. Mathematics and Its Applications, vol 556. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0383-3_10
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DOI: https://doi.org/10.1007/978-94-017-0383-3_10
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6310-6
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