Abstract
In a manner analogous to that in the preceeding chapter, we can also define the notion of a [countably-] complete semimodule over a [countably-] complete semiring R. Refer to [R. Lee, 1979]. Note that if {Mh| h ∈Γ} is a family of [countably-] complete left R-semimodules then the left R-semimodule (math) is also [countably-] complete. Indeed, if |fi|i∈Ω} is a (countable) family of elements of Πh∈Γ M h, we define ∑i∈Ω f i the function from Γ to ∪h ∈Γ M h given by
where f i (h) ∈ M h for all h ∈ Γ and all i ∈ Ω. In particular, we note that a direct product of an arbitrary number of copies of a [countably-] complete semimodule is again [countably-] complete.
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© 1999 Springer Science+Business Media Dordrecht
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Golan, J.S. (1999). Complete Semimodules. In: Semirings and their Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9333-5_23
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DOI: https://doi.org/10.1007/978-94-015-9333-5_23
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5252-0
Online ISBN: 978-94-015-9333-5
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