Complete Semimodules

Jonathan S. Golan
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Jonathan S. Golan: University of Haifa

Chapter 23 in Semirings and their Applications, 1999, pp 259-265 from Springer

Abstract: 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 $$ \sum\limits_{i \in \Omega } {{f_i}:h} \sum\limits_{i \in \Omega } {{f_i}(h)} $$ 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.

Date: 1999
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DOI: 10.1007/978-94-015-9333-5_23

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