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Archimedean decomposition of left S-semimodules and semirings. (English) Zbl 0618.16035

Let \(S=(S,+,\cdot)\) be a semiring with commutative addition and \({}_ SM=(_ SM,+)\) a left S-semimodule. Then the congruence \(\eta\) corresponding to the decomposition of the commutative semigroup \((M,+)\) into its archimedean components [cf. Thm. 4.13 in A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. I (1961; Zbl 0111.03403)] is compatible with the action of each \(s\in S\) on \((M,+)\). Hence the theorem cited above can be extended to S-semimodules \((_ SM,+)\). Applied to \((_ SS,+)\) and \((S_ S,+)\), one obtains: each semiring S as above has a unique decomposition \(S=\cup \{S_{\alpha}|\alpha\in Y\}\), where the semiring \(Y=(Y,+,\cdot)\) is the maximal additively idempotent homomorphic image of \((S,+,\cdot)\) and the archimedean components \((S_{\alpha},+)\) of \((S,+)\) satisfy \(S_{\alpha}+S_{\beta}\subseteq S_{\alpha +\beta}\) and \(S_{\alpha}\cdot S_{\beta}\subseteq S_{\alpha \cdot \beta}\).
Reviewer: H.J.Weinert

MSC:

16Y60 Semirings
20M10 General structure theory for semigroups
20M14 Commutative semigroups

Citations:

Zbl 0111.03403
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