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Keywords:
composite semi-valuation; ordered group; G-homomorphism of ordered groups
composite semi-valuation; ordered group; G-homomorphism of ordered groups
Summary:
In analogy with the notion of the composite semi-valuations, we define the composite $G$-valuation $v$ from two other $G$-valuations $w$ and $u$. We consider a lexicographically exact sequence $(a,\beta ):A_u\rightarrow B_v\rightarrow C_w$ and the composite $G$-valuation $v$ of a field $K$ with value group $B_v$. If the assigned to $v$ set $R_v=\lbrace x\in K/v(x)\ge 0$ or $v(x)$ non comparable to $0\rbrace $ is a local ring, then a $G$-valuation $w$ of $K$ into $C_w$ is defined with its assigned set $R_w$ a local ring, as well as another $G$-valuation $u$ of a residue field is defined with $G$-value group $A_u$.
References:
[1] Kontolatou, A., Stabakis, J.: Embedding groups into linear or lattice structures. Bull. Cal. Math. Soc. 82 (1987), 290-297. MR 1134208
[2] Ohm, J.: Semi-valuations and groups of divisibility. Canad. Journ. of Math. 21 (1969), 576-591. MR 0242819 | Zbl 0177.06501
[3] Ribenboim, P.: Theorie des valuations. Les presses de l’ Universite de Montreal (1968). MR 0249425
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