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In analogy with the notion of the composite semi-valuations, we define the composite -valuation from two other -valuations and . We consider a lexicographically exact sequence and the composite -valuation of a field with value group . If the assigned to set or non comparable to is a local ring, then a -valuation of into is defined with its assigned set a local ring, as well as another -valuation of a residue field is defined with -value group .
Let be a partially ordered abelian group (-group). The construction of the Lorenzen ideal -system in is investigated and the functorial properties of this construction with respect to the semigroup of all -ideal systems defined on are derived, where for and a lower bounded subset , . It is proved that Lorenzen construction is the natural transformation between two functors from the category of -groups with special morphisms into the category of abelian ordered semigroups.
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