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Some notes on the composite G -valuations

Angeliki Kontolatou — 1994

Archivum Mathematicum

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 , β ) : A u → B v → 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 = { x ∈ K / v ( x ) ≥ 0 or v ( x ) non comparable to 0 } 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 .

Some properties of Lorenzen ideal systems

Aleka Kalapodi;  Angeliki Kontolatou;  Jiří Močkoř — 2000

Archivum Mathematicum

Let G be a partially ordered abelian group ( p o -group). The construction of the Lorenzen ideal r a -system in G is investigated and the functorial properties of this construction with respect to the semigroup ( R ( G ) , ⊕ , ≤ ) of all r -ideal systems defined on G are derived, where for r , s ∈ R ( G ) and a lower bounded subset X ⊆ G , X r ⊕ s = X r ∩ X s . It is proved that Lorenzen construction is the natural transformation between two functors from the category of p o -groups with special morphisms into the category of abelian ordered semigroups.

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