Some Notes on the Composite G { Valuations 2732

In analogy with the notion of the composite semi-valuations, we deene the composite G-valuation v from two other G-valuations w and u. We consider a lexicographically exact sequence (a;) : Au ! Bv ! Cw and the composite G-valuation v of a eld K with value group Bv. If the assigned to v set Rv = fx 2 K=v(x) 0 or v(x) non comparable to 0g is a local ring, then a G-valuation w of K into Cw is deened with its assigned set Rw a local ring, as well as another G-valuation u of a residue eld is deened with G-value group Au. It is our main aim to show that under some diierentiations and some adjustments it is possible to transfer the theory of the composite semi-valuations as it is exposed by Ohm in 2], to the case of the G-valuations. So an appropriate homomorphism is introduced, the composite G-valuations are deened by analogy to the former ones and similar conditions are stated under which an ordered exact sequence splits. 1.1. As it is known (e.g..1]) a G-valuation is a function v of the multiplicative group K of a eld K, in an ordered group G such that for all x; y in K : (i) v(xy) = v(x) + v(y) (ii) if v(x) > and v(y) > , then v(x + y) > , for each 2 G (iii) v(?1) = 0 We can extend v on K by specifying that v(0) = 1, where 1 is a symbol such that a < 1 and a + 1 = 1 for all a 2 G. Relation (ii) may be written as (ii) 0 v(x + y) inf ~ G fv(x); v(y)g: In fact, the inf ~ G means the innmum in a concrete order-completion, where the relation a inf ~ G (a 1 ; a 2) gives that a is larger than or equal to the smaller of a 1 ; a 2 , but it would be parallel to the smaller or to both of them.