▁concept ▁writing ▁the ▁group ▁in ▁multip lic ative ▁notation ▁as ▁: ▁ ▁instead ▁of ▁ ∞ , ▁we ▁ad join ▁a ▁formal ▁symbol ▁O ▁to ▁ Γ , ▁with ▁the ▁ordering ▁and ▁group ▁law ▁extended ▁by ▁the ▁rules ▁ ▁for ▁all ▁▁ ∈ ▁ ▁, ▁ ▁for ▁all ▁▁ ∈ ▁. ▁ ▁Then ▁a ▁valu ation ▁of ▁K ▁is ▁any ▁map ▁ ▁satisfying ▁the ▁following ▁properties ▁for ▁all ▁a , ▁b ▁ ∈ ▁K : ▁▁ ▁if ▁and ▁only ▁if ▁, ▁, ▁, ▁with ▁equality ▁if ▁v ( a ) ▁ ≠ ▁v ( b ). ▁ ▁( Note ▁that ▁the ▁directions ▁of ▁the ▁in equal ities ▁are ▁revers ed ▁from ▁those ▁in ▁the ▁add itive ▁notation .) ▁ ▁If ▁▁ Γ ▁is ▁a ▁subgroup ▁of ▁the ▁positive ▁real ▁numbers ▁under ▁multiplication , ▁the ▁last ▁condition ▁is ▁the ▁ul tr am et ric ▁inequality , ▁a ▁stronger ▁form ▁of ▁the ▁triangle ▁inequality ▁, ▁and ▁v ▁is ▁an ▁absolute ▁value . ▁ ▁In ▁this ▁case , ▁we ▁may ▁pass ▁to ▁the ▁add itive ▁notation ▁with ▁value ▁group ▁ ▁by ▁taking ▁v +( a ) ▁= ▁− log ▁v ( a ). ▁ ▁Each ▁valu ation ▁on ▁K ▁defines ▁a ▁corresponding ▁linear ▁pre order : ▁a ▁ <0xE2> <0x89> <0xBC> ▁b ▁ ⇔ ▁v ( a ) ▁ ≤ ▁v ( b ). ▁ ▁Con vers ely , ▁given ▁a ▁' <0xE2> <0x89> <0xBC> ' ▁satisfying ▁the ▁required ▁properties , ▁we ▁can ▁define ▁valu ation ▁v ( a ) ▁= ▁{ b : ▁b ▁