▁of ▁the ▁triangle ▁inequality ▁on ▁metric ▁spaces ▁adapted ▁to ▁an ▁arbitrary ▁ Γ ▁( see ▁Mult ip lic ative ▁notation ▁below ). ▁For ▁valu ations ▁used ▁in ▁geometric ▁applications , ▁the ▁first ▁property ▁implies ▁that ▁any ▁non - empty ▁germ ▁of ▁an ▁analyt ic ▁variety ▁near ▁a ▁point ▁contains ▁that ▁point . ▁▁ ▁The ▁valu ation ▁can ▁be ▁interpreted ▁as ▁the ▁order ▁of ▁the ▁leading - order ▁term . ▁The ▁third ▁property ▁then ▁corresponds ▁to ▁the ▁order ▁of ▁a ▁sum ▁being ▁the ▁order ▁of ▁the ▁larger ▁term , ▁unless ▁the ▁two ▁terms ▁have ▁the ▁same ▁order , ▁in ▁which ▁case ▁they ▁may ▁cancel , ▁in ▁which ▁case ▁the ▁sum ▁may ▁have ▁smaller ▁order . ▁ ▁For ▁many ▁applications , ▁ ▁is ▁an ▁add itive ▁subgroup ▁of ▁the ▁real ▁numbers ▁ ▁in ▁which ▁case ▁ ∞ ▁can ▁be ▁interpreted ▁as ▁+ ∞ ▁in ▁the ▁extended ▁real ▁numbers ; ▁note ▁that ▁ ▁for ▁any ▁real ▁number ▁a , ▁and ▁thus ▁+ ∞ ▁is ▁the ▁unit ▁under ▁the ▁binary ▁operation ▁of ▁minimum . ▁The ▁real ▁numbers ▁( ext ended ▁by ▁+ ∞ ) ▁with ▁the ▁operations ▁of ▁minimum ▁and ▁addition ▁form ▁a ▁sem iring , ▁called ▁the ▁min ▁tropical ▁sem iring , ▁and ▁a ▁valu ation ▁v ▁is ▁almost ▁a ▁sem iring ▁hom omorphism ▁from ▁K ▁to ▁the ▁tropical ▁sem iring , ▁except ▁that ▁the ▁hom omorphism ▁property ▁can ▁fail ▁when ▁two ▁elements ▁with ▁the ▁same ▁valu ation ▁are ▁added ▁together . ▁ ▁Mult ip lic ative ▁notation ▁and ▁absolute ▁values ▁We ▁could ▁define ▁the ▁same