An additive valuation on a field 
is a function 
to a totally ordered Abelian
group, with 
iff 
, 
, and 
. Real-valued additive valuations use
an ordered subgroup of 
. For a non-Archimedean absolute
value, 
gives the additive convention. For example, the order of vanishing along a prime
divisor is an integer-valued additive valuation.
A real-valued additive valuation of the function field of an algebraic variety is centered at a closed point 
when it is nonnegative on
the local ring and strictly positive on its maximal
ideal. Its valuation ideals and log discrepancy
determine the normalized volume of the singularity.
