It has been shown that the cone
C of completely monotonic functions on a commutative semigroup G with identity
induces a vector lattice ordering on the vector space E = C − C spanned by C.
An intrinsic characterization of the absolute value of the functions in E is
desirable. In the present work we offer such a characterization when each
member of G is idempotent, i.e. G is a semilattice. A notion of variation and
bounded variation (BV) of arbitrary functions on G is introduced. We show
that E is precisely the family of BV-functions and that if f ∈ E, then our
concept of variation of f agrees with the usual absolute value as given by
f ∨ (−f).
