Vol. 37, No. 3, 1971

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BV-functions on semilattices

Joseph Edmund Kist and P. H. Maserick

Vol. 37 (1971), No. 3, 711–723

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).

Mathematical Subject Classification 2000
Primary: 26A45
Secondary: 06A35
Received: 15 June 1970
Revised: 11 February 1971
Published: 1 June 1971
Joseph Edmund Kist
P. H. Maserick