New definitions are given for
positivity and bounded variation of functions on a semilattice S so that such
functions extend to measures (respectively, signed measures) on the σ-algebra
generated by some representation of S as a semilattice of sets under intersection. All
such representations lie in a Stone space determined by S. Functions on a
subsemilattice S of a distributive lattice L which extend to isotone valuations on L
are characterized in terms of a partial ordering of finite sequences in S. Functions on
a regulated semilattice which correspond to regular Borel measures on the
associated locally compact space are characterized in terms of inclusion-exclusion
sums.
