The main result of this paper is
that if S is a locally compact semilattice of finite breadth, then every complex
homomorphism of the measure algebra M(S) is given by integration over a Borel
filter (subsemilattice whose complement is an ideal), and that consequently M(S) is a
P-algebra in the sense of S. E. Newman. More generally it is shown that if S is a
locally compact Lawson semilattice which has the property that every bounded
regular Borel measure is concentrated on a Borel set which is the countable union of
compact finite breadth subsemilattices, then M(S) is a P-algebra. Furthermore,
complete descriptions of the maximal ideal space of M(S) and the structure
semigroup of M(S) are given in terms of S, and the idempotent and invertible
measures in M(S) are identified.
