  We give a new construction of a C*-algebra from a cancellative semigroup $P$
via partial isometric representations, generalising the construction from the
second named author's thesis. We then study our construction in detail for the
special case when $P$ is an LCM semigroup. In this case we realize our algebras
as inverse semigroup algebras and groupoid algebras, and apply our construction
to free semigroups and Zappa-Sz\'ep products associated to self-similar groups.
