  We initiate the study of the internal structure of C*-algebras associated to
a left cancellative semigroup in which any two principal right ideals are
either disjoint or intersect in another principal right ideal; these are
variously called right LCM semigroups or semigroups that satisfy Clifford's
condition. Our main findings are results about uniqueness of the full semigroup
C*-algebra. We build our analysis upon a rich interaction between the group of
units of the semigroup and the family of constructible right ideals. As an
application we identify algebraic conditions on S under which C*(S) is purely
infinite and simple.
