  The set of idempotents of any semigroup carries the structure of a biordered
set, which contains a great deal of information concerning the idempotent
generated subsemigroup of the semigroup in question. This leads to the
construction of a free idempotent generated semigroup
$\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered
set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite,
the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of
constraint satisfaction problems involving rational subsets of direct products
of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. As an application,
we obtain decidability of the word problem for an important class of examples.
Also, we prove that for finite $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$ is
always a weakly abundant semigroup satisfying the congruence condition.
