Presentations and word problem for strong semilattices of semigroups

Let $I$ be a semilattice, and $S_i(i\in I)$ be a family of disjoint semigroups. Then we prove that the strong semilattice $S=\mathcal{S} [I,S_i,\phi_{j,i}]$ of semigroups $S_i$ with homomorphisms $\phi _{j,i}:S_j\rightarrow S_i$ $(j\geq i)$ is finitely presented if and only if $I$ is finite and each $S_i$ $(i\in I)$ is finitely presented. Moreover, for a finite semilattice $I$, $S$ has a soluble word problem if and only if each $S_i$ $(i\in I)$ has a soluble word problem. Finally, we give an example of non-automatic semigroup which has a soluble word problem.