  A Markov Additive Process is a bi-variate Markov process
$(\xi,J)=\big((\xi_t,J_t),t\geq0\big)$ which should be thought of as a
multi-type L\'evy process: the second component $J$ is a Markov chain on a
finite space $\{1,\ldots,K\}$, and the first component $\xi$ behaves locally as
a L\'evy process, with local dynamics depending on $J$. In the
subordinator-like case where $\xi$ is nondecreasing, we establish several
results concerning the moments of $\xi$ and of its exponential functional
$I_{\xi}=\int_{0}^{\infty} e^{-\xi_t}\mathrm dt,$ extending the work of Carmona
et al., and Bertoin and Yor.
  We then apply these results to the study of multi-type self-similar
fragmentation processes: these are self-similar analogues of Bertoin's
homogeneous multi-type fragmentation processes Notably, we encode the genealogy
of the process in a tree, and under some Malthusian hypotheses, compute its
Hausdorff dimension in a generalisation of our previous work.
