[@bertsfrag02] has shown that any such fragmentation is also characterized by the same $3$-tuple $(\alpha ,c,\nu )$ as above, meaning that the laws of partition-valued and ranked self-similar fragmentations are in a one-to-one correspondence. In fact, for every $(\alpha ,c,\nu )$, one can construct a version of the partition-valued fragmentation $\Pi $ with parameters $(\alpha ,c,\nu )$, and then $(|\Pi (t)|,t\geq 0)$ is the ranked fragmentation with parameters $(\alpha ,c,\nu )$. Let us build this version now. It is done following [@berthfrag01; @bertsfrag02] by a Poissonian construction. Recall the notation $\rho _{\mathbf{s}}(\mathrm{d}\pi )$, and define $\kappa _{\nu }(\mathrm{d}\pi )=\int_{S}\nu (\mathrm{d}{\mathbf{s}})\rho _{\mathbf{s}}(\mathrm{d}\pi )$. Let $\#$ be the counting measure on $\mathbb{N}$ and let $(\Delta _{t},k_{t})$ be a $\mathcal{P}_{\infty }\times
\mathbb{N}$-valued Poisson point process with intensity $\kappa _{\nu
}\otimes \#$. We may construct a process $(\Pi ^{0}(t),t\geq 0)$ by letting $\Pi ^{0}(0)$ be the trivial partition $(\mathbb{N},\varnothing ,\varnothing
,\ldots )$, and saying that $\Pi ^{0} $ jumps only at times $t$ when an atom $(\Delta _{t},k_{t})$ occurs. When this is the case, $\Pi ^{0}$ jumps from the state $\Pi ^{0}(t-)$ to the following partition $\Pi ^{0}(t)$: replace the block $\Pi _{k_{t}}^{0}(t-)$ by $\Pi _{k_{t}}^{0}(t-)\cap \Delta _{t}$, and leave the other blocks unchanged. 