  The set-indexed fractional Brownian motion (sifBm) has been defined by
Herbin-Merzbach (2006) for indices that are subsets of a metric measure space.
In this paper, the sifBm is proved to statisfy a strenghtened definition of
increment stationarity. This new definition for stationarity property allows to
get a complete characterization of this process by its fractal properties: The
sifBm is the only set-indexed Gaussian process which is self-similar and has
stationary increments.
  Using the fact that the sifBm is the only set-indexed process whose
projection on any increasing path is a one-dimensional fractional Brownian
motion, the limitation of its definition for a self-similarity parameter
0<H<1/2 is studied, as illustrated by some examples. When the indexing
collection is totally ordered, the sifBm can be defined for 0<H<1.
