  We examine analytically and numerically the effect of fractionality on a
saturable bulk and surface impurity embedded in a 1D lattice. We use a
fractional Laplacian introduced previously by us, and by the use of lattice
Green functions we are able to obtain the bound state energies and amplitude
profiles, as a function of the fractional exponent $s$ and saturable impurity
strength $\chi$ for both, surface and bulk impurity. The transmission is
obtained in closed form as a function of $s$ and $\chi$, showing strong
deviations from the standard case, at small fractional exponent values. The
selftrapping of an initially-localized excitation is qualitatively similar for
the bulk and surface mode, but in all cases complete confinement is obtained at
$s\rightarrow 0$, as shown theoretically and observed numerically.
