We test the integrated galactic initial mass function ( IGIMF ) on the chemical evolution of 16 ultra-faint dwarf ( UFD ) galaxies discussing in detail the results obtained for three of them : Boötes I , Boötes II and Canes Venatici I , taken as prototypes of the smallest and the largest UFDs .
These objects have very small stellar masses ( \sim 10 ^ { 3 } -10 ^ { 4 } \mathrm { M _ { \odot } } ) and quite low metallicities ( [ Fe/H ] < -1.0 dex ) .
We consider three observational constraints : the present-day stellar mass , the [ \alpha /Fe ] vs. [ Fe/H ] relation and the stellar metallicity distribution function .
Our model follows in detail the evolution of several chemical species ( H , He , \alpha -elements and Fe ) .
We take into account detailed nucleosynthesis and gas flows ( in and out ) .
Our results show that the IGIMF , coupled with the very low star formation rate predicted by the model for these galaxies ( \sim 10 ^ { -4 } -10 ^ { -6 } \mathrm { M _ { \odot } yr ^ { -1 } } ) , can not reproduce the main chemical properties , because it implies a negligible number of core-collapse SNe and even Type Ia SNe , the most important polluters of galaxies .
On the other hand , a constant classical Salpeter IMF gives the best agreement with data .
We suggest for all the UFDs studied a very short infall time-scale and high galactic wind efficiencies .
Comparing with Galaxy data we suggest that UFDs could not be the building blocks of the entire Galactic halo , although more data are necessary to draw firmer conclusions .