  We model early star forming regions and their chemical enrichment by
Population III (Pop III) supernovae with nucleosynthetic yields featuring high
[C/Fe] ratios and pair-instability supernova (PISN) signatures. We aim to test
how well these chemical abundance signatures are preserved in the gas prior to
forming the first long-lived low-mass stars (or second-generation stars). Our
results show that second-generation stars can retain the nucleosynthetic
signature of their Pop III progenitors, even in the presence of
nucleosynthetically normal Pop III core-collapse supernovae. We find that
carbon-enhanced metal-poor stars are likely second-generation stars that form
in minihaloes. Furthermore, it is likely that the majority of Pop III
supernovae produce high [C/Fe] yields. In contrast, metals ejected by a PISN
are not concentrated in the first star forming haloes, which may explain the
absence of observed PISN signatures in metal-poor stars. We also find that
unique Pop III abundance signatures in the gas are quickly wiped out by the
emergence of Pop II supernovae. We caution that the observed fractions of stars
with Pop III signatures cannot be directly interpreted as the fraction of Pop
III stars producing that signature. Such interpretations require modelling the
metal enrichment process prior to the second-generation stars' formation,
including results from simulations of metal mixing. The full potential of
stellar archaeology can likely be reached in ultra-faint dwarf galaxies, where
the simple formation history may allow for straightforward identification of
second-generation stars.
