  We study algebraic and homological properties of facet ideals of order
complexes of posets which we call ideals of maximal flags of posets or simply
flag ideals. We characterize the unmixed and Cohen-Macaulay flag ideals of
graded posets. We also give structural results for the multigraded Betti
numbers of flag ideals of such posets. The structural results on multigraded
Betti numbers of flag ideals are used to characterize the class of flag ideals
with linear resolutions.
