The projective monoid schemes

For a certain monoid A, let PrA be its associated projective monoid scheme. In this paper,we show that every closed monoid subscheme of PrA can be determined by a certain ideal of the graded monoid S = A[t0, t1, ..., tr]. Furthermore, we show that a monoid scheme Y over M spec(A) is projective if and only if it is isomorphic to M proj(S) for some finitely generated graded monoid S =∨ n⩾0 Sn, where S0 = A