The projective monoid schemes

For a certain monoid $ A$, let $\mathbb{P}_A^{r}$ be its associated projective monoid scheme. In this paper, we show that every closed monoid subscheme of $\mathbb{P}_A^{r}$ can be determined by a certain ideal of the graded monoid $S=A[t_0,t_1,...,t_r]$. Furthermore, we show that a monoid scheme $ Y $ over $ Mspec(A) $ is projective if and only if it is isomorphic to $ Mproj(S) $ for some finitely generated graded monoid $ S=\bigvee_{ n \geqslant 0} S_{n} $, where .$ S_{0}=A$