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 MSPEC (A) IS Projective IF AND ONLY IF IT IS ISOMORPHIC TO MPROJ (S) FOR SOME FINITELY GENERATED GRADED MONOID S=∨N⩾0 SN, WHERE S0=A.