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.

LET (FORMULA) BE A REDUCED POLYNOMIAL OF TOTAL DEGREE AT LEAST TWO AND THE IDEAL JF STANDS FOR JACOBIAN IDEAL OF F WHICH DE NES THE SINGULAR SUBSCHEME OF THE HYPERSURFACE DE NED BY F...

IN THIS PAPER, WE DEFINE COVER OF AN S-POSET A, COESSENTIAL EPIMORPHISM AND projective COVER IN POS-S. ALSO WE SHOW THAT projective COVER IN POS-S IS DETERMINED UNIQUELY UP TO ISOMORPHISM AND THAT projective COVERS OF S-POSETS DO NOT ALWAYS EXIST.

THE ZERO DIVISOR GRAPH OF A PARTIALLY ORDERED SET (POSET, BRIEY) WITH THE LEAST ELEMENT 0, WHICH IS DENOTED BY G* (P), IS AN UNDIRECTED GRAPH WITH VERTEX SET P*=P\ {0} AND, FOR TWO DISTINCT VERTICES X AND Y, X IS ADJACENT TO Y IN G* (P) IF AND ONLY IF {X, Y}L={0}, WHERE, FOR A SUBSET S OF P, SL IS THE SET OF ALL ELEMENTS X 2 P WITH X £S FOR ALL S Î S. IN THIS PAPER WE STUDY THE PROJECTIVITY OF THE ZERO DIVISOR GRAPH G* (P).

projective MANIFOLDS FORM AN IMPORTANT CLASS OF SPACES IN GEOMETRY AND TOPOLOGY.METRIC projective MANIFOLDS ARE ARE TYPICAL EXAMPLES OF SPACES ON WHICH STRAIGHT LINE SEGMENTS ARE THE SHORTEST CONNECTION BETWEEN TWO POINTS, AT LEAST AT A LOCAL SCHEME.RANDERS MANIFOLDS (M, F=A+B) ARE THE UBIQUITOUS IN FINSLERIAN GEOMETRY WITH APPLICATIONS. A NOTABLE SUB-GROUP OF THE projective GROUP PROJ (M, F) WHICH IS DENOTED BY PROJ (M, F) TURNS THE projective FINSLER GEOMETRY TO BE A FINER GEOMETRY CALLED SPECIAL projective GEOMETRY. SOME DIFFICULT RESULTS IN projective FINSLER GEOMETRY WHICH ARE NOT PROVED YET, ARE ESTABLISHED IN THIS FINER projective GEOMETRY; A LICHNEROWICZ-OBATA TYPE RESULT IS PROVED FOR RANDERS MANIFOLDS.