Lattice Theory for Finite Dimensional Hilbert Space with Variables in Zd

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry  were expressed as products of lines in near-linear finite geometry  (where p is a prime). An existence of lattice between any pair of near-linear finite geometry  of  is confirmed. For q|d, a one-to-one correspondence between the set of subgeometry  of  and finite geometry  from the subsets of the set {D(d)} of divisors of d (where each divisor represents a finite geometry) and set of subsystems {∏(q)} (with variables in Zq) of a finite quantum system ∏(d) with variables in Zd and a finite system from the subsets of the set of divisors of d is established.