For quite some time, finite quantum systems with variables in had received enormous attention  with special focus on mutually unbiased bases -. Likewise in recent times, the weak mutually unbiased base ( ) is getting more interest from researchers . This might be due to the fact that they are concepts that have a significant role in quantum computation and information . Previously, most work done on finite geometry is on near-linear geometry. In this type of geometry two lines have at most one point in common -. In this work, we focus our attention on the structure of lattice found in lines in non-near-linear finite geometry and Hilbert space of finite quantum systems. A unique feature of our findings is that, any pair of small size finite geometry of dimensions has a least upper bound (that is the meet) and greatest lower bound (the join). Furthermore, for any two prime dimesional finite geometry there is a reducible join and an irreducible meet. We partition this work into the following sections; the definitions and meaning of notations used in our work was discussed in Section 2 titled preliminaries. Section 3 covers the discussion of the non-near-linear geometry and its subgometries. In Section 4, we discuss the decomposition of lines in non-near-linear finite geometry and its subgometries in relation to group lattice and sublattices. Group lattice and sublattices in the set of finite quantum system is discussed in Section 5. Finally in Section 6, we conclude our work.
1) A POSET means a partially ordered set.
2) represents ring of integer modulo .
3) is where represents the set of invertible elements in and is referred to as Euler Phi function. It is defined as
4) is called the Dedekind psi function where;
5) The notation represents the set of proper divisors of and any pairs of divisors form a lattice in our work. It is a POSET with divisibility as partial order. The number of element in this set is the divisor function . where
It forms a complete lattice in .
6) In this paper, we use the symbol to represent partial order. If is a factor of then and is a subgroup of .
7) We define the set of subgroup of as
It is a partially ordered set with divisibility as partial order. There is a bijection between the set of divisors of and where represents a divisor of . The elements of are embedded in for thus
8) The notations: and denote meet or roof and floor or join respectively. The greatest common divisor of two element and is represented in this work as .
9) We express as
Mathematically, is a cyclic module.
This work focuses on non-near-linear geometry. That is, in this case two lines for example intersects in more than one points. It is related to the fact that is a ring of integer modulo and all the lines in this work are through the origin.
3. Non-Near-Linear Geometry and Its Subgometries
We define the finite geometry as the combination
represents points on a line and represent lines in where
Definition 3.1. A line through the origin is defined as
In this work and represents the same concept, so as a result we use them interchangeably.
Definition 3.2. We define a lattice as a POSET in which every pair of elements have the LUB (or the join, ) and GLB (or the meet, ). In this work each element of the set represents a finite geometry.
From the results of  we confirm the following propositions without proof:
Proposition 3.3. 1) If
then . (10)
then , (11)
We confirmed that is a maximal line in if and is a subline in if .
2) The number of maximal lines in with points each is .
3) Suppose we define a line in finite geometry as in Equation (9), is also
, in (13)
at the same time the line in is a subline of
4) If two maximal lines have points in common where . The points gives a subline where .
If we consider the subgeometry , the subline in is a maximal line in . There is maximal lines in subgeometry of finite geometry .
4. Factorization of Lines in Non-Near-Linear Geometry as Lines in Near-Linear Geometries
In this section, each lines in is decomposed as lines in . Using the concept of Good  two bijective maps were created between the ordinates of each of the lines in non-near-linear geometry. Similar concept was used in the past by  to factorize a large finite dimensional finite quantum systems as products of many small dimensional finite systems that is,
Equation (15) and Equation (16) represent position and momentum states respectively. We used the above bijection in our earlier work  to factorize maximal lines in as prime factor lines . There is a bijection between the set of lines of that is,
and prime factor lines that is
and a prime (20)
We confirmed the existence of maximal lines altogether. Out of which there are only maximal lines are distinct. We also confirm that each distinct lines has equivalent lines whose all its points map the points of each of the distinct lines in the non-near-linear finite geometry and as a result we confirm the existence of an equivalence relation between all the points in each of the maximal lines and other lines in the non-near-linear geometry. Also we discovered that each of the factored lines in is a maximal lines in and at the same time a subline in . In addition, if we take any two arbitrary maximal lines one from each near-linear geometry , the two lines join to form a subline of and at the same time taking the intersection of the near-linear geometries gives a meet which is a subgeometry of the two prime geometries. Hence and form a lattice of .
In this work, the term decomposition is analogous to factorization of non-prime integers as products of their primes. The geometry is related to the set of divisors of , the subline in is related to common divisor between two or more integers and corresponds to a line which contains only one point .
The factorization is related to finding the lowest common multiple (L.C.M.) of a set of integer, the L.C.M. represents the roof. The H.C.F. of any two prime geometries is connected to finding the intersection any two disjoint sets. In this work we call the H.C.F. the floor. As an illustration, we define line in as in Equation (19) thus.
Suppose for a non-prime, not every element in has a multiplicative inverse and so as a result Equation (19) is expressed further as,
here we represent
However, for , the line .
As an illustration, we express all maximal lines in for in terms of its primes discussed in Equation (21) and Equation (23) above by decomposing line .
Using Equation (15) the ordinate 2 in is decomposed as;
also using Equation (16) the ordinate 1 in is decomposed as;
Therefore is decomposed as;
if we relate Equation (26) to Equation (21) and Equation (23), is expressed as
Here we use Equation (15) and Equation (16) to express maximal lines in and its subgometries and as partition in Table 1 where is isomorphic to and . Suppose , , , , , and
Proposition 4.1. 1) Suppose is a non-near-linear finite geometry, then the set of near-linear geometries (for a prime) obtained through factorizing the non-near-linear geometry forms a lattice, and as a result forms a partition.
Table 1. Maximal lines in non-near-linear finite geometries G6 in terms of its prime factor lines.
Proof: Since are near linear geometries and taking the intersection of any two lines and yields only line which is the trivial near-linear geometry . Hence the proof is complete.
2) Two lines in are isomorphic if there is a 1-1 correspondence between the points in and .
Proof: Since and and the existence of bijection between the points in and make itself evident.
4.1. Symplectic Group on
We define the matrices
form a group called symplectic group group.
Suppose we act on all points of line in . This produces all the points of the line . We write it as . Suppose is a prime, acting on the line , we obtain all the lines (maximal lines) through the origin. In this work, we label the lines as
In this work, we take the condition that for , is replaced by .
Thus, is expressed as
where are related in Equation (15) and is related to in Equation (16).
Any pair of geometry in the set form a lattice and the set of all subgometries of is isomorphic to the set .
4.2. Join Reducible and Meet Irreducible in Finite Geometry
In this subsection, we discuss how the union of two or more near-linear geometries forms subgometries of non-near-linear geometry and their intersection produces maximal lines in near-linear geometry via partial ordering and as a result forms a lattice. Suppose we define the set of geometry
If we take the set and : , and .
That is the maximal lines in and are sublines in and the intersection of sublines is isomorphic to maximal lines in . In this case , they form a join in and a meet in . and are called the LUB and GLB and respectively. This forms a complete lattice in . Likewise the three sets, , , and are sublattices in . The subgometries and is isomorphic to and respectively.
The join is analogous to non prime integers which can be expressed as products of prime integers, while the meet is related to the factors of such non prime integers which when one factorizes further it get to a point where there the only factor it will have is the integer 1.
Table 2. A table of maximal lines in non-near-linear finite geometry and its subgometries.
Figure 1. The Hasse diagram showing the non-near-linear geometry and its subgometries, and along with Hilbert spaces of the subsystems of .
5. Lattice Theory for Finite Dimensional Hilbert Space with Variables in
We consider a quantum system with positions and momenta in , which we denote as . For a divisor of , is a subgroup of . In this case we say that is a subsystem of .
Let and be position and momentum states, respectively. Here the , are not variables but rather represent position and momentum respectively, in the -dimensional quantum system. The variables of m belongs to . The Fourier transform is given by:
where We define the displacement operator as
where Equation (36) and Equation (37) satisfy the condition:
The where form a representation of Heisenberg-Weyl group. References  used Equation (15) and Equation (16) to decompose a system with variables in , where is given in Equation (6), in terms of subsystems with variables in . The existence of one-to-one correspondence between and the tensor product is confirmed where
The same analogy is done for momentum basis thus;
Embedding of Small Systems into Large Systems
If then also means that is a subsystem of .
In quantum states, which takes variables in is embedded in which takes values in .
We express it as
The momentum representation is expressed as
Hence the set of subsystems, of is isomorphic to the set and form a complete lattice in .
Our central focus in this work is on the concept of lattice which exists in non-near-linear finite geometry and prime geometries and the finite quantum system and its subsystem with subsystems forming a lattice. More importantly, the complexity shown in this work demonstrates those important relations which exist between stucture and its substructures both in quantum system and geometry in its phase space.
 Shalaby, M. and Vourdas, A. (2013) Mutually Unbiased Projectors and Duality between Lines and Bases in Finite Quantum Systems. Annals of Physics, 337, 208-220.
 Oladejo, S.O., Lei, C. and Vourdas, A. (2014) Partial Ordering of Weak Mutually Unbiased Bases. Journal of Physics A: Mathematical and Theoretical, 47, Article ID: 485204.