Lattice Structure and Matrix Representation in Discrete Order Geometry

Author: Zhang Suhang,

              Luoyang, Henan

Abstract

Discrete Order Geometry (DOG) takes finite discrete lattice points and order nesting as its ontology, without presupposing continuous coordinates or metrics. This paper establishes, within DOG, a correspondence between local order lattices and matrix representations. It defines DOG local order lattices, order coupling matrices, and topological coupling matrices, and introduces the order Laplacian matrix. It proves that a finite DOG local order lattice uniquely determines an order coupling matrix given neighborhoods and order weights; that the topological coupling matrix can be derived from the order coupling matrix and reconstructs the direct order-coupling topological relations; and that the supremum and infimum of a local lattice provide additional order constraints for matrix closure. The conclusion is: the DOG lattice is the geometric ontology, while the matrix is its algebraic carrier; the two correspond but are not identical.

Keywords: Discrete Order Geometry; local order lattice; order coupling matrix; topological coupling matrix; order Laplacian matrix

1 Introduction

Discrete Order Geometry (DOG) abandons the continuous manifold assumption and takes finite discrete lattice point sets and order nesting as its underlying ontology. In DOG, geometric degrees of freedom arise from order associations among lattice points, rather than from a pre-given differential structure.

This paper discusses only two basic structures within DOG:

1. Lattice structure: describes order nesting and local order relations among lattice points;

2. Matrix structure: describes order associations and coupling relations among lattice points.

The goal is to establish a correspondence between the two: the DOG lattice provides the geometric ontology, and the matrix provides the algebraic representation.

2 DOG Lattice Points and Local Order Lattices

2.1 Set of Lattice Points

Define the set of DOG lattice points as a finite set:

\mathcal{L}=\{p,q,r,\dots\}

No continuous coordinates or metrics are intrinsically defined in \mathcal{L}.

2.2 Neighborhood Structure

For each lattice point p\in\mathcal{L}, define its neighborhood:

N(p)\subseteq \mathcal{L}

The neighborhood satisfies local association: a lattice point p has direct order associations only with lattice points in N(p).

2.3 Local Order Relation

On N(p), define an order relation:

\preceq

If for any q,r\in N(p), there exist a supremum q\vee r and an infimum q\wedge r, then N(p) under \preceq is called a local order lattice.

2.4 DOG Local Order Lattice

Define a DOG local order lattice as the triple:

\mathcal{G}=(\mathcal{L},\{N(p)\}_{p\in\mathcal{L}},\preceq)

It satisfies three basic properties:

1. Finiteness: \mathcal{L} is a finite set;

2. Locality: order relations are directly defined only within the neighborhood N(p);

3. Order nesting: each neighborhood N(p) internally forms a local order lattice.

3 Definitions and Algebraic Structure of Matrices

3.1 Definition of the Order Coupling Matrix

For a DOG local order lattice \mathcal{G}, define the order coupling matrix:

\mathbf{M}=(M_{pq})_{p,q\in\mathcal{L}}

where M_{pq} denotes the strength of the direct order association between lattice points p and q.

If q\notin N(p), we set:

M_{pq}=0

Therefore, the zero pattern of the matrix \mathbf{M} directly encodes the direct order-coupling relations of DOG.

3.2 Topological Coupling Matrix

From the order coupling matrix \mathbf{M}, define the topological coupling matrix:

C_{pq}=

\begin{cases}

1, & M_{pq}\neq 0,\\

0, & M_{pq}=0.

\end{cases}

The topological coupling matrix C records whether a direct order coupling exists between lattice points, stripping away coupling strength information and retaining only the topological information of whether coupling exists.

3.3 Degree Matrix and Order Laplacian Matrix

Define the degree matrix:

D=\operatorname{diag}(d_p),\qquad

d_p=\sum_{q\in\mathcal{L}}|M_{pq}|.

Define the order Laplacian matrix:

\mathbf{L}=D-\mathbf{M}.

Note: This order Laplacian matrix is constructed from the order coupling matrix and is a generalization of weighted spectral graph theory within the DOG framework, distinct from the standard graph Laplacian based on a binary topological matrix. The matrix \mathbf{L} describes the structure of order propagation and diffusion on DOG lattice points.

4 Correspondence from Lattice to Matrix

4.1 Construction

Given a DOG local order lattice \mathcal{G}, choose a basis vector e_p for each lattice point p. Define an order weight function:

w:N(p)\times N(p)\to \mathbb{R}

Let:

M_{pq}=

\begin{cases}

w(p,q), & q\in N(p),\\

0, & q\notin N(p).

\end{cases}

Thus the order coupling matrix \mathbf{M} is obtained.

4.2 Uniqueness Theorem

Theorem 1: Let \mathcal{G} be a finite DOG local order lattice. Given the neighborhood structure and the order weight function w, there exists a unique order coupling matrix \mathbf{M} such that M_{pq}=w(p,q) when q\in N(p), and M_{pq}=0 otherwise.

Proof: Since \mathcal{L} is finite, the matrix entries M_{pq} are uniquely determined term by term from w and the neighborhood relation. Therefore, \mathbf{M} exists and is unique.

4.3 Reconstruction from Matrix to Coupling Relations

Theorem 2: Given the order coupling matrix \mathbf{M}, define the topological coupling matrix C by C_{pq}=1 if and only if M_{pq}\neq0. Then C reconstructs the direct order-coupling relations of DOG.

If \mathbf{M} is symmetric, then the direct order-coupling relation is symmetric.

4.4 Local Lattice Constraints

Theorem 3: In each neighborhood N(p), if \preceq forms a local order lattice, then for any q,r\in N(p), there exist a supremum q\vee r and an infimum q\wedge r. The zero pattern of the matrix \mathbf{M} must be consistent with the coupling relations in N(p); however, \mathbf{M} itself generally does not uniquely determine q\vee r and q\wedge r. Local lattice operations require additional order data.

Therefore, the DOG lattice and the order coupling matrix are in a corresponding encoding relation, rather than a complete isomorphism.

5 Matrix Operations and Order Nesting

5.1 Matrix Powers and Order Propagation

The entries of the matrix power \mathbf{M}^k:

(\mathbf{M}^k)_{pq}

represent the total strength of k-step order propagation from lattice point p to lattice point q.

5.2 Order Laplacian Matrix and Connected Components

If \mathbf{M} is symmetric and nonnegative, then the dimension of the null space of the order Laplacian matrix \mathbf{L}=D-\mathbf{M} equals the number of connected components of the DOG lattice. This conclusion inherits from weighted spectral graph theory and provides an algebraic criterion for the non-global connectivity of DOG.

5.3 Local Supremum and Matrix Closure

In the local order lattice N(p), if q,r\in N(p), then there exists a supremum s=q\vee r satisfying:

q\preceq s,\qquad r\preceq s.

In the matrix, this can appear as an order path closure from q,r to s. The infimum t=q\wedge r appears as a reverse closure.

6 Examples

6.1 One-Dimensional Chain

Let:

N(p)=\{p-1,p,p+1\}.

The order coupling matrix is tridiagonal:

M_{p,p-1}\neq0,\qquad M_{p,p}\neq0,\qquad M_{p,p+1}\neq0.

The topological coupling matrix corresponds to a one-dimensional chain structure.

6.2 Three-Element Local Lattice

Let:

N(p)=\{a,b,c\},\qquad a\preceq c,\qquad b\preceq c.

Then c=a\vee b is the supremum. The order coupling matrix is nonzero at the positions (a,c) and (b,c), and may be zero at (a,b). The zero pattern of the matrix is consistent with the local order coupling.

6.3 Disconnected DOG Lattice

If \mathcal{L} is divided into two mutually unassociated subsets, then \mathbf{M} takes the block diagonal form:

\mathbf{M}=

\begin{pmatrix}

\mathbf{M}_1 & 0\\

0 & \mathbf{M}_2

\end{pmatrix}.

The null space dimension of the order Laplacian matrix \mathbf{L} is 2, corresponding to two connected components.

7 Conclusion

This paper establishes, within DOG, a correspondence between lattice structure and matrix representation:

1. A DOG local order lattice consists of finite lattice points, neighborhood structure, and local order relations;

2. The order coupling matrix \mathbf{M} completely encodes the strength of direct order associations among lattice points;

3. The topological coupling matrix C reconstructs the direct order-coupling topological relations;

4. The order Laplacian matrix \mathbf{L} describes order propagation and connected components;

5. The supremum and infimum of a local lattice provide additional order constraints for matrix closure;

6. The DOG lattice and the order coupling matrix correspond but are not identical: the lattice is the geometric ontology, and the matrix is the algebraic carrier.

This framework provides a foundational mathematical basis for the subsequent dynamics and algebraic construction of DOG.

References

Omitted.