Foundations of Discrete Order Geometry (DOG): System Definition, Morphisms and Order Isomorphism

 

— First Cornerstone Draft of the DOG Framework

 

Author: Zhang Suhang

(Luoyang, Henan)

 

 

 

Remarks

 

This paper lays the first foundation for the Discrete Order Geometry (DOG) framework: rigorous definitions of DOG systems, DOG morphisms and order isomorphisms.

 

Without this foundation, DOG cannot address:

 

- When two systems belong to the same type;

- Mappings between DOG systems;

- Equivalence classes and invariants of DOG;

- Precise interfaces with order theory, graph theory and topology.

 

All definitions are tentative framework attempts. No claim is made to uniqueness or optimality. The sole purpose is to transform core DOG concepts from qualitative descriptions into mathematically tractable objects.

 

 

 

1 Introduction

 

The fundamental tenet of DOG (Discrete Order Geometry) is:

Geometric systems may be characterized primarily by hierarchical nesting, order isomorphism and continued‑fraction scales, independent of spatial connectedness.

 

Under this premise, the basic questions are not about computation, but:

 

1. What is a DOG system?

2. When are two DOG systems regarded as structurally identical?

3. How do we construct mappings between DOG systems?

4. Which quantities are preserved under DOG mappings?

 

This paper answers these four questions by providing:

 

- The 5‑tuple definition of a DOG system;

- Definition of DOG morphisms (including monotonicity and functional relations);

- Definition of order isomorphism;

- Proof that order isomorphism is an equivalence relation;

- The first DOG invariant: rank of hierarchy;

- Complexity analysis for the order‑isomorphism decision problem.

 

This work constitutes the mathematical foundation of DOG, rather than isolated technical definitions.

 

 

 

2 Definition of a DOG System

 

Definition 2.1 (DOG System)

A DOG system is a 5‑tuple

\mathcal{S}=(\mathcal{L},\,\preceq,\,\mathcal{R},\,r,\,C)

where:

 

1. \mathcal{L} is a finite or countably infinite discrete set, whose elements are called lattice points;

2. \preceq is a partial order on \mathcal{L}, termed the hierarchy order;

3. \mathcal{R}\subseteq \mathcal{L}\times\mathcal{L} is a binary relation, called adjacency relation;

4. r:\mathcal{L}\to\mathbb{R}_{>0} is a scale function assigning a positive real scale to each lattice point;

5. C:\mathcal{L}\to\mathbb{N} is a hierarchy‑coefficient function assigning a continued‑fraction coefficient to each lattice point.

 

Definition 2.2 (Compatibility Condition)

The adjacency relation \mathcal{R} is compatible with the hierarchy order \preceq if:

 

1. If x\preceq y, then (x,y)\notin\mathcal{R};

2. If (x,y)\in\mathcal{R}, then x,y lie in the same hierarchy level or adjacent levels;

3. \mathcal{R} is locally finite: every lattice point is adjacent to finitely many other lattice points.

 

Remarks

 

Structural data Description 

  Nesting relation 

  Cross‑coupling relation 

  Scale 

  Continued‑fraction hierarchy 

 

These four components together form the complete structural data of a DOG system.

 

 

 

3 DOG Morphisms

 

Definition 3.1 (DOG Morphism)

Let

\mathcal{S}_1=(\mathcal{L}_1,\preceq_1,\mathcal{R}_1,r_1,C_1),\quad

\mathcal{S}_2=(\mathcal{L}_2,\preceq_2,\mathcal{R}_2,r_2,C_2)

be two DOG systems.

 

A DOG morphism is a single‑valued function

\phi:\mathcal{L}_1\to\mathcal{L}_2

satisfying four conditions:

 

1. Monotonicity: \phi is monotone with respect to the hierarchy order:

x\preceq_1 y\implies\phi(x)\preceq_2\phi(y);

2. Adjacency preservation: If (x,y)\in\mathcal{R}_1, then (\phi(x),\phi(y))\in\mathcal{R}_2;

3. Scale‑ratio preservation: There exists a constant \lambda>0 such that for all x\in\mathcal{L}_1,

r_2(\phi(x))=\lambda\cdot r_1(x);

4. Continued‑fraction‑coefficient preservation: For all x\in\mathcal{L}_1,

C_2(\phi(x))=C_1(x).

 

Remarks

 

- \phi is a single‑valued function;

- Condition 1 is standard order‑preservation, i.e., non‑strict monotonicity;

- Condition 3 permits global scaling, forbidding local scale distortion;

- Condition 4 requires the continued‑fraction hierarchy to be preserved.

 

 

 

4 Order Isomorphism

 

Definition 4.1 (Order Isomorphism)

Two DOG systems \mathcal{S}_1,\mathcal{S}_2 are order‑isomorphic if there exists a DOG morphism

\phi:\mathcal{L}_1\to\mathcal{L}_2

such that:

 

1. \phi is bijective;

2. \phi^{-1} is also a DOG morphism.

 

We write \mathcal{S}_1\cong_{\mathrm{DOG}}\mathcal{S}_2, and call \phi an order isomorphism.

 

Remarks

 

- Since both \phi and \phi^{-1} preserve order, \phi is a strictly monotone bijection;

- Strict monotonicity implies that \phi not only preserves order but also reflects order:

\phi(x)\preceq_2\phi(y)\implies x\preceq_1 y;

- Similarly, \phi both preserves and reflects adjacency.

 

 

 

5 Basic Properties of Order Isomorphism

 

Proposition 5.1 (Reflexivity)

\mathcal{S}\cong_{\mathrm{DOG}}\mathcal{S}.

 

Proposition 5.2 (Symmetry)

If \mathcal{S}_1\cong_{\mathrm{DOG}}\mathcal{S}_2, then \mathcal{S}_2\cong_{\mathrm{DOG}}\mathcal{S}_1.

 

Proposition 5.3 (Transitivity)

If \mathcal{S}_1\cong_{\mathrm{DOG}}\mathcal{S}_2 and \mathcal{S}_2\cong_{\mathrm{DOG}}\mathcal{S}_3, then \mathcal{S}_1\cong_{\mathrm{DOG}}\mathcal{S}_3.

 

Corollary 5.4

Order isomorphism is an equivalence relation on the class of DOG systems.

 

Proof. Follows directly from Propositions 5.1‑5.3. \square

 

 

 

6 Weakened Versions: Homomorphisms and Embeddings

 

The bijectivity requirement is overly restrictive in practical settings, so we introduce weaker notions.

 

Definition 6.1 (Order Homomorphism)

A DOG morphism \phi:\mathcal{L}_1\to\mathcal{L}_2 is called an order homomorphism when bijectivity is not required.

 

Definition 6.2 (Order Embedding)

An order homomorphism \phi is an order embedding if:

 

1. \phi is injective;

2. \phi is strictly monotone: x\prec_1 y\implies\phi(x)\prec_2\phi(y);

3. \phi reflects hierarchy order: if \phi(x)\preceq_2\phi(y), then x\preceq_1 y;

4. \phi reflects adjacency: if (\phi(x),\phi(y))\in\mathcal{R}_2, then (x,y)\in\mathcal{R}_1.

 

Remarks

An embedding preserves the original structure, while allowing the target system to contain extra lattice points.

 

Comparison of three types of morphisms:

 

Type Monotonicity Adjacency Bijectivity 

Order homomorphism Non‑strictly monotone Preserves adjacency No 

Order embedding Strictly monotone + reflects order Preserves and reflects adjacency Injective 

Order isomorphism Strictly monotone Preserves and reflects adjacency Bijective 

 

 

 

7 First DOG Invariant: Rank of Hierarchy

 

Definition 7.1 (Rank of Hierarchy)

Given \mathcal{S}=(\mathcal{L},\preceq,\mathcal{R},r,C), its rank of hierarchy is

\mathrm{rank}(\mathcal{S})=\sup\{n\in\mathbb{N}:\exists\,x_0\prec x_1\prec\cdots\prec x_n\}.

i.e., the maximum length of a chain under the partial order \preceq.

 

Proposition 7.2

Order isomorphism preserves rank of hierarchy:

\mathcal{S}_1\cong_{\mathrm{DOG}}\mathcal{S}_2\implies \mathrm{rank}(\mathcal{S}_1)=\mathrm{rank}(\mathcal{S}_2).

Proof. Order‑preserving bijections preserve chain lengths. \square

Remarks
Rank of hierarchy is the first DOG invariant, yet it is coarse‑grained. Stronger DOG invariants are to be constructed in future work.

 

8 Decision Problem for Order Isomorphism

Problem 8.1
Given two finite DOG systems \mathcal{S}_1,\mathcal{S}_2, determine whether they are order‑isomorphic.

Remarks

- When DOG systems are encoded as labelled graphs, this problem reduces to the graph‑isomorphism problem;
​
- The computational complexity of graph isomorphism remains unsettled;
​
- Hence the complexity of the order‑isomorphism decision problem is at least that of graph isomorphism.

This constitutes a concrete mathematical problem open for investigation.

 

9 Open Problems

1. Stronger DOG invariants: Rank of hierarchy is too coarse; invariants with finer discriminatory power need to be developed.
​
2. Continued‑fraction‑spectrum invariants: What role do the coefficients C play under order isomorphism?
​
3. Continuous limit: Does a DOG system converge to continuous geometry as lattice‑point density tends to infinity, and with respect to which topology does convergence hold?
​
4. Relations with established theories: Clarify precise correspondences between DOG isomorphism and order isomorphism, graph isomorphism, homeomorphism and homotopy.
​
5. Computable examples: Construct non‑trivial hand‑computable examples of order isomorphism.

 

10 Conclusion

This paper establishes the foundational framework for DOG:

- DOG systems are defined via the 5‑tuple (\mathcal{L},\preceq,\mathcal{R},r,C);
​
- DOG morphisms are defined as single‑valued functions satisfying monotonicity, adjacency preservation, scale‑ratio preservation and continued‑fraction‑coefficient preservation;
​
- Order isomorphism is defined as a bijection whose inverse is also a DOG morphism;
​
- Order isomorphism is proven to be an equivalence relation;
​
- A comparison table for three morphism types is provided;
​
- The first invariant, rank of hierarchy, is introduced;
​
- The decision problem for order isomorphism is shown to reduce to graph isomorphism.

These definitions provide the first mathematically tractable interface for DOG. Whether they represent the optimal choice awaits further investigation.

 

References

None

 

Note: This is a draft on system definitions and order isomorphism within the DOG framework. All definitions are exploratory and do not constitute final conclusions. Connections with combinatorial topology, order theory and graph theory constitute a key direction for subsequent research.