Group Structures in Discrete Order Geometry

Author: Zhang Suhang
(Luoyang, Henan)

Abstract

Discrete Order Geometry (DOG) takes finite discrete lattice points, nested order and local correlations as its ontology. This paper constructs group structures intrinsically within DOG. The DOG order group, local order group and hierarchical group are defined. It is proved that order automorphisms on the DOG lattice set form a group, order-preserving transformations within local neighbourhoods form subgroups, and the hierarchical structure induces a homomorphism of hierarchical groups. The conclusion is that the DOG group serves as the abstract parent structure for order transformations, and together with the geometric ontology of DOG, they form a two-layer order–symmetry system.

Keywords: Discrete Order Geometry; order group; local group; hierarchical group; order automorphism

1 Introduction

Discrete Order Geometry (DOG) adopts finite discrete lattice sets and nested order as its underlying ontology. This paper establishes group structures inside DOG.

In DOG, groups do not arise from external symmetry assumptions but from transformation rules inherent to the order structure:

- The order relation \preceq among lattice points determines which transformations preserve order;
- Neighbourhood structures determine local transformation groups;
- Hierarchical structures determine hierarchical transformation groups.

The objectives of this paper are:

1. To define the DOG order group;
2. To define the DOG local order group;
3. To define the DOG hierarchical group;
4. To establish the hierarchical relation between groups and the DOG ontology.

2 DOG Order Transformations

2.1 Order Automorphism

Let a DOG local order lattice be
\mathcal{G}=(\mathcal{L},\{N(p)\}_{p\in\mathcal{L}},\preceq).
Define a map
\varphi:\mathcal{L}\to\mathcal{L}.
If \varphi is bijective and for all p,q\in\mathcal{L},
p\preceq q \;\Longleftrightarrow\; \varphi(p)\preceq\varphi(q),
then \varphi is called a pure order automorphism, preserving only the order relation.

2.2 Neighbourhood-Preserving Condition

If the pure order automorphism \varphi additionally satisfies
\varphi(N(p))=N(\varphi(p)),
then \varphi is termed a DOG neighbourhood-preserving automorphism.

2.3 Order Transformations

A DOG order transformation is a bijection preserving both the order relation and the neighbourhood structure:
\varphi\in \operatorname{Aut}(\mathcal{G}).

3 DOG Order Group

3.1 Definition

The DOG order group is defined as
G_{\mathrm{DOG}}=\operatorname{Aut}(\mathcal{G}),
the collection of all DOG neighbourhood-preserving order automorphisms.

3.2 Group Structure Theorem

Theorem 1: G_{\mathrm{DOG}} forms a group under composition of mappings.

Proof:

1. Closure: If \varphi,\psi preserve \preceq and neighbourhoods, then their composition \varphi\circ\psi also preserves them.
2. Associativity: Mapping composition satisfies associativity naturally.
3. Identity element: The identity map \mathrm{id} preserves order and neighbourhoods.
4. Inverse element: If \varphi is bijective and preserves order and neighbourhoods, its inverse map \varphi^{-1} preserves order and neighbourhoods as well.

Thus G_{\mathrm{DOG}} constitutes a group.

3.3 Order Invariants

Quantities invariant under the action of G_{\mathrm{DOG}} are called DOG order invariants, examples include:

- Hierarchical depth of order;
- Neighbourhood size;
- Number of connected order components.

Remark: Such geometric invariants are not equivalent to dynamical conservation laws. No direct application of Noether’s theorem for conservation relations is valid within the discrete framework.

4 DOG Local Order Group

4.1 Definition

For a lattice point p, its local order group is defined by
G_p=\{\varphi\in G_{\mathrm{DOG}}\mid \varphi(p)=p\},
the subgroup of order automorphisms fixing the lattice point p.

4.2 Neighbourhood Restriction

G_p acts on the neighbourhood N(p) and preserves the structure of the local order lattice:
\varphi(N(p))=N(p).

4.3 Subgroup Theorem

Theorem 2: G_p is a subgroup of G_{\mathrm{DOG}}.

Proof: G_p is closed under mapping composition, contains the identity element, and the inverse of any element remains in the set. Hence it is a subgroup.

4.4 Local Order Invariants

Quantities invariant under G_p are local order invariants, for instance:

- Local supremum and infimum;
- Local order dimension;
- Local adjacency pattern.

5 DOG Hierarchical Group

5.1 Hierarchical Structure

Suppose DOG lattice points are stratified by order hierarchy:
\mathcal{L}=\bigcup_{\ell\in\Lambda}\mathcal{L}_\ell,
where \Lambda denotes the index set of hierarchy levels.

5.2 Hierarchy-Preserving Transformations

A hierarchy-preserving transformation satisfies
\varphi(\mathcal{L}_\ell)=\mathcal{L}_\ell,
meaning transformations act only within the same hierarchy and do not mix lattice points across different levels.

5.3 Hierarchical Group

The DOG hierarchical group is defined by
G_\Lambda=\{\varphi\in G_{\mathrm{DOG}}\mid \varphi(\mathcal{L}_\ell)=\mathcal{L}_\ell,\;\forall \ell\in\Lambda\}.

5.4 Hierarchical Homomorphism

There exists a natural homomorphism
\pi:G_{\mathrm{DOG}}\to \operatorname{Perm}(\Lambda).
Every order automorphism induces a permutation on the index set of hierarchy levels. The kernel of this homomorphism is exactly the hierarchical group G_\Lambda, i.e. \ker\pi=G_\Lambda.

6 Hierarchical Relation between Groups and DOG Ontology

6.1 Two-Layer Structure

1. Group layer: G_{\mathrm{DOG}}, abstract rules for order transformations;
2. Lattice layer: \mathcal{G}, geometric carrier of the order structure.

6.2 Group Action on the Lattice

G_{\mathrm{DOG}}\times \mathcal{L}\to \mathcal{L}.
Group elements transform lattice points while preserving the order relation.

6.3 Correspondence without Identification

- A group is a collection of abstract transformation rules;
- A lattice is the carrier upon which the group acts.

They correspond to each other but are not identical.

7 Examples

7.1 One-Dimensional Cyclic Chain

\mathcal{L}=\mathbb{Z}_N,\qquad N(p)=\{p-1,p,p+1\}.
When only translational order transformations are considered, the corresponding subgroup is cyclic: G_{\text{trans}}\cong \mathbb Z_N. If reflection transformations are included, the full order automorphism group is the dihedral group D_N.

7.2 Three-Element Local Lattice

N(p)=\{a,b,c\},\qquad a\preceq c,\qquad b\preceq c.
The local group fixing c swaps a and b:
G_c\cong \mathbb{Z}_2.

7.3 Disconnected DOG Lattice

If \mathcal{L} splits into two disconnected order components, then
G_{\mathrm{DOG}}\cong G_1\times G_2,
the direct product of the order groups for each connected component.

8 Conclusion

This paper constructs group structures intrinsically within DOG:

1. Order automorphisms of DOG form the order group G_{\mathrm{DOG}};
2. Order automorphisms fixing a lattice point form the local order group G_p;
3. Hierarchy-preserving transformations form the hierarchical group G_\Lambda;
4. Groups act on the lattice and preserve the order structure;
5. Groups and lattices constitute a two-layer abstract–carrier system for DOG.

This framework provides a fundamental group-theoretic foundation for the symmetry, conservation quantities and representation theory of DOG.

References

(Omitted)