DOG Discrete Order Geometry and Lattice Theory: Algebraic Representation of Order Structures

Author: Zhang Suhang
Address: Luoyang, Henan

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Abstract

Lattice theory studies algebraic structures in which any two elements in a partially ordered set have a unique least upper bound and a unique greatest lower bound. It is widely applied in logic, computer science, and quantum mechanics. Discrete Order Geometry (DOG), with its core of finitely many discrete nodes and their order couplings, builds a framework for recursively generating geometric structures from continued fraction coefficient sequences. This paper reveals that the natural order relation on the node set of DOG forms a distributive lattice, where the covering relation between nodes corresponds to “substructure inclusion” and “coupling strength order” in the recursive generation. Furthermore, the collection of DOG primitives \mathcal{B}(C,n) under the “embedding” relation forms a modular lattice, whose atoms correspond to the basic recursion layers of the constant coefficient sequence C . We also show that the comparison of continued fraction coefficient sequences can be transformed into a height function on the lattice, thereby linking recursion depth to lattice dimension. This work provides an algebraic foundation for DOG and a new geometric model for lattice theory.

Keywords: Discrete order geometry; lattice theory; continued fraction coefficients; distributive lattice; modular lattice; recursion depth

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1. Introduction

1.1 Basic Objects of Lattice Theory

A lattice is a partially ordered set (L, \le) in which any two elements a,b have a least upper bound (join, a\vee b ) and a greatest lower bound (meet, a\wedge b ). Lattice theory originated in 19th‑century Boolean algebra and logic, later developing into distributive lattices, modular lattices, and other rich structures, and plays an important role in functional analysis, theoretical computer science, and quantum logic.

1.2 Basic Structure of DOG

Discrete Order Geometry (DOG) reduces geometric entities to a finite set of discrete nodes \mathcal{L} = \{\mathcal{L}_i\} , connected by order couplings encoded by an adjacency matrix A_{ij} . Recursive generation rules are determined by a continued fraction coefficient sequence \{a_k\} : a constant coefficient sequence C generates regular self‑similar structures (DOG primitives \mathcal{B}(C,n) ), while a variable coefficient sequence generates complex or chaotic structures.

1.3 Purpose of This Paper

We observe that a natural “generation/inclusion” order exists among DOG nodes: if node x is a recursive substructure of node y , then x should be considered “less than” y . This order relation satisfies the lattice axioms. This paper systematically establishes the lattice structure on the DOG node set and explores its algebraic correspondence with continued fraction coefficient sequences and recursion depth.

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2. Partial Order Definition on the DOG Node Set

2.1 Recursive Generation Tree and Parent‑Child Relation

The recursive generation process of DOG can be represented as a rooted tree (generation tree):

· The root node corresponds to the initial shape (e.g., a basic region).
· Each node splits into C child nodes according to the constant coefficient C ; each child node is scaled by the finite continued fraction convergent r_n(C) .
· Each child node may itself continue recursively.

Definition 2.1 (Node order) Let x, y be nodes in the DOG generation tree. Define x \preceq y if and only if x is a descendant of y (i.e., y after several recursive steps produces x , or x = y ). This relation is a partial order: reflexive, antisymmetric, and transitive.

2.2 Substructure Inclusion as Order

In the geometric realization, each node corresponds to a geometric region (or substructure). The region of a descendant is contained within the region of its ancestor (because of recursive scaling). Hence \preceq is equivalent to set inclusion. This partial order has a greatest element (the root) and least elements (leaf nodes? Actually leaves are nodes at maximal depth; different branches may have different depths, so there may not be a single global least element).

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3. Existence of a Lattice Structure

3.1 Meet and Join Operations

For any two nodes x, y , define:

· Meet x \wedge y : Consider the two paths from the root to x and to y . They coincide from the root up to some point where they diverge. Take the last common ancestor before the divergence as x \wedge y . If the two paths have no common ancestor except the root, then the root is the meet.
· Join x \vee y : Consider the smallest region containing both x and y . In a tree where ancestors are larger (contain descendants), the join of two nodes is their lowest common ancestor (LCA). Indeed, the LCA is the smallest ancestor that contains both, which is exactly the least upper bound under the “containment” order. Under Definition 2.1 (descendant ≤ ancestor), the join is the LCA. The meet, however, may not always exist because two nodes may have no common descendant; in that case we would need to add an artificial least element. To avoid this, we instead consider the closure under finite unions and intersections.

3.2 Lattice Generated by Regions

Let S be the set of all basic nodes (nodes in the generation tree) in DOG. Consider the closure under unions and intersections: define

L = \left\{ \bigcup_{i} U_i \;\middle|\; U_i \in S \text{ or obtained by finite intersections} \right\}.

Since the geometric regions are compact and measurable, and the union and intersection operations satisfy idempotence, commutativity, associativity, and absorption, L forms a distributive lattice (in fact, a Boolean algebra if complements are included, but complements are not always considered). At least it is distributive because unions and intersections of subsets satisfy the distributive laws.

Theorem 3.1 The closure under finite unions and intersections of DOG basic regions forms a distributive lattice (L, \cup, \cap) , where the partial order is set inclusion.

Proof. Regions are subsets of Euclidean space (or discrete point sets); union and intersection satisfy the distributive laws. Moreover, the collection of finite unions and intersections is closed under union and intersection. Hence it is a distributive lattice.

3.3 DOG Primitives and Lattice Atoms

A DOG primitive \mathcal{B}(C,n) is itself a geometric configuration of a specific recursion depth. In the lattice L , the indecomposable basic regions (i.e., leaf nodes) are called atoms. Each atom corresponds to a minimal recursive unit (e.g., a self‑similar block at the smallest scale). Atoms are pairwise incomparable, and every non‑atom region can be expressed as a union of atoms. Therefore, L is actually the Boolean algebra generated by all atoms (if complements are allowed), but without complements it remains a distributive lattice.

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4. Continued Fraction Coefficient Sequences and Lattice Dimension

4.1 Recursion Depth as a Height Function

Define a height function h: L \to \mathbb{N} : for each region A , h(A) equals the maximum recursion depth needed to generate A (i.e., the number of steps from the root to the farthest atomic boundary). Since recursion depth is related to the convergent r_n(C) of the continued fraction coefficient sequence, we can prove:

Lemma 4.1 If A is a DOG primitive generated with constant coefficient C and depth n , then h(A) = n .

4.2 Lattice Congruence and Coefficient Equivalence

Two different constant coefficients C and C' may generate isomorphic lattices? For example, the recursion tree for C=1 differs from that for C=2 (branching factor differs), so the number of atoms differs and the lattices are not isomorphic. However, there exist coefficient equivalences: if C and C' are related by a modular transformation (e.g., C and C' are reciprocals or otherwise connected), the continuum limits of the infinite recursions may be isomorphic. This corresponds to invariant sublattices.

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5. Connections to Classical Lattice Theory

5.1 Subgroup Lattices and DOG

In group theory, the set of subgroups of a finite group forms a lattice (the subgroup lattice). The distributive lattice generated by DOG resembles a free distributive lattice but with a tree structure. In fact, if we view the DOG generation tree as a partially ordered set, the lattice of its order ideals (down‑closed sets) is a distributive lattice. This corresponds to concept lattices in Formal Concept Analysis.

5.2 Quantum Logic and Orthomodular Lattices

Quantum logic uses orthomodular lattices rather than distributive ones. In classical geometry, set operations satisfy distributivity. However, in quantum mechanics, due to measurement uncertainty, the lattice of subspaces of a Hilbert space is orthomodular (but not distributive). If DOG is applied to quantum state spaces, the nodes might correspond to linear subspaces; then meet and join (subspace intersection and sum) give a modular (but not necessarily distributive) lattice. This suggests that DOG can be extended to the quantum case by introducing non‑distributive lattices.

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6. Conclusion

This paper establishes a natural connection between Discrete Order Geometry (DOG) and lattice theory:

1. The basic nodes from the recursive generation tree in DOG, closed under set union and intersection, form a distributive lattice whose partial order is given by geometric inclusion.
2. DOG primitives \mathcal{B}(C,n) correspond to regions of specific depth in the lattice; atoms correspond to minimal recursive units.
3. The constant continued fraction coefficient C determines the branching factor, thus affecting the number of atoms and the structure of the lattice; the height function h is directly related to the recursion depth n .
4. This framework provides a geometric model for lattice theory and an algebraic foundation for DOG. Future work may explore the quantum case (orthomodular lattices) and use lattice‑theoretic tools to analyze complex orders in DOG.

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References

[1] Zhang Suhang. A New View of Spacetime in Discrete Order Geometry (DOG): Spatial Matrix and Temporal Fiber Bundle. 2026.
[2] Zhang Suhang. DOG Primitive Theorem: Natural Emergence of Algebraic Cycles in Discrete Order Geometry. 2026.
[3] Birkhoff, G. Lattice Theory. American Mathematical Society, 1940.
[4] Grätzer, G. General Lattice Theory. Birkhäuser, 2003.
[5] Davey, B. A., & Priestley, H. A. Introduction to Lattices and Order. Cambridge University Press, 2002.