Axis-Aligned Rule Regions

• Axis-Aligned Rule Regions are geometric constructs describing how axis-parallel boxes intersect to form combinatorial codes.
• They are characterized by the product theorem, which constructs high-dimensional box codes from 1D interval codes and determines box embedding dimensions.
• These structures have practical applications in computational geometry, neural code realization, and data filtering in multidimensional systems.

Axis-aligned rule regions are geometric constructs central to the study of combinatorial codes arising from collections of axis-parallel boxes in Euclidean space. These structures capture the set-theoretic intersections determined by the placement of such boxes and form the foundation for distinguishing a key class of convex codes—those that can be realized as unions of axis-aligned box regions. The subject intersects discrete geometry, data structures for computational filtering problems, and the theory of neural code realization.

1. Combinatorial Codes and Axis-Aligned Boxes

Let $U = \{U_1, \ldots, U_n\}$ be an ordered collection of subsets of $\mathbb{R}^d$. The combinatorial code of $U$ is defined as

$$\operatorname{code}(U) = \{\sigma \subseteq [n] \mid \exists\, p \in \mathbb{R}^d \text{ such that } p \in U_i \ \Leftrightarrow\ i \in \sigma\}.$$

For axis-aligned boxes—Cartesian products $[a_1, b_1] \times \cdots \times [a_d, b_d]$—this code succinctly specifies the combinatorial overlap pattern produced by the boxes. Given a collection $B = \{B_1, \ldots, B_n\}$ of such boxes, the code $\operatorname{code}(B)$ aggregates all index sets realized somewhere in $\mathbb{R}^d$ via region intersections.

A code $C$ is box-convex (alternatively, a box code) in dimension $d$ if there exists a realization by $d$-dimensional axis-parallel boxes, with the minimal such $d$ termed the box embedding dimension, $\operatorname{bdim}(C)$ (Benitez et al., 2022).

2. The Intersection-Product and Product Theorem

A central construction for box codes is the intersection-product of codes. If $C, D \subseteq 2^{[n]}$ are codes, their intersection-product is

$$C \wedge D = \{c_1 \cap c_2 \mid c_1 \in C,\, c_2 \in D\}.$$

The Product Theorem articulates the key structure:

Product Theorem. Let $U = \{U_1, \ldots, U_n\} \subset \mathbb{R}^{d_1}$ and $V = \{V_1, \ldots, V_n\} \subset \mathbb{R}^{d_2}$, and define $$W_i = U_i \times V_i \subset \mathbb{R}^{d_1 + d_2}$$. Then

$$\operatorname{code}(W) = \operatorname{code}(U) \wedge \operatorname{code}(V).$$

This theorem enables the explicit construction of higher-dimensional box codes from 1-dimensional interval codes by repeated intersection-product operations, establishing a correspondence between the complexity of a box code and its factorizability into interval codes (Benitez et al., 2022).

3. Characterization of Box-Convex Codes and Forbidden Configurations

A code is called an interval code if it admits realization by closed intervals in $\mathbb{R}^1$. The corollary to the Product Theorem asserts:

A code $C$ is box-convex in $\mathbb{R}^d$ if and only if $C = C^{(1)} \wedge \cdots \wedge C^{(d)}$ for some collection of interval codes $C^{(j)}$.

The minimal $d$ required in such a decomposition is the box embedding dimension $\operatorname{bdim}(C)$. Codes that do not admit such a factorization are called irreducible non-interval codes and cannot be box-convex in any dimension. Explicit examples on three elements include $C_1 = \{123, 12, 13, 23, \varnothing\}$, which is convex but not box-convex (Benitez et al., 2022).

4. Comparison with General Convex Codes

The inclusion structure among codes is as follows:

• All box-convex codes are closed-convex codes, i.e., realizable by unions of closed convex sets.
• Every open-convex code is closed-convex, but not vice versa.
• The reverse containments fail in general; some convex codes (including open-convex ones) are not box-convex, and there exist closed-convex codes with no open-convex realization.

Concrete evidence is provided by codes like $C_1$, which is open-convex and closed-convex in $\mathbb{R}^2$ but not box-convex in any dimension. This demonstrates box codes form a strict subclass of convex codes, representing a genuine refinement in the combinatorial-geometric hierarchy (Benitez et al., 2022).

5. Structural Properties and Weak Monotonicity

Define $\Delta(C)$ as the set of all faces of maximal codewords of $C$. For any closed-convex code $C \subseteq 2^{[n]}$ and downward-closed family $D \subseteq \Delta(C)\setminus C$, it holds that $C \cup D$ is closed-convex and

$$\operatorname{cdim}(C \cup D) \leq \operatorname{cdim}(C) + 2.$$

This "weak monotonicity" property extends monotonicity results for open-convex codes to closed-convex codes, with an explicit upper bound on convex embedding dimension growth under downward-closed augmentation (Benitez et al., 2022).

6. Illustrative Examples

Example	Construction	Key Properties
Four-rectangle code in $\mathbb{R}^2$	$B_1 = [0,2]\times[1,3]$, $B_2 = [1,3]\times[0,2]$, $B_3 = [0,2]\times[0,1]$, $B_4 = [2,3]\times[1,2]$	Code contains maximal words $123$ and $24$; factors into two interval codes via the product theorem.
Sunflower code $F_n$ $(n=2k\geq 4)$	$F_n = \{[n], 1,2,\ldots,n, \varnothing\}$	$$\operatorname{odim}(F_n) = \operatorname{cdim}(F_n) = 2$$, but $\operatorname{bdim}(F_n) = k$; box embedding dimension can be arbitrarily large even when convex dimension remains bounded.

These examples demonstrate the expressiveness, factorizability, and subtleties distinguishing box-convex codes from broader convex code classes (Benitez et al., 2022).

7. Impact and Applications

The structural classification of axis-aligned rule regions has far-reaching implications, especially for the efficient representation and querying of set intersection patterns in computational geometry and theoretical neuroscience. In packet filtering and routing, ranges across multiple axes map naturally to axis-aligned rectangles, facilitating efficient data structure design. The factorization of box codes via 1-dimensional interval codes yields both a succinct theoretical characterization and explicit construction techniques for high-dimensional rule-based partitioning (Benitez et al., 2022). A plausible implication is that advances in the understanding of box codes may refine methods for conflict detection, priority resolution, and combinatorial analysis in multidimensional rule systems.