The realm of nite lattices in combination with a new dimension

TL;DR

The paper introduces the large inductive dimension $Ind$ for finite lattices and develops its theory via minimal covers and pseudocomplements. It defines $Ind$ inductively, proves isomorphism invariance, and shows that every $k\in\mathbb{N}$ is realizable as $Ind(L)$ for some finite lattice, including constructive methods to adjust $Ind$ and analyze sublattices. It then situates $Ind$ relative to existing lattice notions, demonstrating that $Ind$ can differ markedly from $ind$, $ abla$, $Kdim$, and the height, and that gaps between $Ind$ and these dimensions can be arbitrarily large. The paper further analyzes how $Ind$ behaves under linear sums and various products (Cartesian, lexicographic, rectangular), revealing asymmetries, bounds such as $Ind(L_1\times L_2) \le \max\{Ind(L_1),Ind(L_2)\}$, and SP-property-based refinements that bound $Ind$ under more complex constructions. Collectively, these results establish $Ind$ as a distinct and robust dimension notion for finite lattices with intricate interactions under lattice operations, with potential applications to digital imagery and discrete topology modeling.

Abstract

The Ordered Set Theory is a branch of Mathematics that studies partially ordered sets (usually posets) and lattices. The meaning of dimension is one of the main parts of this eld. Dimensions of partially ordered sets and lattices have been studied in various researches. In particular, the covering dimension, the Krull dimension and the small inductive dimension have been studied extensively for the class of nite lattices. In this paper, we insert new meaning of dimension for nite lattices called large inductive dimension and denoted by Ind. We study various of its properties based on minimal covers. Also, given two nite lattices, we study the dimension Ind of their linear sum, Cartesian, lexicographic and rectangular product, investigating the behavior of this dimension. In addition, we study relations of this new dimension with the small inductive dimension, covering dimension and Krull dimension, presenting various facts and examples that strengthen the corresponding results.

Figures (20)

• Figure 1: The finite lattices $(L_1,\leqslant_1)$ and $(L_2,\leqslant _2)$
• Figure 2: The finite lattice $L^{\prime}$
• Figure 3: The lattice $(L,\leqslant)$
• Figure 4: The finite lattice $(L,\leqslant)$
• Figure 5: The finite lattice $(L,\leqslant)$

Theorems & Definitions (43)

• Definition 3.1
• Example 3.2
• Proposition 3.3
• Theorem 3.4
• Theorem 3.5
• Proposition 3.6
• Definition 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4