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arXiv · 2503.22007

The realm of nite lattices in combination with a new dimension

D. Georgiou·Y. Hattori·A. Megaritis·F. Sereti
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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.

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D. Georgiou, Y. Hattori, A. Megaritis, F. Sereti. 2025-03-27. The realm of nite lattices in combination with a new dimension. https://arxiv.org/abs/2503.22007

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