Finite lattices and large inductive dimension

Abstract

[EN] 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 field. In particular, the covering dimension, the Krull dimension and the small inductive dimension have been studied extensively for the class of finite lattices. In this paper,we insert a new meaning of dimension for finite lattices called large inductive dimension. We study various of its properties based on minimal covers. Also, given two finite 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 andexamples that strengthen the corresponding results.