Year: 2024
Type: preprint
Abstract: We consider the convex subset $[A,B]$ of all elements between two levels $A$ and $B$ of a finite distributive lattice, as a union of (or covered by) intervals $[a,b]$. A 1988 result of Voigt and Wegener shows that for such convex subsets of finite Boolean lattices, covers using $\max(|A|,|B|)$ intervals (the minimum possible number) exist. In 1992 Bouchemakh and Engel pointed out that this result holds more generally for finite products of finite chains. In this paper we show that covers of size $\max(|A|,|B|)$ exist for $[A,B]$ when $A$ is the set of atoms and $B$ the set of coatoms of any finite distributive lattice. This is a consequence of a more general result for finite partially ordered sets. We also speculate on the situation when other levels of finite distributive lattices are considered, and prove a couple of theorems supporting these speculations.
Source: arXiv (Cornell University)
Authors: Duffus, Dwight, Sands, Bill
Institutions: Emory University, University of Calgary
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4391376293