# The best extending cover-preserving geometric lattices of semimodular lattices

## TL;DR

This paper introduces an algorithm to identify optimal cover-preserving geometric lattices for finite semimodular lattices, characterizing their size and atom count to understand minimal embeddings.

## Contribution

It proposes a method to compute all minimal cover-preserving geometric lattices for any finite semimodular lattice, establishing key properties of these embeddings.

## Key findings

- The length of the geometric lattice equals that of the semimodular lattice.
- The number of atoms in the geometric lattice equals the number of non-zero join-irreducible elements.
- The algorithm effectively finds all best extending cover-preserving geometric lattices.

## Abstract

In 2010, G\'{a}bor Cz\'{e}dli and E. Tam\'{a}s Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices, Advances in Mathematics 225 (2010) 2455-2463]. That is to say: What are the geometric lattices $G$ such that a given finite semimodular lattice $L$ has a cover-preserving embedding into $G$ with the smallest $|G|$? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices $G$ of a given semimodular lattice $L$ and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice $G$ equal the length of $L$ and the number of non-zero join-irreducible elements of $L$, respectively. Therefore, we comprehend the best cover-preserving embedding of a given semimodular lattice.