The best extending cover-preserving geometric lattices of semimodular lattices

Abstract

In 2010, Gábor Czédli and E. Tamás 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.

26