Medians are below joins in semimodular lattices of breadth 2

TL;DR

This paper proves that in certain finite-length semimodular lattices with breadth at most 2, medians are always below the join of the elements, establishing a specific median property.

Contribution

It establishes the $c_1$-median property for upper semimodular lattices of finite length with breadth ≤ 2, and shows this result is sharp.

Findings

Lattices with breadth ≤ 2 satisfy the $c_1$-median property.

Construction methods for semimodular lattices are provided.

The median property does not hold if breadth exceeds 2.

Abstract

Let $L$ be a lattice of finite length and let $d$ denote the minimum path length metric on the covering graph of $L$. For any $\xi=(x_1,\dots,x_k)\in L^k$, an element $y$ belonging to $L$ is called a median of $\xi$ if the sum $d(y,x_1)+\cdots+d(y,x_k)$ is minimum. The lattice $L$ satisfies the $c_1$-median property if, for any $\xi=(x_1,\dots,x_k)\in L^k$ and for any median $y$ of $\xi$, $y\leq x_1\vee\dots\vee x_k$. Our main theorem asserts that if $L$ is an upper semimodular lattice of finite length and the breadth of $L$ is less than or equal to $2$, then $L$ satisfies the $c_1$-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that $2$ cannot be replaced by $3$.