Medians are below joins in semimodular lattices of breadth 2

• Published in: arXiv.org
• Main Authors: Czédli, Gábor, Powers, Robert C, White, Jeremy M
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 05.11.2019
• Subjects: Lattice modules; Lattices; Theorems
• ISSN: 2331-8422

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 \(3\).