Medians are Below Joins in Semimodular Lattices of Breadth 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 ξ if the sum d(y,x1) + ⋯ + d(y,xk) is minimal. The lattice L satisfies the c1-median property if, for any xi =(x_{1},dots ,x_{k})in L^{k} and for any median y of ξ, yleq 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 c1-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.