Medians are Below Joins in Semimodular Lattices of Breadth 2

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), ... , x(k)) is an element of L-k, an element y belonging to L is called a median of xi if the sum d(y, x(1)) + center dot center dot center dot + d(y, x(k)) is minimal. The lattice L satisfies the c(1)-median property if, for any xi = (x(1), ... , x(k)) is an element of L-k and for any median y of xi, y <= x(1) boolean OR center dot center dot center dot boolean OR 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.