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, \(y\leq x_{1}\vee \dots \vee x_{k}\). 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 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.
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Czédli, G., Powers, R.C. & White, J.M. Medians are Below Joins in Semimodular Lattices of Breadth 2. Order 38, 351–363 (2021). https://doi.org/10.1007/s11083-020-09544-1
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DOI: https://doi.org/10.1007/s11083-020-09544-1