AbstractLet L be a lattice of finite length and let d denote the minimum path length metric on the covering graph of L. For any $\xi =(x_{1},\dots ,x_{k})\in L^{k}$
ξ
=
(
x
1
,
…
,
x
k
)
∈
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 $\xi =(x_{1},\dots ,x_{k})\in L^{k}$
ξ
=
(
x
1
,
…
,
x
k
)
∈
L
k
and for any median y of ξ, $y\leq x_{1}\vee \dots \vee x_{k}$
y
≤
x
1
∨
⋯
∨
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.
Binary trees, fringe thickness and minimum path length
