Geodesic
Interdependence between carathйodory numbers and $n$-distributivity in lattices
Matematičeskie zametki, Tome 52 (1992) no. 3, pp. 44-47.

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For a lattice $L$ with zero a subset $F\subseteq L$ is called a (lower) spanning tree if for any y $y\in L/\{0\}$ there exists $x\in F$ such that $0$. The main goal of the present note is a proof of two theorems, one of which is the following: THEOREM 1. Suppose that the spanning tree of an algebraic lattice $L$ consists of completely join-irreducible elements and that each element $x\in L$ is the union of some subset (in general, infinite) of $F$. Then the Caratheodory number of $L$ relative to the spanning tree $F$ is equal to the distributivity number of this lattice. The second theorem states the same result as the first, though under different conditions on the lattice $L$ and the spanning tree $F$. 
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     author = {A. P. Zolotarev},
     title = {Interdependence between carath{\cyrishrt}odory numbers and $n$-distributivity in lattices},
     journal = {Matemati\v{c}eskie zametki},
     pages = {44--47},
     publisher = {mathdoc},
     volume = {52},
     number = {3},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1992_52_3_a4/}
}
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A. P. Zolotarev. Interdependence between carathйodory numbers and $n$-distributivity in lattices. Matematičeskie zametki, Tome 52 (1992) no. 3, pp. 44-47. http://geodesic.mathdoc.fr/item/MZM_1992_52_3_a4/