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 Mat. Zametki, 1992, Volume 52, Issue 3, Pages 44–47 (Mi mz4699)

Interdependence between carathéodory numbers and $n$-distributivity in lattices

A. P. Zolotarev

Moscow Institute of Engineering and Physics, Second Division

Abstract: 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 $0<x\leqslant y$. 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 $F$.

Full text: PDF file (857 kB)

English version:
Mathematical Notes, 1992, 52:3, 903–906

Bibliographic databases:

UDC: 512.56

Citation: A. P. Zolotarev, “Interdependence between carathéodory numbers and $n$-distributivity in lattices”, Mat. Zametki, 52:3 (1992), 44–47; Math. Notes, 52:3 (1992), 903–906

Citation in format AMSBIB
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