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Distributivity and decomposability on the lattices satisfying the chain conditions

✍ Scribed by Wenchang Chu

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 159 KB
Volume: 174
Category: Article
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(96)00320-2

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✦ Synopsis

This short note will confirm the converse of Birkhoff's theorem and establish the following: Suppose L is a lattice satisfying the ascending chain condition (resp., the descending chain condition). Then L is distributive if and only if each element of L has a unique finite irredundant decomposition into irreducibles. This conclusion has improved some related results of Dilworth (1961). In addition, some interesting corrollaries are presented.

On the decomposability of lattices satisfying the chain condition, the classical theorem is due to Birkhoff [1,2], and concerns distributive lattices. The following result is its converse theorem.

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