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Summary
DescriptionBirkhoff representation theorem.gif
Example of Birkhoff's representation theorem for distributive lattices. The join-irreducible elements (nodes) of the distributive example lattice are shadowed, and named a,...,g for reference purposes. They satisfy the relations f < d < a, f < c < a, g < e < b, g < c < b. By Birkhoff's theorem, the set of all down-sets built of these elements, ordered by set inclusion, is isomorphic to the original lattice. For each node, the down-set it corresponds to is shown in blue, with set braces {} omitted for brevity.
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Own work, inspired by Fig.8.3 (p.167) of B.A. Davey and H.A. Priestley (1990) Introduction to Lattices and Order, Cambridge Mathematical Textbooks, Cambridge University Press
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Jochen Burghardt
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19:13, 29 January 2016
423 × 723 (7 KB)
Jochen Burghardt
tried to improve readability: boldface for shadowed nodes, darker colors
17:15, 18 October 2014
423 × 723 (7 KB)
Jochen Burghardt
{{Information |Description=Example of en:Birkhoff's representation theorem for lattices. |Source={{own}}, inspired by Fig.8.3 (p.167) of {{cite book | author=B.A. Davey and H.A. Priestley | title=Introduction to Lattices and Order | publisher=Camb...
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