File:Birkhoff representation theorem.gif

Summary

Description 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.
Date
Source 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
Author Jochen Burghardt
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Category:Hasse diagrams Category:Files by User:Jochen Burghardt Category:Lattice theory Category:Garrett Birkhoff Category:Graphs with 18 vertices
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