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Which distributive lattices are lattices of closed sets?

Published online by Cambridge University Press:  24 October 2008

S. Papert
S. Papert
Affiliation:
Paris
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Extract

1. An elegant theorem due to Tarski states that a completely distributive complete Boolean algebra is isomorphic with a lattice of sets, and in fact the lattice of all the subsets of some aggregate. The obvious generalization of the question underlying this theorem is to ask whether one can pick out by means of a distributivity condition those lattices (not necessarily Boolean algebras) which are isomorphs of lattices of sets. The answer is no. The real numbers with their natural order form a complete lattice which satisfies the strongest possible distributivity conditions and yet is not iso-morphic with any lattice of sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 2 , April 1959 , pp. 172 - 176
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

REFERENCE

(1)Birkhoff, G.Lattice theory (New York, 1948).Google Scholar