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Suppose i have a complete lattice (meet and join exist for any two subsets of the lattice) and for every element of the lattice, there is a unique element that is its complement (hence a Boolean negation). If such a lattice is also modular, then it is Boolean. Negation is a strong requirement.. If i don't know it is modular, then what kind of lattice is this? What if the lattice is also atomic?

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up vote 2 down vote accepted

There are lattices with unique complements that are not Boolean: In his article "Lattices with unique complements", R.P. Dilworth proves that every lattice can be embedded into a lattice with unique complements. So take any non-modular lattice (e.g. $N_5$). This may be embdedded into a lattice $L$ with unique complements. Then $L$ cannot be modular since it contains a non-modular sublattice.

In this book review of "Lattices with Unique Complements" by V. N. Salii the author mentions at the end that the question whether there exist complete non-distributive lattices with unique complements is still unsolved (by 1990).

However it is true that every atomic complete lattice with unique complements is Boolean (and isomorphic to the power set lattice of its atoms). For a proof, see Theorem 18 in §6 of G. Birkhoff's book "Lattice theory".

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Thank you very much for your informative reply. That i could not find anything about it, had to mean it was either a silly or a very difficult question. ( Now i know!) I downloaded your references and will look at them . Thanks once more! – Joseph Aug 24 '12 at 12:44

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