# Complete Lattice with unique negation

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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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.