Department of Mathematics,
University of Manitoba,
Winnipeg, Manitoba R3T 2N2,
Mathematics and Computer Science Division,
Argonne National Laboratory,
Argonne, Illinois 60439-4844,
Department of Computer Science,
University of New Mexico,
Albuquerque, New Mexico 87131,
Lattice Laws Forcing Distributivity Under Unique Complementation.
We give several new lattice identities valid in
nonmodular lattices such that a uniquely complemented lattice
satisfying any of these identities is necessarily Boolean. Since
some of these identities are consequences of modularity as well,
these results generalize the classical result of Birkhoff and von
Neumann that every uniquely complemented modular lattice is Boolean.
In particular, every uniquely complemented lattice in
M ∨ V(N5), the least nonmodular variety, is Boolean.