Lattices with unique complementation

Lattices with Unique Complements

Introduction. For'several years one of the outstanding problems of lattice theory has been the following: Is every lattice with unique complements a Boolean algebra? Any number of weak additional restrictions are sufficient for an affirmative answer. For example, if a lattice is modular (G. Bergman [l](1)) or ortho-complemented (G. Birkhoff [l]) or atomic (G. Birkhoff and M. Ward [l]), then uni...

Complementation in normal subgroup lattices

The goal of this survey is to study some classes of groups determined by different types of complementation of their normal subgroup lattices. The groups whose normal subgroup lattices are Stone lattices, boolean lattices or ortholattices will be also investigated. MSC (2000): Primary 20D30, Secondary 20E15, 06C15, 06D15.

Atomic Lattices with Unique Comparable Complements

Lattice Laws Forcing Distributivity Under Unique Complementation

In this paper, 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 part...

Compactly Generated Lattices with a Unique Essential Element

Lattices with a unique essential element are investigated. It is shown that, a compactly generated lattice with a unique essential element different from the top element, and with finite Goldie dimension has finite length. Mathematics Subject Classification: Primary 06C15, secondary 16P20