The class of uniquely complemented lattices properly contains all Boolean lattices. However, no explicit example of a non-Boolean lattice of this class has been found. In addition, the question of whether this class contains any complete non-Boolean lattices remains unanswered. This book focuses on these classical problems of lattice theory and the various attempts to solve them. Requiring no specialized knowledge, the book is directed at researchers and students interested in general algebra and mathematical logic.

Table of Contents

• Ordered sets
• Lattices
• Modular lattices
• Distributive lattices
• Complemented lattices
• Boolean algebras
• Axiomatics of Boolean algebras
• Do there exist nondistributive uniquely complemented lattices?
• There exist nondistributive uniquely complemented lattices
• Complete lattices
• Completion by cuts
• Representations of complete lattices
• Algebraic lattices
• Continuous lattices
• Collectively complemented lattices
• Orthogonal systems and lattice transformations of sets
• Regular elements and decomposition theorems for complete uniquely complemented lattices