Book Review: Lattices with Unique Complements

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...

Atomic Lattices with Unique Comparable Complements

Reductions and Algorithms for Lattices with Complements

We present tractable algorithms for deciding the universal theories of lattices with different notions of complementation by reduction to more standard procedures. The reductions use basic structural properties of lattices. For Boolean algebras, generalised Boolean algebras and atomic distributive lattices we reduce to SAT-procedures. For Brouwerian algebras we reduce to the uniform word proble...

Book Review

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Formality of the Complements of Subspace Arrangements with Geometric Lattices

We show that, for an arrangement of subspaces in a complex vector space with geometric intersection lattice, the complement of the arrangement is formal. We prove that the Morgan rational model for such an arrangement complement is formal as a differential graded algebra.