Orthomodular Lattices

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. Bowever, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programmi ng profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-s.cale order", which are almost impossible to fit into the existing classifica­ tion schemes. They draw upon widely different sections of mathe­ matics.

Contenu

I: Introduction.- II: Elementary Theory of Orthomodular Lattices.- 1. Ortholattices.- 2. Commutativity.- 3. Orthomodular lattices.- 4. Properties of commutativity in orthomodular lattices.- 5. Characteristic properties of orthomodular lattices.- 6. Interval algebra.- Exercises.- III: Structure of Orthomodular Lattices.- 1. Skew operations.- 2. Free orthomodular lattice F2.- 3. Introduction to Hilbert spaces.- 4. Projection lattice of a Hilbert space.- Exercises.- IV: Amalgams.- 1. Amalgams of posets.- 2. Amalgams of lattices.- 3. Amalgams of orthomodular lattices.- 4. Atomic amalgams of Boolean algebras.- Exercises.- V: Generalized Orthomodular Lattices.- 1. Orthogonality relation.- 2. Janowitz's embedding.- 3. Congruence relations.- 4. Congruence relations and p-ideals.- 5. Commutators.- Exercises.- VI: Solvability of Generalized Orthomodular Lattices.- 1. Reflective and coreflective congruences.- 2. Projective allelomorph.- 3. Commutator sublattices.- 4. Solvability in equational classes of lattices.- Exercises.- VII: Special Properties of Orthomodularity.- 1. Commutators of n elements.- 2. Finitely generated orthomodular lattices.- 3. Formulas for orthomodular lattices.- 4. Exchange theorems.- 5. Center of an orthomodular lattice.- 6. Identities and operations.- 7. Analogues of Foulis-Holland Theorem.- Exercises.- VIII: Application.- 1. Orthomodularity and experimental propositions.- 2. Compatibility.- 3. Dimension theory.- 4. Orthologics.- Exercises.- Answers to Exercises.- Solutions to Exercises of Chapter II.- Solutions to Exercises of Chapter III.- Solutions to Exercises of Chapter IV.- Solutions to Exercises of Chapter V.- Solutions to Exercises of Chapter VI.- Solutions to Exercises of Chapter VII.- Solutions to Exercises of Chapter VIII.- References.