Orthomodular Lattices: Algebraic Approach
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.
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Generalized Orthomodular Lattices 1 Orthogonality relation e
Janowitzs embedding e e e
Congruence relations e e
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a a b a a c a s b a v b a v b v c a v c acbi atomic amalgam b a a b a c base set Boolean algebra Boolean lattice c a d c v d closed subspace complete lattice conditions are equivalent congruence relation Corollary defined denote distributive lattice equational class exist elements finite following conditions Hence Hilbert space ideal implies infimum interval isomorphic lattice and let lattice G lattice of Figure least element Lemma Let G Math modular lattice nonvoid subset operations ortho orthocomplemented poset orthogonal ortholattice orthologic orthomodular lattice orthomodular poset p-ideal polynomial Proof prove quotient relatively complemented lattice Remark s a t s v t satisfies skew lattice subalgebra Suppose supremum Theorem Theorem II.4.2 vector x a y x v y