Cyclic atoms in orthomodular lattices

Author:
Donald E. Catlin

Journal:
Proc. Amer. Math. Soc. **30** (1971), 412-418

MSC:
Primary 06.40

DOI:
https://doi.org/10.1090/S0002-9939-1971-0285457-3

MathSciNet review:
0285457

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the projection lattice of a separable Hilbert space *H*. For each , let denote the projection onto the one dimensional subspace generated by *x*. If *B* is a Boolean sublattice of , then it is a theorem that whenever *B* is maximal in there exists a vector , called a *cyclic vector for B*, such that the join in of all the as *Q* ranges through *B* is the identity operator *I*. In this paper we show that this theorem is an immediate corollary of a more general theorem in orthomodular lattice theory. In addition, a final theorem in the paper makes clear the necessity for the separability assumption on *H*.

**[1]**Donald E. Catlin,*Irreducibility conditions on orthomodular lattices*, J. Natur. Sci. and Math.**8**(1968), 81–87. MR**0233743****[2]**David J. Foulis,*A note on orthomodular lattices*, Portugal. Math.**21**(1962), 65–72. MR**0148581****[3]**R. J. Greechie,*On the structure of orthomodular lattices satisfying the chain condition*, J. Combinatorial Theory**4**(1968), 210–218. MR**0227056****[4]**M. F. Janowitz,*Quantifiers and orthomodular lattices*, Pacific J. Math.**13**(1963), 1241–1249. MR**0156800****[5]**M. F. Janowitz,*Residuated closure operators*, Portugal. Math.**26**(1967), 221–252. MR**0249331****[6]**G. W. Mackey,*The mathematical foundations of quantum mechanics*, Benjamin, New York, 1963, pp. 86-87. MR**27**#5501.**[7]**Otton Martin Nikodým,*The mathematical apparatus for quantum-theories, based on the theory of Boolean lattices*, Die Grundlehren der mathematischen Wissenschaften, Band 129, Springer-Verlag New York, Inc., New York, 1966. MR**0224338**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0285457-3

Keywords:
Separable Hilbert space,
cyclic vector,
orthomodular lattice,
Sasaki projection,
atomic lattice,
complete lattice,
atomic bisection property,
atomic projection property,
block,
cyclic atom,
hyperoctant property

Article copyright:
© Copyright 1971
American Mathematical Society