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Decomposition of Projections on Orthomodular Lattices(1)

Published online by Cambridge University Press:  20 November 2018

G. T. Rüttimann
G. T. Rüttimann*
Affiliation:
University of Calgary, Calgary, Alberta, Canada
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The set of projections in the BAER*-semigroup of hemimorphisms on an orthomodular lattice L can be partially ordered such that the subset of closed projections becomes an orthocomplemented lattice isomorphic to the underlying lattice L. The set of closed projections is identical with the set of Sasaki-projections on L (Foulis [2]). Another interesting class of (in general nonclosed) projections, first investigated by Janowitz [4], are the symmetric closure operators. They map onto orthomodular sublattices where Sasaki-projections map onto segments of the lattice L.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 2 , June 1975 , pp. 263 - 267
Copyright
Copyright © Canadian Mathematical Society 1975

Footnotes

(2)

Present address: Department of Mathematics, University of Massachusetts, Amhurst, MA 01002.

(1)

Work supported by The Canada Council.

References

1. Birkhoff, G., Lattice Theory, American Mathematical Society, 3rd ed. (1967).Google Scholar
2. Foulis, D. J., BAER*-Semigroups, Proceedings of the Amerian Mathematical Society 11, 648-654 (1960).Google Scholar
3. Foulis, D. J., A Note on Orthomodular Lattices, Portugaliae Mathematica 21, 65-72 (1962).Google Scholar
4. Janowitz, M. F., Residuated Closure Operators. Portugaliae Mathematica 26,221-252 (1967).Google Scholar
5. Nakamura, M., The Permutability in a Certain Orthocomplemented Lattice, Kodai Math. Sem. Rep. 9, 158-160 (1957).Google Scholar
6. Sasaki, U., On Orthocomplemented Lattices Satisfying the Exchange Axiom. Hiroshima Japan University Journal of Science Ser. A17, 293-302 (1954).Google Scholar