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ARCHIMEDEAN AND BLOCK-FINITE LATTICE EFFECT ALGEBRAS

ARCHIMEDEAN AND BLOCK-FINITE LATTICE EFFECT ALGEBRAS D E M O N S T R A T I O MATHEMATICAVol. XXXIIINo 32000Zdenka RiecanováARCHIMEDEAN A N D BLOCK-FINITELATTICE E F F E C T A L G E B R A SA b s t r a c t . We show t h a t every complete effect algebra is Archimedean. Moreover, ablock-finite lattice effect algebra has the MacNeille completion which is a complete effectalgebra iff it is Archimedean. We apply our results to orthomodular lattices.1. Basic definitionsEffect algebras (introduced by Foulis D.J. and Bennett M.K. in [7], 1994)are important for modelling unsharp measurements in Hilbert space: Theset of all effects is the set of all self-adjoint operators T on a Hilbert spaceH with 0 < T < 1. In a general algebraic form an effect algebra is definedas follows:1.1. A structure ( £ ; © , 0 , 1 ) is called an effect-algebraif 0, 1are two distinguished elements and © is a partially defined binary operationon P which satisfies the following conditions for any a, b, c £ E:DEFINITION(Ei) 6ffio = a f f i 6 i f a © 6 i s defined,(Eii) (a © b) © c = a © (b © http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ARCHIMEDEAN AND BLOCK-FINITE LATTICE EFFECT ALGEBRAS

Demonstratio Mathematica , Volume 33 (3): 10 – Jul 1, 2000

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References (6)

Publisher
de Gruyter
Copyright
© by Zdenka Riecanová
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2000-0302
Publisher site
See Article on Publisher Site

Abstract

D E M O N S T R A T I O MATHEMATICAVol. XXXIIINo 32000Zdenka RiecanováARCHIMEDEAN A N D BLOCK-FINITELATTICE E F F E C T A L G E B R A SA b s t r a c t . We show t h a t every complete effect algebra is Archimedean. Moreover, ablock-finite lattice effect algebra has the MacNeille completion which is a complete effectalgebra iff it is Archimedean. We apply our results to orthomodular lattices.1. Basic definitionsEffect algebras (introduced by Foulis D.J. and Bennett M.K. in [7], 1994)are important for modelling unsharp measurements in Hilbert space: Theset of all effects is the set of all self-adjoint operators T on a Hilbert spaceH with 0 < T < 1. In a general algebraic form an effect algebra is definedas follows:1.1. A structure ( £ ; © , 0 , 1 ) is called an effect-algebraif 0, 1are two distinguished elements and © is a partially defined binary operationon P which satisfies the following conditions for any a, b, c £ E:DEFINITION(Ei) 6ffio = a f f i 6 i f a © 6 i s defined,(Eii) (a © b) © c = a © (b ©

Journal

Demonstratio Mathematicade Gruyter

Published: Jul 1, 2000

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