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ON COMPLETIONS OF ORTHOPOSETS

ON COMPLETIONS OF ORTHOPOSETS DEMONSTRATIO MATHEMATICAVol. XXVIINo 3-41994Zdenka RiecanováO N C O M P L E T I O N S OF O R T H O P O S E T SDedicated to Professor TadeuszTraczykIntroductionThe well known fact is that every poset Ρ can be embedded into complete lattice called the MacNeille completion or completion by cuts (abbreviated MC(P)). This completion has a misbehaviour with respect toorthomodular posets since MC(P) of an orthomodular poset is not alwaysorthomodular. V. Palko [6] introduced the MacNeille orthocompletion (abbreviated MOC(P)) of an orthoposet Ρ and showed that in some caseswhen MC(P) is not orthomodular MOC(P) can still be an orthomodularposet. We show that if MC(P) of an orthoposet Ρ is orthomodular thenMOC(P) is isomorphic to MC(P); but the isomorphism of MOC(P) withMC(P) does not imply the orthomodularity of MC(P). Moreover MOC(P)(resp. σ-orthocompletion ΜΟ σ Ο(Ρ)) need not be orthomodular even if Ρis an orthomodular lattice. Furthermore we give some necessary and sufficient conditions for orthoposet Ρ to have MOC(P), respectively MC(P)orthomodular. Some examples are shown.1. PreliminariesRecall that an orthoposet Ρ is a poset with zero and unit elements suchthat there exists a mapping 1 ; Ρ —• Ρ having the following properties forall χ, y £ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON COMPLETIONS OF ORTHOPOSETS

Riecanová, Zdenka
Demonstratio Mathematica , Volume 27 (3-4): 8 – Jul 1, 1994
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Publisher
de Gruyter
Copyright
© by Zdenka Riecanová
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1994-3-406
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXVIINo 3-41994Zdenka RiecanováO N C O M P L E T I O N S OF O R T H O P O S E T SDedicated to Professor TadeuszTraczykIntroductionThe well known fact is that every poset Ρ can be embedded into complete lattice called the MacNeille completion or completion by cuts (abbreviated MC(P)). This completion has a misbehaviour with respect toorthomodular posets since MC(P) of an orthomodular poset is not alwaysorthomodular. V. Palko [6] introduced the MacNeille orthocompletion (abbreviated MOC(P)) of an orthoposet Ρ and showed that in some caseswhen MC(P) is not orthomodular MOC(P) can still be an orthomodularposet. We show that if MC(P) of an orthoposet Ρ is orthomodular thenMOC(P) is isomorphic to MC(P); but the isomorphism of MOC(P) withMC(P) does not imply the orthomodularity of MC(P). Moreover MOC(P)(resp. σ-orthocompletion ΜΟ σ Ο(Ρ)) need not be orthomodular even if Ρis an orthomodular lattice. Furthermore we give some necessary and sufficient conditions for orthoposet Ρ to have MOC(P), respectively MC(P)orthomodular. Some examples are shown.1. PreliminariesRecall that an orthoposet Ρ is a poset with zero and unit elements suchthat there exists a mapping 1 ; Ρ —• Ρ having the following properties forall χ, y £

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 1994

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