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Order-topological separable complete modular ortholattices admit order continuous faithful valuations
Author(s):
Zdenka
Riecanová
Journal:
Proc. Amer. Math. Soc.
126
(1998),
231-237.
MSC (1991):
Primary 03G12, 06C15, 06F30
MathSciNet review:
1415337
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Abstract:
We prove that on every separable complete atomic modular ortholattice (i.e.order topological) there exists an order continuous faithful valuation. We also give a construction of the existing order continuous faithful valuation. For separable atomic modular ortholattices we give a necessary and sufficient condition to admit an order continuous faithful valuation and we show that it is equivalent with the condition to have a modular MacNeille completion. We improve one statement on complete metric lattices from Birkhoff's Lattice Theory.
References:
- [1]
- D. H. Adams, The completions by cuts of an orthocomplemented modular lattice, Bull. Austral. Math. Soc. 1 (1969), 279-280. MR 40:7162; MR 41:3339
- [2]
- G. Birkhoff, Lattice Theory, 3rd edition, Amer. Math. Soc., Providence, Rhode Island, 1967. MR 37:2638
- [3]
- G. Bruns, Free ortholattices, Canadian J. Math. 28 (1976), 977-985. MR 54:7335
- [4]
- G. Bruns, R. Greechie, J. Harding, and M. Roddy, Completions of orthomodular lattices, Order 7 (1990), 67-76. MR 92b:06029
- [5]
- M. Erné and Z. Rie\v{c}anová, Order topological complete orthomodular lattices, Topology and its Applications 61 (1995), 215-227. MR 95k:06019
- [6]
- J. Harding, Orthomodular lattice whose MacNeille completions are not orthomodular, Order 8 (1991), 93-103. MR 92k:06015
- [7]
- J. Harding, Completions of orthomodular lattices II, Order 10 (1993), 283-294. MR 95c:06018
- [8]
- M. F. Janowitz, Indexed orthomodular lattices, Math. Z. 119 (1971), 28-32. MR 45:127
- [9]
- G. Kalmbach, Orthomodular Lattices, Academic Press, London, 1983. MR 85f:06012
- [10]
- G. Kalmbach, Measures and Hilbert lattices, World Scientific, Singapore, 1986. MR 88a:06013
- [11]
- H. Kirchheimová, Some remarks on
 -convergence, Proc. of the First Winter School on Measure Theory, Liptovský Ján (1990), SAV, Bratislava, Slovakia, pp. 110-113. MR 92d:06008 - [12]
- I. Kaplansky, Any orthocomplemented complete modular lattice is a continuous geometry, Annals of Mathematics 61 no. 3 (1955), 524-541. MR 19:524f
- [13]
- F. Maeda, Kontinuierliche Geometrien, Springer-Verlag, Berlin, 1958. MR 19:833c
- [14]
- V. Palko, Embeddings of orthoposets into orthocomplete posets, Tatra Mountains Math. Publ. 3 (1993), 7-12. MR 95h:06002
- [15]
- Z. Rie\v{c}anová, Applications of topological methods to the completions of atomic orthomodular lattices, Demonstratio Math. 24 (1991), 331-341. MR 92m:06020
- [16]
- Z. Rie\v{c}anová, Measures and topologies on atomic quantum logics, Mathematical Research, Vol. 66, Topology, Measures and Fractals, Akademie Verlag, 1992, pp. 154-161. MR 94g:06015
- [17]
- Z. Rie\v{c}anová, Topological and order-topological orthomodular lattices, Bull. Austral. Math. Soc. 47 (1993), 509-518. MR 93k:06017
- [18]
- Z. Rie\v{c}anová and S. Pulmannová, Logics with separating sets of measures, Math. Slovaca 41 no. 2, (1991), 167-177. MR 92d:06025
- [19]
- T. A. Sarymsakov, S. A. Ajupov, D. Chadzhijev, and V. J. Chilin, Ordered algebras, FAB, Tashkent, 1983. (in Russian) MR 86h:46004
- [20]
- J. Schmidt, Zur Kennzeichnung der Dedekind-Mac Neilleschen Hulle einer Geordneten Menge, Archiv d. Math. 7 (1956), 241-249. MR 18:868a
- [21]
- J. Von Neumann, Continuous geometry, Proc. Nat. Acad. Sci. 22 (1936), 92-100.
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Additional Information:
Zdenka
Riecanová
Affiliation:
Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak Technical University, Ilkovicova 3, 812 19 Bratislava, Slovak Republic
Email:
zriecan@elf.stuba.sk
DOI:
10.1090/S0002-9939-98-04072-6
PII:
S 0002-9939(98)04072-6
Keywords:
Order convergence,
order topology,
order-topological,
modular ortholattice,
valuation,
strongly compactly atomistic,
MacNeille completion
Received by editor(s):
March 12, 1996
Received by editor(s) in revised form:
June 26, 1996
Communicated by:
Lance W. Small
Copyright of article:
Copyright
1998,
American Mathematical Society
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