Partially additive states on orthomodular posets

Josef Tkadlec

Colloquium Mathematicae (1991)

  • Volume: 62, Issue: 1, page 7-14
  • ISSN: 0010-1354

Abstract

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We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also be viewed as a generalization of [6]. Then we prove an extension theorem for B-states giving, as a by-product, a topological proof of a classical Boolean result.

How to cite

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Tkadlec, Josef. "Partially additive states on orthomodular posets." Colloquium Mathematicae 62.1 (1991): 7-14. <http://eudml.org/doc/210102>.

@article{Tkadlec1991,
abstract = {We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also be viewed as a generalization of [6]. Then we prove an extension theorem for B-states giving, as a by-product, a topological proof of a classical Boolean result.},
author = {Tkadlec, Josef},
journal = {Colloquium Mathematicae},
keywords = {extension of states; orthomodular poset; partially additive state; representation; compact Hausdorff closure space; Stone representation; Boolean algebra},
language = {eng},
number = {1},
pages = {7-14},
title = {Partially additive states on orthomodular posets},
url = {http://eudml.org/doc/210102},
volume = {62},
year = {1991},
}

TY - JOUR
AU - Tkadlec, Josef
TI - Partially additive states on orthomodular posets
JO - Colloquium Mathematicae
PY - 1991
VL - 62
IS - 1
SP - 7
EP - 14
AB - We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also be viewed as a generalization of [6]. Then we prove an extension theorem for B-states giving, as a by-product, a topological proof of a classical Boolean result.
LA - eng
KW - extension of states; orthomodular poset; partially additive state; representation; compact Hausdorff closure space; Stone representation; Boolean algebra
UR - http://eudml.org/doc/210102
ER -

References

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  1. [1] J. Binder and P. Pták, A representation of orthomodular lattices, Acta Univ. Carolin.--Math. Phys. 31 (1990), 21-26. Zbl0778.06010
  2. [2] E. Čech, Topological Spaces, Publ. House of the Czechoslovak Academy of Sciences, Prague, and Interscience, London 1966. 
  3. [3] R. J. Greechie, Orthomodular lattices admitting no states, J. Combin. Theory Ser. A 10 (1971), 119-132. Zbl0219.06007
  4. [4] S. P. Gudder, Stochastic Methods in Quantum Mechanics, North-Holland, New York 1979. Zbl0439.46047
  5. [5] A. Horn and A. Tarski, Measures in Boolean algebras, Trans. Amer. Math. Soc. 64 (1948), 467-497. Zbl0035.03001
  6. [6] L. Iturrioz, A representation theory for orthomodular lattices by means of closure spaces, Acta Math. Hungar. 47 (1986), 145-151. Zbl0608.06008
  7. [7] G. Kalmbach, Orthomodular Lattices, Academic Press, London 1983. 
  8. [8] M. J. Mączyński, Probability measures on a Boolean algebra, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 849-852. Zbl0249.60002
  9. [9] R. Mayet, Une dualité pour les ensembles ordonnés orthocomplémentés, C. R. Acad. Sci. Paris Sér. I 294 (1982), 63-65. Zbl0484.06002
  10. [10] R. Mayet, Varieties of orthomodular lattices related to states, Algebra Universalis 20 (1985), 368-396. Zbl0581.06006
  11. [11] P. Pták, Extensions of states on logics, Bull. Polish Acad. Sci. Math. 33 (1985), 493-497. Zbl0589.03040
  12. [12] P. Pták, Weak dispersion-free states and the hidden variables hypothesis, J. Math. Phys. 24 (1983), 839-840. Zbl0508.60006
  13. [13] N. Zierler and M. Schlessinger, Boolean embeddings of orthomodular sets and quantum logic, Duke Math. J. 32 (1965), 251-262. Zbl0171.25403

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