Skip to main content
SpringerLink
Log in

On Finch’s Conditions for the Completion of Orthomodular Posets

  • Published:
Foundations of Science Aims and scope Submit manuscript

Abstract

In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57–62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Since the notion of a Kleene lattice predates its non-distributive generalisation, the prefix “pseudo” was used in Chajda (2016) and in Giuntini et al. (2017) to distinguish the latter from the former.

References

  • Adams, D. H. (1969). A note on a paper of D. P. Finch. Bulletin of the Australian Mathematical Society, 1, 63–64.

    Google Scholar 

  • Amemiya, I., & Araki, H. (1967). A remark on Piron’s paper. Publications of the Research Institute for Mathematical Sciences, Kyoto University Series A, 2, 423–427.

    Article  Google Scholar 

  • Banaschewski, B. (1956). Hüllensysteme und Erweiterung von Quasi-Ordnungen. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 2, 117–130.

    Article  Google Scholar 

  • Banaschewski, B., & Bruns, G. (1967). Categorical characterization of the MacNeille completion. Archiv der Mathematik, 18, 369–377.

    Article  Google Scholar 

  • Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.

    Article  Google Scholar 

  • Blyth, T. S. (2005). Lattices and ordered algebraic structures. Berlin: Springer.

    Google Scholar 

  • Blyth, T. S., & Janowitz, M. F. (1972). Residuation theory. Oxford: Pergamon Press.

    Google Scholar 

  • Bruns, G., Greechie, R., Harding, J., & Roddy, M. (1990). Completions of orthomodular lattices. Order, 7, 61–76.

  • Bruns, G., & Harding, J. (2000). Algebraic aspects of orthomodular lattices. In B. Coecke, D. Moore, & A. Wilce (Eds). Current research in operational quantum logic. Fundamental theories of physics (Vol. 111). Dordrecht: Springer.

  • Burris, S., & Sankappanavar, H. P. (1981). A course in universal algebra. Berlin: Springer.

    Book  Google Scholar 

  • Chajda, I. (2016). A note on pseudo-Kleene algebras. Acta Universitatis Palackianae Olomucensis, 55(1), 39–45.

    Google Scholar 

  • Chajda, I., Halaš, R., & Kühr, J. (2009). Many-valued quantum algebras. Algebra Universalis, 60, 63–90.

    Article  Google Scholar 

  • Chajda, I., Länger, H., & Paseka, J. (2018). Residuated operators and Dedekind–MacNeille completion. arXiv:1812.09616.

  • Dalla Chiara, M. L., Giuntini, R., & Greechie, R. (2004). Reasoning in quantum theory. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Dvurečenskij, A., & Pulmannová, S. (2000). New trends in quantum structures. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Finch, P. D. (1970a). Orthogonality relations and orthomodularity. Bulletin of the Australian Mathematical Society, 2, 125–128.

    Article  Google Scholar 

  • Finch, P. D. (1970b). On orthomodular posets. Bulletin of the Australian Mathematical Society, 2, 57–62.

    Article  Google Scholar 

  • Foulis, D. J., & Bennett, M. K. (1994). Effect algebras and unsharp quantum logic. Foundations of Physics, 24, 1331–1352.

    Article  Google Scholar 

  • Giuntini, R., & Greuling, H. (1989). Toward a formal language for unsharp properties. Foundations of Physics, 19, 931–945.

    Article  Google Scholar 

  • Giuntini, R., Ledda, A., & Paoli, F. (2017). On some properties of \(\text{ PBZ}^{\ast }\)-lattices. International Journal of Theoretical Physics, 56(12), 3895–3911.

    Article  Google Scholar 

  • Gudder, S. (1974). Inner product spaces. American Mathematical Monthly, 81, 29–36.

    Article  Google Scholar 

  • Gudder, S. (1975a). Correction to inner product spaces. American Mathematical Monthly, 82, 251–252.

    Google Scholar 

  • Gudder, S. (1975b). Second correction to inner product spaces. American Mathematical Monthly, 82, 818.

    Article  Google Scholar 

  • Habil, E. (1994). Orthosummable orthoalgebras. International Journal of Theoretical Physics, 33(10), 1957–1984.

    Article  Google Scholar 

  • Harding, J. (1991). Orthomodular lattices whose MacNeille completions are not orthomodular. Order, 8, 93–103.

    Article  Google Scholar 

  • Harding, J. (1993). Completions of orthomodular lattices II. Order, 10(3), 283–294.

    Article  Google Scholar 

  • Harding, J. (1996). Decompositions in quantum logic. Transactions of the American Mathematical Society, 348(5), 1839–1862.

    Article  Google Scholar 

  • Höhle, U. (1995). Commutative, residuated \(\ell \)-monoids. In U. Höhle & E. P. Klement (Eds.), Non-classical logics and their applications to fuzzy subsets (pp. 53–106). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Janowitz, M. F. (1971). Indexed orthomodular lattices. Mathematische Zeitschrift, 119, 28–32.

    Article  Google Scholar 

  • Kalman, J. A. (1958). Lattices with involution. Transactions of the American Mathematical Society, 87, 485–491.

    Article  Google Scholar 

  • Kalmbach, G. (1983). Orthomodular lattices, London Mathematical Society Monographs (Vol. VIII). London: Academic Press.

    Google Scholar 

  • Kôpka, F., & Chovanec, F. (1997). Boolean D-posets. Tatra Mountains Mathematical Publications, 10, 183–197.

    Google Scholar 

  • MacKey, G. W. (2004). Mathematical foundations of quantum mechanics. New York: Dover Publications.

    Google Scholar 

  • MacLaren, M. D. (1964). Atomic orthocomplemented lattices. Pacific Journal of Mathematics, 14, 597–612.

    Article  Google Scholar 

  • Niederle, J., & Paseka, J. (2016). On realization of effect algebras. Mathematica Slovaca, 66(2), 343–358.

    Article  Google Scholar 

  • Piron, C. (1964). Axiomatique quantique. Helvetica Physica Acta, 37, 439–468.

    Google Scholar 

  • Riecǎnová, Z. (1993). Atomic orthoposets with orthomodular MacNeille completions. Tatra Mountain Mathematical Publications, 2, 203–207.

    Google Scholar 

  • Riečanová, Z. (2000a). Archimedean and block-finite lattice effect algebras. Demonstratio Mathematica, 33, 443–452.

    Article  Google Scholar 

  • Riečanová, Z. (2000b). MacNeille completions of D-posets and effect algebras. International Journal of Theoretical Physics, 39, 859–869.

    Article  Google Scholar 

  • Riečanová, Z. (2001a). Proper effect algebras admitting no states. International Journal of Theoretical Physics, 40, 1683–1691.

    Article  Google Scholar 

  • Riečanová, Z. (2001b). Orthogonal sets in effect algebras. Demonstratio Mathematica, 3, 525–532.

    Google Scholar 

  • Riečanová, Z. (2001c). Sub-effect algebras and Boolean sub-effect algebras. Soft Computing, 5, 400–403.

    Article  Google Scholar 

  • Riečanová, Z., & Zajac, M. (2012). Hilbert space effect-representations of effect algebras. Reports on Physics, 3(70), 283–290.

    Article  Google Scholar 

  • Schmidt, J. (1972a). Universal and internal properties of some extensions of partially ordered sets. Journal Reine Angewandte Mathematik, 253, 28–42.

    Google Scholar 

  • Schmidt, J. (1972b). Universal and internal properties of some completions of \(k\)-join-semilattices and \(k\)-join-distributive partially ordered sets. Journal Reine Angewandte Mathematik, 255, 8–22.

    Google Scholar 

  • Schroeder, B. (2016). Ordered sets (2nd ed.). Berlin: Springer.

    Google Scholar 

Download references

Acknowledgements

Two anonymous referees greatly helped to improve the presentation of this paper; we thank them for their suggestions. The authors gratefully acknowledge the following funding sources: Project “Per un’ estensione semantica della Logica Computazionale Quantistica-Impatto teorico e ricadute implementative”, Regione Autonoma della Sardegna, (RAS: RASSR40341), L.R. 7/2017, annualità 2017-Fondo di Sviluppo e Coesione (FSC) 2014–2020; MIUR, within the Projects PRIN 2017: “Logic and cognition. Theory, experiments, and applications”, CUP: 2013YP4N3, and PRIN 2017: “ Theory and applications of resource sensitive logics”, CUP: 20173WKCM5.

Author information

Authors and Affiliations

  1. Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, via Is Mirrionis 1, 09123, Cagliari, Italy

    D. Fazio, A. Ledda & F. Paoli

Authors
  1. D. Fazio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Ledda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Paoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Fazio.

Ethics declarations

Conflict of interest

The authors declare that they do not have any conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fazio, D., Ledda, A. & Paoli, F. On Finch’s Conditions for the Completion of Orthomodular Posets. Found Sci 28, 419–440 (2023). https://doi.org/10.1007/s10699-020-09702-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10699-020-09702-z

Keywords

Mathematics Subject Classification

Search

Navigation