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Completions of orthomodular lattices II

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Abstract

If\(\mathcal{K}\) is a variety of orthomodular lattices generated by a set of orthomodular lattices having a finite uniform upper bound om the length of their chains, then the MacNeille completion of every algebra in\(\mathcal{K}\) again belongs to\(\mathcal{K}\).

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References

  1. D. H. Adams (1969) The completion by cuts of an orthocomplemented modular lattice,Bull. Austral. Math. Soc. 1, 279–280.

    Google Scholar 

  2. D. R. Balbes and P. Dwinger (1974)Distributive Lattices, University of Missouri Press.

  3. B. Banaschewski (1956) Hullensysteme und Erweiterungen von Quasi-Ordnungen,Z. Math. Logik Grundl. Math. 2, 35–46.

    Google Scholar 

  4. G. Bruns, R. J. Greechie, J. Harding, and M. Roddy (1990) Completions of orthomodular lattices,Order 7, 67–76.

    Google Scholar 

  5. S. Burris and H. P. Sankappanavar (1981)A Course in Universal Algebra, Springer.

  6. A. Carson (1989)Model Completions, Ring Representations and the Topology of the Pierce Sheaf, Pitman research notes in mathematics series 209.

  7. S. D. Comer (1971) Representations by algebras of sections over Boolean spaces,Pacific J. of Math. 38, 29–38.

    Google Scholar 

  8. B. A. Davey (1973) Sheaf spaces and sheaves of universal algebras,Math. Zeit. 134, 275–290.

    Google Scholar 

  9. R. J. Greechie, private communication.

  10. P. R. Halmos (1967)Lectures on Boolean Algebras, Van Nostrand.

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

    Google Scholar 

  12. J. Harding (1992) Irreducible orthomodular lattices which are simple,Algebra Universalis 29, 556–563.

    Google Scholar 

  13. J. Harding (1994) Any lattice can be embedded into the MacNeille completion of a distributive lattice (to appear inThe Houston Journal of Math.).

  14. P. T. Johnstone (1982)Stone Spaces, Cambridge University Press.

  15. B. Jónsson (1967) Algebras whose congruence lattices are distributive,Math. Scand. 21, 110–121.

    Google Scholar 

  16. G. Kalmbach (1983)Orthomodular Lattices, Academic Press.

  17. A. Macintyre (1973) Model-completeness for sheaves of structures,Fund. Math. 81, 73–89.

    Google Scholar 

  18. M. D. MacLaren (1964) Atomic orthocomplemented lattices,Pac. J. Math. 14, 597–612.

    Google Scholar 

  19. H. M. MacNeille (1937) Partially ordered sets,Trans. AMS 42, 416–460.

    Google Scholar 

  20. R. S. Pierce (1967) Modules over commutative regular rings,Mem. Amer. Math. Soc. 70.

  21. J. Schmidt (1956) Zur Kennzeichnung der Dedekind MacNeilleschen Hulle einer geordneten Menge,Archiv d. Math. 7, 241–149.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 37240, Nashville, Tennessee, U.S.A.

    John Harding

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  1. John Harding
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Communicated by B. Jónsson

The author gratefully acknowledges the support of NSERC.

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Harding, J. Completions of orthomodular lattices II. Order 10, 283–294 (1993). https://doi.org/10.1007/BF01110549

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