Abstract
We introduce so-called weakly orthomodular and dually weakly orthomodular lattices which are lattices with a unary operation satisfying formally the orthomodular law or its dual although neither boundedness nor complementation is assumed. It turns out that lattices being both weakly orthomodular and dually weakly orthomodular are in fact complemented but the complementation need not be neither antitone nor an involution. Moreover, every modular lattice with complementation is both weakly orthomodular and dually weakly orthomodular. The class of weakly orthomodular lattices and the class of dually weakly orthomodular lattices form varieties which are arithmetical and congruence regular. Connections to left residuated lattices are presented and commuting elements are introduced. Using commuting elements, we define a center of such a (dually) weakly orthomodular lattice and we provide conditions under which such lattices can be represented as a non-trivial direct product.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beran, L.: Orthomodular Lattices. Algebraic Approach. Reidel, Dordrecht (1985). ISBN 90-277-1715-X
Birkhoff, G.: Lattice Theory. AMS, Providence (1979). ISBN 0-8218-1025-1
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. of Math. 37, 823–843 (1936)
Bonzio, S., Chajda, I.: A note on orthomodular lattices. Internat. J. Theoret. Phys. 56, 3740–3743 (2017)
Bruns, G., Harding, J.: Algebraic aspects of orthomodular lattices. Fund. Theories Phys. 111, 37–65 (2000). Kluwer, Dordrecht
Chajda, I., Eigenthaler, G., Länger, H.: Congruence Classes in Universal Algebra. Heldermann, Lemgo (2012). ISBN 3-88538-226-1
Chajda, I., Länger, H.: Residuation in orthomodular lattices. Topological Algebra Appl. 5, 1–5 (2017)
Chajda, I., Länger, H.: Orthomodular lattices can be converted into left residuated l-groupoids. Miskolc Math. Notes (to appear)
Husimi, K.: Studies of the foundation of quantum mechanics. I. Proc. Phys.-Math. Soc. Japan 19, 766–789 (1937)
Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983). ISBN 0-12-394580-1
Acknowledgments
Open access funding provided by Austrian Science Fund (FWF). Support of the research of both authors by ÖAD, project CZ 04/2017, as well as by IGA, project PřF 2018 012, is gratefully acknowledged. Support of the second author by the Austrian Science Fund (FWF), project I 1923-25 entitled “New perspectives on residuated posets”, is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chajda, I., Länger, H. Weakly Orthomodular and Dually Weakly Orthomodular Lattices. Order 35, 541–555 (2018). https://doi.org/10.1007/s11083-017-9448-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-017-9448-x