Abstract
PBZ\(^{*}\)–lattices are lattices with additional operations that arise in the context of the unsharp approach to quantum logic. They include orthomodular lattices and Kleene algebras with an extra unary operation. We study in the framework of PBZ\(^{*}\)–lattices two constructions—the ordinal sum construction and the horizontal sum construction—that have been widely used in the investigation of both quantum structures and residuated structures. We provide axiomatisations of the varieties generated by certain sums of PBZ\(^{*}\)–lattices, in particular of the variety generated by all horizontal sums of an orthomodular lattice and an antiortholattice.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
That the spectral ordering is indeed a lattice ordering has been essentially shown by Olson [28] and de Groote [21], who also proved that it coincides with the more familiar ordering of effects induced via the trace functional when both orderings are restricted to the set of projection operators of the same Hilbert space. The same ordering has also been given an algebraic treatment, in a different context, in [12].
- 2.
See however [15] for the several distinct notions of sharp element that collapse in the context of PBZ\(^{*}\)–lattices.
References
P. Aglianó, F. Montagna, Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181, 105–129 (2003)
L. Beran, Orthomodular Lattices: Algebraic Approach (Reidel, Dordrecht, 1985)
M. Bergmann, An Introduction to Many-Valued and Fuzzy Logic (Cambridge University Press, Cambridge, 2008)
G. Bruns, J. Harding, Algebraic aspects of orthomodular lattices, in Current Research in Operational Quantum Logic, ed. by B. Coecke, et al. (Springer, Berlin, 2000), pp. 37–65
S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, vol. 78, Graduate Texts in Mathematics (Springer, New York, 1981)
G. Cattaneo, G. Nisticò, Brouwer-Zadeh posets and three-valued Łukasiewicz posets. Fuzzy Sets Syst. 33(2), 165–190 (1989)
I. Chajda, A note on pseudo-Kleene algebras. Acta Univ. Palacky Olomouc 55(1), 39–45 (2016)
I. Chajda, H. Länger, Horizontal sums of bounded lattices. Mathematica Pannonica 20(1), 1–5 (2009)
C.C. Chen, G. Grätzer, Stone lattices. I: Construction theorems. Cana. J. Math. 21, 884–894 (1969)
M.L. Dalla Chiara, R. Giuntini, R.J. Greechie, Reasoning in Quantum Theory (Kluwer, Dordrecht, 2004)
A. Dvurečenskij, Aglianó-Montagna type decomposition of pseudo hoops and its applications. J. Aust. Math. Soc. 211, 851–861 (2007)
A. Dvurečenskij, Olson order of quantum observables. Int. J. Theor. Phys. 55, 4896–4912 (2016)
R. Freese, R. McKenzie, Commutator Theory for Congruence-modular Varieties, London Mathematical Society Lecture Note Series, vol. 25 (Cambridge University Press, Cambridge, 1987)
L. Fuchs, Partially Ordered Algebraic Systems (Pergamon Press, Oxford, 1963)
R. Giuntini, A. Ledda, F. Paoli, A new view of effects in a Hilbert space. Studia Logica 104, 1145–1177 (2016)
R. Giuntini, A. Ledda, F. Paoli, On some properties of PBZ\(^{\ast }\)-lattices. Int. J. Theor. Phys. 56(12), 3895–3911 (2017)
R. Giuntini, C. Mureşan, F. Paoli, “PBZ\(^{\ast }\)–lattices: Structure Theory and Subvarieties”. Rep. Math. Logic 55, 3–39 (2020)
G. Grätzer, General Lattice Theory (Birkhäuser Akademie-Verlag, Basel, 1978)
G. Grätzer, Universal Algebra, 2nd edn. (Springer Science+Business Media, LLC, New York, 2008)
R.J. Greechie, On the structure of orthomodular lattices satisfying the chain condition. J. Comb. Theory 4(3), 210–218 (1968)
H.F. De Groote, On a canonical lattice structure on the effect algebra of a von Neumann algebra (2005), ar**v:math-ph/0410018v2
B. Jónsson, Congruence-distributive varieties. Math. Japonica 42(2), 353–401 (1995)
J.A. Kalman, Lattices with involution. Trans. Amer. Math. Soc. 87, 485–491 (1958)
E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978)
P.S. Mostert, A.L. Shields, On the structure of semigroups on a compact manifold with boundary. Ann. Math. 65, 117–143 (1957)
C. Mureşan, Cancelling congruences of lattices, while kee** their filters and ideals, ar**v:1710.10183v2 [math.RA]
C. Mureşan. Some properties of lattice congruences preserving involutions and their largest numbers in the finite case, ar**v:1802.05344v2 [math.RA]
M.P. Olson, The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc. Amer. Math. Soc. 28, 537–544 (1971)
A. Salibra, A. Ledda, F. Paoli, T. Kowalski, Boolean-like algebras. Algebra Universalis 69(2), 113–138 (2013)
D.W. Stroock, A Concise Introduction to the Theory of Integration, 3rd edn. (Birkhäuser, Basel, 1998)
Acknowledgements
We thank Davide Fazio and Antonio Ledda for insightful discussions on the topics of this paper. This work was supported by the research grants “Proprietà d‘Ordine Nella–Semantica Algebrica delle Logiche Non–classiche”, Università degli Studi di Cagliari, Regione Autonoma della Sardegna, L. R. 7/2007, n. 7, 2015, CUP: F72F16002920002 and “Per un’estensione semantica della Logica Computazionale Quantistica - Impatto teorico e ricadute implementative”, RAS: SR40341. Moreover, all authors gratefully acknowledge the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Giuntini, R., Mureşan, C., Paoli, F. (2021). PBZ\(^{*}\)–Lattices: Ordinal and Horizontal Sums. In: Fazio, D., Ledda, A., Paoli, F. (eds) Algebraic Perspectives on Substructural Logics. Trends in Logic, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-030-52163-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-52163-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-52162-2
Online ISBN: 978-3-030-52163-9
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)