Completely hereditarily atomic OMLS
-
John Harding
and
Andre Kornell
Abstract
An irreducible complete atomic oml of infinite height cannot be algebraic and have the covering property. However, modest departure from these conditions allows infinite-height examples. We use an extension of Kalmbach’s construction to the setting of infinite chains to provide an example of an infinite-height, irreducible, algebraic oml with the 2-covering property, and Keller’s construction provides an example of an infinite-height, irreducible, complete oml that has the covering property and is completely hereditarily atomic. Completely hereditarily atomic omls generalize algebraic omls suitably to quantum predicate logic.
The first listed author gratefully acknowledges support of US Army grant W911NF-21-1-0247 and NSF grant DMS-2231414. The second author gratefully acknowledges support of Air Force Office of Scientific Research grant FA9550-21-1-0041.
Acknowledgement
The second author thanks Wolfgang Rump for suggesting Keller’s Hermitian space as a potential example.
Communicated by Mirko Navara
References
[1] Baer, R.: Linear Algebra and Projective Geometry, Academic Press, Inc., New York, 1952.Search in Google Scholar
[2] Balbes, D. R.—Dwinger, P.: Distributive Lattices, University of Missouri Press, Columbia, 1974.Search in Google Scholar
[3] Birkhoff, G.: Lattice Theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, AMS, Providence, 1967.Search in Google Scholar
[4] Bruns, G.—Harding, J.: Algebraic Aspects of Orthomodular Lattices. Current Research in Operational Quantum Logic: Algebras, Categories, Languages (B. Cooke, D. Moore and A. Wilce, eds.), Kluwer, 2000.10.1007/978-94-017-1201-9_2Search in Google Scholar
[5] Crawley, P.—Dilworth, R. P.: Algebraic Theory of Lattices, Prentice-Hall, 1973.Search in Google Scholar
[6] Day, G. W.: Free complete extensions of Boolean algebras, Pacific J. of Math. 15(4) (1965), 1145–1151.10.2140/pjm.1965.15.1145Search in Google Scholar
[7] Foulis, D.—Bennet, M.: Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325–1346.10.1007/BF02283036Search in Google Scholar
[8] Gierz, G.—Hoffmann, K. H.—Keimel, K.—Lawson, J. D.—Mislove, M.—Scott, D. S.: A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980.10.1007/978-3-642-67678-9Search in Google Scholar
[9] Gross, H.—Künzi, U. M.: On a class of orthomodular quadratic spaces, Enseign. Math. 31 (1985), 187–212.Search in Google Scholar
[10] Harding, J.: Orthomodular lattices whose MacNeille completions are not orthomodular, Order 8(1) (1991), 93–103.10.1007/BF00385817Search in Google Scholar
[11] Harding, J.: Remarks on concrete orthomodular lattices, Internat. J. Theoret. Phys. 43(10) (2004), 2149–2168.10.1023/B:IJTP.0000049016.83846.72Search in Google Scholar
[12] Van der Hoeven, J.: Transseries and Real Differential Algebra. Lecture Notes in Math., Vol. 1888, Springer-Verlag, Berlin, 2006.10.1007/3-540-35590-1Search in Google Scholar
[13] Husimi, K.: Studies on the foundations of quantum mechanics I, Proc. Phys.-Math. Soc. Japan 19 (1937), 766–789.Search in Google Scholar
[14] Jenča, G.: Effect algebras are the Eilenberg-Moore category for the Kalmbach monad, Order 32(3) (2015), 439–448.10.1007/s11083-014-9344-6Search in Google Scholar
[15] Jónsson, B.—Tarski, A.: Boolean algebras with operators. Part I, Amer. J. Math. 73 (1951), 891–939.10.2307/2372123Search in Google Scholar
[16] Kalmbach, G.: Orthomodular lattices do not satisfy any special lattice equations, Arch. Math. (Basel) 28(1) (1977), 7–8.10.1007/BF01223881Search in Google Scholar
[17] Kalmbach, G.: Orthomodular Lattices, Academic Press, 1983.Search in Google Scholar
[18] Keller, H. A.: Ein nicht-klassischer Hilbertscher Raum, Math. Z. 172 (1980), 41–49.10.1007/BF01182777Search in Google Scholar
[19] Keller, H. A.—Ochsenius, A.: Spectral Decompositions of Operators on Non-Archimedean Orthomodular Spaces, Internat. J. Theoret. Phys. 34(8) (1995), 1507–1517.10.1007/BF00676261Search in Google Scholar
[20] Kornell, A.: Quantum sets, J. Math. Phys. 61 (2020), Art. ID 102202.10.1063/1.5054128Search in Google Scholar
[21] Kornell, A.: Discrete quantum structures I: Quantum predicate logic, J. Noncommut. Geom. 18(1) (2024), 337–382.10.4171/jncg/531Search in Google Scholar
[22] Kornell, A.: Axioms for the category of sets and relations, https://doi.org/10.48550/arXiv.2302.14153.Search in Google Scholar
[23] Maeda, F.—Maeda, S.: Theory of Symmetric Lattices. Grundlehren Math. Wiss., Vol. 173, 1970.10.1007/978-3-642-46248-1Search in Google Scholar
[24] Piziak, R.: Orthomodular lattices and quadratic spaces: A survey, Rocky Mountain J. Math. 21(3) (1991), 951–992.10.1216/rmjm/1181072924Search in Google Scholar
[25] Solèr, M. P.: Characterization of Hilbert spaces with orthomodular spaces, Comm. Algebra 23(1) (1995), 219–243.10.1080/00927879508825218Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- On monoids of endomorphisms of a cycle graph
- The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
- Completely hereditarily atomic OMLS
- New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
- On the monogenity and Galois group of certain classes of polynomials
- Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
- Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
- Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
- Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
- The discrete fractional Karamata theorem and its applications
- Complex Generalized Stancu-Schurer Operators
- New oscillatory criteria for third-order differential equations with mixed argument
- Cauchy problem for a loaded hyperbolic equation with the Bessel operator
- Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
- Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
- On the relation of Kannan contraction and Banach contraction
- Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
- Residual Inaccuracy Extropy and its properties
- Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
Articles in the same Issue
- On monoids of endomorphisms of a cycle graph
- The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
- Completely hereditarily atomic OMLS
- New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
- On the monogenity and Galois group of certain classes of polynomials
- Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
- Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
- Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
- Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
- The discrete fractional Karamata theorem and its applications
- Complex Generalized Stancu-Schurer Operators
- New oscillatory criteria for third-order differential equations with mixed argument
- Cauchy problem for a loaded hyperbolic equation with the Bessel operator
- Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
- Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
- On the relation of Kannan contraction and Banach contraction
- Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
- Residual Inaccuracy Extropy and its properties
- Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
