Abstract
All published work on orthomodular lattice theory has appeared within the last fifteen years; no more than thirty people have ever worked on it; no more than fifty papers dealing explicitly with it have ever appeared: Orthomodular lattice theory is therefore a newly uncovered very small corner of mathematics.
Reprinted from Trends in Lattice Theory (ed. by J. C. Abbott), Van Nostrand Reinhold Math. Studies #31, Van Nostrand Reinhold, New York, 1970.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Part 1. Works Referenced in the Text
Amemiya, I. and Halperin, I., ‘Complemented Modular Lattices’, Canad. J. Math. 11 (1950), 481–520.
Amemiya, I. and Araki, H., ‘A Remark on Piron’s Paper’, Publ. Res. Inst. Math. Sci. Ser. A, 12 (1966/67), 423–427.
Birkhoff, G., ‘Lattices in Applied Mathematics’, from Lattice Theory, Proc. Symp. Pure Math. Vol. 11, Amer. Math. Soc., Providence 1961.
Birkhoff, G., Lattice Theory, 3rd ed., Amer. Math. Soc. Colloq. Publ. Vol. 25 (1966).
Birkhoff, G. and Neumann, J. von, ‘The Logic of Quantum Mechanics’, Ann. of Math. 37 (1936), 823–842.
Bodiou, G., Théorie dialectique des probabilités, Gauthier-Villars, Paris, 1964.
Dilworth, R. P., ‘On Complemented Lattices’, Tohoku Math. J. 47 (1940), 18–23.
Dixmier, J., ‘Position relative de deux variétés linéaires fermées dans un espace de Hilbert’, La Revue Scientifique, Fasc. 7, 86 (1948), 387–399.
Dixmier, J., Les algèbres d’opérateurs dans l’espace hilbertien, Gauthier-Villars, Paris, 1957.
Dye, H. A., ‘On the Geometry of Projections in Certain Operator Algebras’, Ann. of Math. 61 (1955), 73–89.
Fillmore, P. A., ‘Perspectivity in Projection lattices’, Proc. Amer. Math. Soc. 16 (1965), 383–387.
Foulis, D. J., ‘Baer *-Semigroups’, Proc. Amer. Math. Soc. 11 (1960), 648–654.
Foulis, D. J., ‘A Note on Orthomodular Lattices’, Portugal. Math. 21 (1962), 65–72.
Foulis, D. J., ‘Notes on Orthomodular Lattices’, lecture notes, Univ. of Florida, 1963 (unpublished).
Foulis, D. J., ‘Lattices and Semigroups’, mimeographed, unpublished.
Gardner, L. T., ‘On Isomorphisms of C*-Algebras’, Amer. J. Math. 87 (1965), 384–396.
Gleason, A. M., ‘Measures on the Closed Subspaces of a Hilbert Space’, J. Math. Mech. 6 (1957), 885–894.
Greechie, R. J., ‘On the Structure of Orthomodular Lattices Satisfying the Chain Condition’, (to appear).
Gudder, S. P., ‘Spectral Methods for a Generalized Probability Theory’, Trans. Amer. Math. Soc. 119 (1965), 428–442.
Halmos, P. R., Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea, New York, 1957.
Halperin, I., ‘Complemented Modular Lattices’, from Lattice Theory, Proc. Symp. Pure Math. Vol. II, Amer. Math. Soc., Providence, 1961.
Holland, Jr., S. S., ‘A Radon-Nikodym Theorem in Dimension Lattices’, Trans. Amer. Math. Soc. 108 (1963), 66–87.
Holland, Jr., S. S., ‘Distributivity and Perspectivity in Orthomodular Lattices’, Trans. Amer. Math. Soc. 112 (1964), 330–343.
Janowitz, M. F., ‘A Semigroup Approach to Lattices’, Canad. J. Math. 18 (1966), 1212–1223.
Janowitz, M. F., ‘Notes on Orthomodular Lattices’, Univ. of New Mexico, 1965 (unpublished).
Janowitz, M. F., ‘On Conditionally Continuous Lattices’, Univ. of New Mexico Tech. Report no. 107 (1966).
Kaplansky, I., ‘Any Orthocomplemented Complete Modular Lattice is a Continuous Geometry’, Ann. of Math. 61 (1955), 524–541.
Kaplansky, I., ‘Rings of Operators’, University of Chicago Mimeographed Notes, 1955.
Loomis, L. H., ‘The Lattice Theoretic Background of the Dimension Theory of Operator Algebras’, Mem. Amer. Math. Soc., 18 (1955), 36 pp.
Ludwig, G., ‘Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeiner physikalischer Theorien’, Z. Physik 181 (1964), 233–260 (mimeographed English translation available).
Mackey, G. W., ‘On Infinite-Dimensional Linear Spaces’, Transl. Amer. Math., Soc. 57 (1945), 155–207.
Mackey, G. W., Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1963.
Mac Laren, M. D., ‘Nearly Modular Orthocomplemented Lattices’, Trans. Amer. Math. Soc. 114 (1965), 401–416.
Mac Laren, M. D., ‘Notes on Axioms for Quantum Mechanics’, Argonne National Lab. Rept. ANL-7065, July 1965 (available from the Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U.S. Dept. of Commerce, Springfield, Virginia, $ 1.00).
Maeda, F., ‘Relative Dimensionality in Operator Rings’, J. Sci. Hiroshima Univ. Ser. A 11 (1941), 1–6.
Maeda, F., Kontinuierliche geometrien, Springer, Berlin, 1958.
Maeda, F. and Maeda, S., Theory of Symmetric Lattices, book in press.
Maeda, S., ‘Dimension Functions on Certain General Lattices’, J. Sci. Hiroshima Univ. Ser. A-1 Math. 19 (1955), 211–237.
Maeda, S., ‘On Conditions for the Orthomodularity’, Proc. Japan Acad. 42 (1966), 247–251.
Murray, F. J. and Neumann, J. von, ‘On Rings of Operators’, Ann. of Math. 37 (1936), 116–229.
Nakamura, M., ‘The Permutability in a Certain Orthocomplemented Lattice’, Kodai Math. Series. Rep. 9 (1957), 158–160.
Neumann, J. von, ‘Continuous Geometry’, Proc. Natl. Acad. Sci. U.S.A. 22 (1936), 92–100.
Neumann, J. von, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton, 1955.
Neumann, J. von, Continuous Geometry, Princeton Univ. Press, Princeton 1960.
Ramsay, A., ‘Dimension Theory in Complete Orthocomplemented Weakly Modular Lattices’, Trans. Amer. Math. Soc. 116 (1965), 9–31.
Randall, C. H., ‘A Mathematical Foundation for Empirical Science — with Special Reference to Quantum Theory. Part 1, A Calculus of Experimental Propositions’, Knolls Atomic Power Lab. Report. KAPL-3147. June 1966. Available from the Clearinghouse for Federal Scientific and Technical Information, National Bureau of Standards, U.S. Department of Commerce, Springfield. Virginia, $6.00.
Segal, I. E., ‘A Non-Commutative Extension of Abstract Integration’, Ann. of Math. (2) 57 (1953), 401–457.
Simmons, G. F., Topology and Modern Analysis, McGraw-Hill, 1963.
Stone, M. H., ‘Linear Transformations in Hilbert Space’, Amer. Math. Soc., Colloq. Publ. Vol. 15, Amer. Math. Soc., Providence, 1932.
Topping, D. M., ‘Asymptoticity and Semimodularity in Projection Lattices’, to appear.
Varadarajan, V. S., ‘Probability in Physics and a Theorem on Simultaneous Observability’, Comm. Pure and Appl. Math. 15, (1962), 189–217.
Zierler, N. ‘Axioms for Non-Relativistic Quantum Mechanics’, Pacific J. Math. 11 (1961), 1151–1169.
Zierler, N., ‘On the Lattice of Closed Subspaces of Hilbert Space’, Pacific J. Math. 19 (1966), 583–586.
Part 2. Works Not Referenced in the Text
Berberian, S. K., ‘On the Projection Geometry of a Finite AW*-Algebra’, Trans. Amer. Math. Soc. 83 (1956), 493–509.
Bodiou, G., ‘Probabilité sur une trellis non modulair’, Publ. Inst. Statist., Univ. Paris 6 (1957), 11–25.
Bodiou, G., Recherches sur les fondements du calcul quantique des probabilité dans les cas purs, Masson, Paris, 1950.
Croisot, R., ‘Applications Residuées’, Ann. Sci. Ecole Norm. Sup. 73 (1956), 453–474.
Dilworth, R. P., ‘The Structure of Relatively Complemented Lattices’, Ann. of Math. 51 (1950), 348–359.
Davis, C., ‘Separation of Two Linear Subspaces’, Acta Sci. Math. (Szeged) 19 (1958), 172–187.
Dye, H. A., ‘The Radon-Nikodym Theorem for Finite Rings of Operators’, Trans. Amer. Math. Soc. 72 (1952), 243–280.
Foulis, D. J., ‘Conditions for the Modularity of an Orthomodular Lattice’, Pacific J. Math. 11 (1961), 889–895.
Foulis, D. J., ‘Relative Inverses in Baer *-Semigroups’, Mich. Math. J. 10 (1963), 65–84.
Foulis, D. J., ‘Semigroups Coordinatizing Orthomodular Geometries’, Canad. J. Math. 17 (1960), 40–51.
Gratzer, G. and Schmidt, E. T., ‘Ideals and Congruence Relations in Lattices’, Acta. Math. Acad. Sci. Hungar 9 (1958), 137–175.
Gratzer, G. and Schmidt, E. T., ‘Standard Ideals in Lattices’, Acta. Math. Acad. Sci. Hungar 12 (1961), 17–86.
Halmos, P. R., Algebraic Logic, Chelsea, New York, 1962.
Halperin, I., ‘Introduction to von Neumann Algebras and Continuous Geometry’, Canad. Math. Bull. 3 (1960), 273–288.
Iqbalunnisa, ‘Neutrality in Weakly Modular Lattices’, Acta. Math. Acad. Sci. Hunger. 26 (1965), 325.
Iqbalunnisa, ‘On Neutral Elements in a Lattice’, J. Indian Math. Soc. (New series) 28 (1964), 25–31.
Janowitz, M. F., ‘Quasi-Orthomodular Lattices’, Univ. of New Mexico Tech. Report No. 42 (1963).
Janowitz, M. F., ‘AC-Lattices’, Univ. of New Mexico Tech. Report No. 45 (1963).
Janowitz, M. F., ‘Quantifiers and Orthomodular Lattices’, Pacific J. Math. 13 (1963), 1241–1249.
Janowitz, M. F., ‘On the Antitone Mappings of a Poset’, Proc. Amer. Math. Soc. 15 (1964), 529–533.
Janowitz, M. F., ‘Projective Ideals and Congruence Relations’, Univ. of New Mexico Tech. Report No. 51 (1964).
Janowitz, M. F., ‘Projective Ideals and Congruence Relations IF, Univ. of New Mexico Tech. Report No. 63 (1964).
Janowitz, M. F., ‘Residuated Closure Operators’, Univ. of New Mexico Tech. Report No. 79 (1965).
Janowitz, M. F., ‘A Note on Normal Ideals’, Univ. of New Mexico Tech. Report No. 95 (1965).
Janowitz, M. F., ‘A Characterization of Standard Ideals’, Acta. Math. Acad. Sci. Hungar 26 (1965), 289–301.
Janowitz, M. F., ‘Quantifier Theory on Quasi-Orthomodular Lattices’, Illinois J. Math. 9 (1965), 660–676.
Janowitz, M. F., ‘Baer-Semigroups’, Duke Math. J. 32 (1965), 85–96.
Janowitz, M. F., ‘Independent Complements in Lattices’, Univ. of New Mexico Tech. Report No. 87 (1965).
Janowitz, M. F., ‘The Center of a Complete Relatively Complemented Lattice is a Complete Sublattice’, Univ. of New Mexico Tech. Report No. 105 (1966).
Kaplansky, I., ‘Projections in Banach Algebras’, Ann. of Math. 53 (1951), 235–249.
Mac Laren, M. D., ‘Atomic Orthocomplemented Lattices’, Pacific J. Math. 14 (1964), 597–612.
Maeda, F., ‘Direct Sums and Normal Ideals of Lattices’, J. Sci. Hiroshima Univ. Ser. A-I Math. 14 (1949), 85–92.
Maeda, F., ‘Representations of Orthocomplemented Lattices’, J. Sci. Hiroshima Univ. Ser. A 14 (1950), 1–4.
Maeda, F., ‘Decomposition of General Lattices into direct Summands of Types I, II and III’, J. Sci. Hiroshima Univ. Ser. A 23 (1959), 151–170.
Maeda, S., ‘On the Lattices of Projections of a Baer *-Ring’, J. Sci. Hiroshima Univ. Ser. A 22 (1958), 76–88.
Maeda, S., ‘On Relatively Semi-Orthocomplemented Lattices’, J. Sci. Hiroshima Univ. Ser. A 24 (1960), 155–161.
Maeda, S., ‘On a Ring whose Principal Right Ideals Generated by Idempotents Form a Lattice’, J. Sci. Hiroshima Univ. Ser. A 24 (1960), 510–525.
Maeda, S., ‘Dimension Theory on Relatively Semi-Orthocomplemented Complete Lattices’, J. Sci. Hiroshima Univ. Ser. A 25 (1961), 369–404.
Maeda, S., ‘On the Symmetry of the Modular Relation in Atomic Lattices’, J. Sci. Hiroshima Univ. Ser. A-I 29 (1965), 165–170.
Morgado, J., ‘On the Automorphisms of the Lattice of Closure Operators of a Complete Lattice’, Rev. Un. Mat. Argentina 20 (1960), 188–193.
Morgado, J., ‘Some Results on Closure Operators of Partially Ordered Sets’, Portugal Math. 19 (1960), 101–139.
Morgado, J., ‘Note on the Automorphisms of the Lattice of Closure Operators of a Complete Lattice’, Nederl. Akad. Wetensch. Proc. Ser. A 64 (1961), 211–218.
Morgado, J., ‘Some Remarks on Quasi-Isomorphisms between Finite Lattices’, Portugal Math. 20 (1961), 137–145.
Morgado, J., ‘Quasi-Isomorphisms between Complete Lattices’, Portugal Math. 20 (1961), 17–31.
Nakamura, N., ‘Center of Closure Operators and a Decomposition of a Lattice’, Math. Japan 4 (1964), 49–52.
Neumann, J. von, Collected Works, 6 vols., Pergamon, 1961.
Sachs, D., ‘Partition and Modulated Lattices’, Pacific J. Math. 11 (1961), 325–345.
Sasaki, U., ‘On an Axiom of Continuous Geometry’, J.’ Sci. Hiroshima Univ. Ser. A (1950), 100–101.
Sasaki, U., ‘On Orthocomplemented Lattices Satisfying the Exchange Axiom’, J. Sci. Hiroshima Univ. Ser. A 17 (1954), 293–302.
Sasaki, U., ‘Lattices of Projections in AW*-Algebra’, J. Sci. Hiroshima Univ. Ser. A 19 (1955), 1–30.
Schmidt, E. T., ‘Remark on a Paper of M. F. Janowitz’, Acta. Math. Acad. Sci. Hungar. 16 (1965), 435.
Schreiner, E. A., ‘Modular Pairs in Orthomodular Lattices’, Pacific J. Math. 19 (1966), 519–528.
Topping, D. M., ‘Jordan Algebra of Self-Adjoint Operators’, Mem. Amer. Math. Soc. 53 (1965), 48 pp.
Wright, F. B., ‘A Reduction for Algebras of Finite Type’, Ann. of Math. 60 (1954), 560–570.
Wright, F. B., ‘Some Remarks on Boolean Duality’, Portugal Math. 16 (1957), 109–117.
Zierler, N. and Schlessinger, M., ‘Boolean Embeddings of Orthomodular Sets and Quantum Logic’, Duke Math. J. 32 (1965), 251–262.
Part 3. Some Recent Papers on Quantum Mechanics Not Listed in [34] or [46]
Jordan, P., ‘Quantenlogik und das kommutative Gesatz’, from The Axiomatic Method (ed. by L. Henkin, P. Suppes, and A. Tarski, Amsterdam 1959.
Ludwig, G., ‘An Axiomatic Foundation of Quantum Mechanics on a Non-Subjective Basis’, Marburg/L. Mimeographed, unpublished.
Ludwig, G., ‘Attempt of an Axiomatic Foundation of Quantum Mechanics and More General Physical Theories II’, mimeographed, unpublished.
Rose, G., ‘Zur Orthomodularität von Wahrscheinlichkeitsfeldern’, Z. Physik 181 (1964), 331–332.
Weizsacker, C. G. von, ‘Komplementarität und Logik I’, Naturwissenschaften 42 (1955), 521.
Weizsacker, C. G. von, ‘Komplementarität und Logik II’, Z. Naturforsch. 13a (1958), 245.
Weizsacker, C. G. von, ‘Komplementarität und Logik III’, Z. Naturforsch. 13a (1958), 705.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1975 D. Reidel Publishing Company, Dordrecht, Holland
About this chapter
Cite this chapter
Holland, S.S. (1975). The Current Interest in Orthomodular Lattices. In: Hooker, C.A. (eds) The Logico-Algebraic Approach to Quantum Mechanics. The University of Western Ontario Series in Philosophy of Science, vol 5a. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1795-4_25
Download citation
DOI: https://doi.org/10.1007/978-94-010-1795-4_25
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-277-0613-3
Online ISBN: 978-94-010-1795-4
eBook Packages: Springer Book Archive