DOI: 10.2478/s12175-012-0061-6 Math. Slovaca 62 (2012), No. 6, 1007–1018

DAVID JAMES FOULIS Richard J. Greechie Dedicated to Prof. David J. Foulis on the occasion of his 80th birthday (Communicated by Sylvia Pulmannov´ a) ABSTRACT. This paper is in honour of Dave Foulis’s 80th birthday. A brief account of some of his outstanding academic contributions to the fields of Ordered Structures, Orthostructures, Foundations of Quantum Mechanics, Foundations of Statistics and Operator Theory is present here. c
2012 Mathematical Institute Slovak Academy of Sciences

The following papers are dedicated to Dave Foulis in celebration of a lifetime of outstanding academic contributions to the fields of Ordered Structures, Orthostructures, Foundations of Quantum Mechanics, Foundations of Statistics and Operator Theory. Dave was born in Hinsdale, Illinois, in 1930. He is married with three children, David, Dean and Scott, and five grandchildren. He was an undergraduate student at The University of Miami, where he graduated in 1952, Magma Cum Laude. Dave received his graduate training at the University of Miami, the University of Chicago, and Tulane University where he earned a Ph.D. in 1958. While at the University of Chicago, he was strongly influenced by Irving Kaplansky and Paul Halmos, both expositors par excellence. That influence persists in his oral presentations as well as his written publications, and manifests itself in a much admired polished clarity in which the presentation of mathematical ideas becomes a kind of poetry. This expository excellence is available to undergraduates in Dave’s 7 books. They span the spectrum of undergraduate mathematics. (One, a large red calculus book, made its way as a prop in the movie, “The Sure Thing”, directed by Rob Reiner, and has appeared elsewhere in the greater cultural landscape.) I present only my favorites — the more advanced books: 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 01A70; Secondary 00B30. K e y w o r d s: Baer *-semigroup, Foulis semigroup, orthomodular lattice, effect algebra, quantum logic, manual.

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Fundamental Concepts of Mathematics, Prindle, Weber, and Schmidt, Boston, 1969, 212 pp., ISBN 87150-075-4. Calculus, (with M. Munem), Worth Publishers, Inc. N.Y., 1978, 993 pp., ISBN 0-87901-087-8. After Calculus: Algebra, (with M. Munem), Dellen/Macmillan, N.J., 1988, 488 pp., ISBN 0-02-384790-5. After Calculus: Analysis, (with M. Munem), Dellen/Macmillan, N.J., 1989, 556 pp., ISBN 0-02-339130-8. One of his professors at Miami, Wayman Strother, later became the Head of the Department of Mathematics and Statistics at the University of Massachusetts, Amherst. Wayman invited Dave to join the Math Faculty at UMA “to revise the undergraduate curriculum”. Within 5 years, the undergraduate curriculum was substantially revised and the world’s largest group of specialists in orthomodular lattice theory was centered in Amherst. Wayman once confided to me that he had made an observation when Dave was one of his Advanced Calculus students in Miami; he said, “Dave is someone who does not know how to make a bad proof”. During 50 years of working under and then with Dave, I’ve often thought how prescient this observation was. Dave began his academic career by studying multiplicative structures. His dissertation, written under Fred Wright at Tulane University, was entitled Involution Semigroups. In it he initiated research into what he called Baer *semigroups, which later came to be called Foulis semigroups. These structures abstracted the multiplicative features of operator algebras. Through Dave’s penetrating arguments, they captured more of the ring theory of operator algebras than experts might reasonably have expected. Dave always had a compelling interest in “understanding” quantum physics. These demands of understanding led him from being an undergraduate physics major to focusing on mathematics in graduate school. He developed models (many unpublished) for the foundations of quantum mechanics. When Dave learned of the innovative and independent ideas of Charlie Randall, a close collaboration was born. This led to an important thrust called “The Amherst School” in which he and Charlie, along with other colleagues and students, made profound progress in our understanding of mathematical foundations of empirical studies, in particular of quantum mechanics. Studying projections in Baer *-semigroups led to the study of orthomodular lattices. The representation of orthomodular lattices which admit an order determining set of states (or generalized probability measures) led to the study of 1008

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partially ordered abelian groups, thus changing the focus of Dave’s research from multiplicative structures to additive structures. Studying empirical logic led to the abstraction called effect algebras, with its focus on a partially defined “plus”. The transition from studying orthomodular lattices to studying effect algebras allowed for the study of not only sharp quantum logics but possibly unsharp quantum logics. Dave was not the only researcher to study these structures, but it is not unfair to say that he was a, perhaps the, driving force in their creation. Dave has held employment positions at Lehigh University, Wayne State University, University of Florida and University of Massachusetts where he retired as Professor Emeritus in 1997. It is interesting to note that almost half of his over 100 publications are dated after he retired. A quick papers/year analysis might indicate that his productivity has increased. But much of the difference may be accounted for by the attention he paid to his students. Each of us has had our own special interactions with Dave. My own favorite is the story of how, when I was an aspiring (first year) graduate student who had shown an interest in projections on a Hilbert Space, Dave offered to tell me how (an abstraction of) these structures played a role in the foundations of quantum mechanics. He proceeded to give me a one-on-one lecture that lasted from 7 pm on a Friday evening till 1 am. At 1, Dave said that he couldn’t finish that evening and would I like to meet again “tomorrow”. The lecture resumed exactly 6 hours later! Dave is a member and a former President of the International Quantum Structures Association. In a strong sense, IQSA is his academic home. His wife, Hyla, enjoys listening to his lectures and, during an IQSA Meeting, can usually be seen in the back of the lecture hall during Dave’s presentation. There is no doubt in my mind that she plays a major role as facilitator, which has been an important factor in Dave’s outstanding productivity. Hyla wrote the solutions manual to Dave’s first Calculus book, providing solutions to approximately 5000 problems, an impressive feat before the availability of graphing calculators. I suspect that Hyla also plays an editorial role before Dave’s papers are submitted, and have often been amused by her comments after a lecture, over dinner, such as, “You changed that part about the connection with operator theory”. Dave has had 23 Ph.D. students and according to the current on-line database, The Mathematics Genealogy Project, he has 68 descendants. There is no count of the number of dissertations, directed by others, which contain statements of appreciation to Dave for advice and/or direction; there is no count of the number of research papers acknowledging Dave, nor of the number of anonymous referee’s reports which were far more elaborate than the profession demands — some longer than the submitted paper itself (but this author knows of many of each type). Here is a list of Dave’s Ph.D. students. The first 4 listed graduated from Wayne State University, the next 3 graduated from the University of Florida 1009

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and the last 16 graduated from the University of Massachusetts, where Dave spent the major part of his teaching career. It is followed by a list of his published articles to date, along with co-authors and Math Reviews Numbers, when available. All this is evidence of an extremely productive and influential career. Dave’s colleagues and friends, many through the papers presented herein, sincerely hope that Dave maintains his health and continues to enrich our community. Dave’s Ph.D. students 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 1010

Melvin F. Janowitz, Quantifiers on Quasi-orthomodular Lattices, 1963. Erik A. Schreiner, Modular Pairs in Orthomodular Lattices, 1964. Linda Brown, Span Spaces, 1965. Jean Claude Derderian, Residuated Mappings, 1965. Jean H. Bevis, Quantifiers and Dimension Equivalence Relations on Orthomodular Lattices, 1965. Donald E. Catlin, Implicativity and Irreducibility in Orthomodular Lattices, 1965. Richard J. Greechie, Orthomodular Lattices, 1966. Mary K. Bennett, Convex, Affine, and Projective Geometries, 1966. Edwin L. Marsden, Jr., The Commutator and Irreducibility Conditions on Orthomodular Lattices, 1967. James C. Dacey Jr., Orthomodular Spaces, 1968. Robert J. Weaver, Orthogonality Spaces and the Free Orthogonality Monoid, 1969. Robert Piziak, An Algebraic Generalization of Hilbert Space Geometry, 1969. Louis M. Herman, Semi-Orthogonality in Rickart Rings, 1970. Walter R. Collins, A Category of Sample Spaces, 1970. David P. Sumner, Indecomposable Graphs, 1970. Norman K. Roth, Extending Weights on Generalized Sample Spaces, 1970. Barbara Jeffcott, Orthologics, 1971. Ron Wright, Projection Valued States, 1977. Karen Benbury, Dimension *-Semigroups, 1979. Patricia Frazer Lock, Categories of Manuals, 1981. Andrew S. Golfin, Jr., Representations and Products of Lattices, 1987. Alexander G. Wilce, The Signed Weight Space of a Tensor Product, 1989. Nathan Ritter, Order Unit Intervals in Unigroups, 2001.

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Dave’s publications [1] (with Pulmannov´ a, S.): Logical connectives on lattice effect algebras, Studia Logica, 2012 (To appear). [2] (with Pulmannov´ a, S.): Quotients of dimension effect algebras, Algebra Universalis 67 (2012), 81–104. MR2885515 [3] (with Pulmannov´ a, S. and Vincekov´a, E.): Lattice pseudo-effect algebras as double residuated structures, Soft Comput. 15 (2011), 2479–2488. [4] (with Pulmannov´ a, S.): The exocenter of a generalized effect algebra, Rep. Math. Phys. 68 (2011), 347–371. MR2900853 [5] (with Pulmannov´ a, S.): Hull mappings and dimension effect algebras, Math. Slovaca 61 (2011), 485–522. MR2796257 [6] (with Pulmannov´ a, S.): Regular elements in generalized Hermitian algebras, Math. Slovaca 61 (2011), 155–172. MR2786691 [7] (with Pulmannov´ a, S. and Vincekov´a, E.): Type-decomposition of a pseudo-effect algebra, J. Aust. Math. Soc. 89 (2010), 335–358. MR2785909 [8] (with Pulmannov´ a, S.): Centrally orthocomplete effect algebras, Algebra Universalis 64 (2010), 283–307. MR2781080 [9] (with Pulmannov´ a, S.): Type-decomposition of an effect algebra, Found. Phys. 40 (2010), 1543–1565. MR2726371 [10] Synaptic algebras, Math. Slovaca 60 (2010), 631–654. MR2728528 [11] (with Pulmannov´ a, S.): Projections in a synaptic algebra, Order 27 (2010), 235–257. MR2660730 [12] (with Pulmannov´ a, S.): Generalized Hermitian algebras, Int. J. Theor. Phys. 48 (2009), 1320–1333. MR2518515 [13] (with Pulmannov´ a, S.): Spin factors as generalized Hermitian algebras, Found. Phys. 39 (2009), 237–255. MR2486542 1011

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[14] (with Pulmannov´ a, S.): Spectral resolution in an order-unit space, Rep. Math. Phys. 62 (2008), 323–344. MR2490137 [15] (with Greechie, R. J.): Quantum logic and partially ordered abelian groups. Handbook of quantum logic and quantum structures, Elsevier Sci. B. V., Amsterdam, 2007, pp. 215–283. MR2423609 [16] Observables, states, and symmetries in the context of CB-effect algebras, Rep. Math. Phys. 60 (2007), 329–346. MR2374826 [17] Effects, observables, states, and symmetries in physics, Found. Phys. 37 (2007), 1421–1446. MR2356223 [18] (with Pulmannov´ a, S.): Polar decomposition in e-rings, J. Math. Anal. Appl. 333 (2007), 1024–1035. MR2331711 [19] Square roots and inverses in e-rings, Rep. Math. Phys. 58 (2006), 357–373. MR2293420 [20] Sharp and fuzzy elements of an RC-group, Math. Slovaca 56 (2006), 525–541. MR2293585 [21] The universal group of a Heyting effect algebra, Stud. Log. 84 (2006), 407–424. MR2290116 [22] Rings with effects, (2006), pp. 20. arXiv:quant-ph/0609181 [23] (with Pulmannov´ a, S.): Monotone σ-complete RC-groups, J. Lond. Math. Soc. 73 (2006), 304–324. MR2225488 [24] Comparability groups, Demonstr. Math. 39 (2006), 15–32. MR2223870 [25] Compression bases in unital groups, Int. J. Theor. Phys. 44 (2005), 2191–2198. MR2234751 [26] Logic and Partially Ordered Abelian Groups. (2005) pp. 26. arXiv:math/0504553 [27] Compressible groups with general comparability, Math. Slovaca 55 (2005), 409–429. MR2181781 [28] (with Greechie, R. J.): Semisimplicial unital groups, Int. J. Theor. Phys. 43 (2004), 1689–1704. MR2108305 [29] Spectral resolution in a Rickart comgroup, Rep. Math. Phys. 54 (2004), 229–250. MR2107868

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[30] Compressions on partially ordered abelian groups, Proc. Am. Math. Soc. 132 (2004), 3581–3587. (electronic). MR2084080 [31] (with Cook, TH. A.): The base-normed space of a unital group, Math. Slovaca 54 (2004), 69–85. MR2074031 [32] Compressible groups, Math. Slovaca 53 (2003), 433–455. MR2038512 [33] Representation of a unital group having a finite unit interval, Demonstr. Math. 36 (2003), 793–805. MR2018699 [34] Removing the torsion from a unital group, Rep. Math. Phys. 52 (2003), 187–203. MR2016215 [35] Archimedean unital groups with finite unit intervals, Int. J. Math. Math. Sci. 2003 (2003), 2787–2801. MR2003789 [36] Sequential probability models and transition probabilities, Atti Semin. Mat. Fis. Univ. Modena 50 (2002), 225–249. MR1910789 [37] (with Gudder, S. P.): Observables, calibration, and effect algebras, Found. Phys. 31 (2001), 1515–1544. MR1873264 [38] (with Wilce, A.): Free extensions of group actions, induced representations, and the foundations of physics. Current research in operational quantum logic, (2000) 139–165, Fund. Theories Phys., 111, Kluwer Acad. Publ., Dordrecht. MR1907159 [39] Representations on unigroups. Current research in operational quantum logic, (2000) 115–138, Fund. Theories Phys., 111, Kluwer Acad. Publ., Dordrecht. MR1907158 [40] MV and Heyting effect algebras, Found. Phys. 30 (2000), 1687–1706. MR1810197 [41] Algebraic measure theory, Atti Semin. Mat. Fis. Univ. Modena 48 (2000), 435–461. MR1811545 [42] (with Greechie, R. J.): Specification of finite effect algebras, Int. J. Theor. Phys. 39 (2000), 665–676. MR1790903 [43] A half-century of quantum logic. What have we learned?, Quantum structures and the nature of reality, 1–36, Einstein Meets Magritte, 7, Kluwer Acad. Publ., Dordrecht, 1999. MR1787387 [44] (with Greechie, R. J.): Probability weights and measures on finite effect algebras, Int. J. Theor. Phys. 38 (1999), 3189–3208. MR1764458 1013

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[45] (with Bennett, M. K.): A generalized Sasaki projection for effect algebras, Tatra Mt. Math. Publ. 15 (1998), 55–66. MR1655078 [46] (with Greechie, R. J. and Bennett, M. K.): The transition to unigroups, Int. J. Theor. Phys. 37 (1998), 45–63. MR1637148 [47] Mathematical metascience, J. Nat. Geom. 13 (1998), 1–50. MR1484048 [48] (with Bennett, M. K.): Interval and scale effect algebras, Adv. Appl. Math. 19 (1997), 200–215. MR1459498 [49] (with Bennett, M. K. and Greechie, R. J.): Quotients of interval effect algebras, Int. J. Theor. Phys. 35 (1996), 2321–2338. MR1423409 [50] (with Greechie, R. J. and Dalla Chiara, M. L. and Giuntini, R.): Quantum Logic, Encyclopedia of Applied Physics 15 (1996), 229–255. [51] (with Bennett, M. K. and Greechie, R. J.): Test groups and effect algebras, Int. J. Theor. Phys. 35 (1996), 1117–1140. MR1392825 [52] (with Pt´ ak, P.): On absolutely compatible elements and hidden variables in quantum logics, Ric. Mat. 44 (1995), 19–29. MR1470184 [53] (with Bennett, M. K.): Phi-symmetric effect algebras, Found. Phys. 25 (1995), 1699–1722. MR1377109 [54] (with Greechie, R. J.): The Transition to effect algebras, Int. J. Theor. Phys. 34 (1995), 1369–1382. MR1353682 [55] (with Greechie, R. J. and Pulmannov´a, S.): The center of an effect algebra, Order 12 (1995), 91–106. MR1336539 [56] (with Pt´ ak, P.): On the tensor product of a Boolean algebra and an orthoalgebra, Czech. Math. J. 45 (1995), 117–126. MR1314534 [57] (with Greechie, R. J. and Bennett, M. K.): Sums and products of interval algebras, Int. J. Theor. Phys. 33 (1994), 2119–2136. MR1311152

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[58] (with Bennett, M. K.): Tensor products of quantum logics, The Interpretation of Quantum Theory: Where Do We Stand? (Luigi Accardi, ed.), Istituto Della Enciclopedia Italiana, Rome (1994) 303-316, ISBN 88-12-00019-3. [59] (with Bennett, M. K.): Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1331–1352. MR1304942 [60] (with Bennett, M. K.): Tensor products of orthoalgebras, Order 10 (1993), 271–282. MR1267193 [61] (with Greechie, R. J. and R¨ uttimann, G. T.): Logicoalgebraic structures. II: Supports in test spaces, Int. J. Theor. Phys. 32 (1993), 1675–1690. MR1255375 [62] (with Greechie, R. J. and R¨ uttimann, G. T.): Filters and supports in orthoalgebras, Int. J. Theor. Phys. 31 (1992), 789–807. MR1162623 [63] (with Bennett, M. K.): Superposition in quantum and classical mechanics, Found. Phys. 20 (1990), 733–744. MR1067801 [64] (with Schroeck jr., F. E.): Stochastic quantum mechanics viewed from the language of manuals, Found. Phys. 20 (1990), 823–858. MR1067217 [65] (with Kl¨ay, M. P.): Maximum likelihood estimation on generalized sample spaces: an alternative resolution of Simpson’s paradox, Found. Phys. 20 (1990), 777–799. MR1008686 [66] (with Bennett, M. K.): Charles Hamilton Randall: 1928–1987, Found. Phys. 20 (1990), 473–476. MR1060618 [67] Coupled physical systems, Found. Phys. 19 (1989), 905–922. MR1013911 [68] (with Kl¨ay, M. and Randall, C.): Tensor products and probability weights, Int. J. Theor. Phys. 26 (1987), 199–219. MR0892411 [69] (with Randall, C. H.): Dirac revisited. Symposium on the foundations of modern physics, 97–112, World Sci. Publishing, Singapore, 1985. MR0843846

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[70] (with Randall, C. H.): Stochastic entities. Recent developments in quantum logic, 265–284, Grundlagen Exakt. Naturwiss., 6, Bibliographisches Inst., Mannheim, 1985. MR0805578 [71] (with Randall, C. H.): A note on misunderstandings of Piron’s axioms for quantum mechanics, Found. Phys. 14 (1984), 65–88. MR0781028 [72] (with Randall, C. H.): Properties and operational propositions in quantum mechanics, Found. Phys. 13 (1983), 843–857. MR0788063 [73] (with Piron, C. and Randall, C.): Realism, operationalism, and quantum mechanics, Found. Phys. 13 (1983), 813–841. MR0788062 [74] (with Randall, C. H.): A mathematical language for quantum physics, Transactions of the 25 Cours de Perfectionnement de l’Association Vaudoise des Chercheurs in Physique; Les Fondements de la Mecanique Quantique, Montana, Switzerland (C. Gruber, C. Piron, T. Minhtom, R. Weil, eds.) (1983), 193-226. [75] (with Randall, C. H.): What are quantum logics and what ought they to be? Current issues in quantum logic (Ettore Majorana Internat. Sci. Ser.: Phys. Sci., 8), Plenum, New York-London, 1981, pp. 35–52. MR0723148 [76] (with Randall, C. H.): Operational statistics and tensor products. Interpretations and foundations of quantum theory, pp. 21–28, Grundlagen Exakt. Naturwiss., 5, Bibliographisches Inst., Mannheim, 1981. MR0683889 [77] (with Randall, C. H.): Empirical logic and tensor products. Interpretations and foundations of quantum theory, pp. 9–20, Grundlagen Exakt. Naturwiss., 5, Bibliographisches Inst., Mannheim, 1981. MR0683888 [78] (with Frazer, P. J. and Randall, C. H.): Weight functions on extensions of the compound manual, Glasgow Math. J. 21 (1980), 97–101. MR0582117 [79] (with Randall, C. H.): The operational approach to quantum mechanics. Physical theory as logicooperational structure, pp. 167–201, Univ. Western Ontario Ser. Philos. Sci., 7, Reidel, Dordrecht-Boston, Mass., 1979. MR0535608

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[80] (with Randall, C. H.): Tensor products of quantum logics do not exist, Notices Amer. Math. Soc. 26 (1979), A-557. [81] (with Randall, C. H.): Manuals, morphisms and quantum mechanics. Mathematical foundations of quantum theory, pp. 105–126. Academic Press, New York, 1978. MR0495822 [82] (with Randall, C. H.): A mathematical setting for inductive reasoning. With discussion. Foundations of probability theory, statistical inference, and statistical theories of science, Vol. III, pp. 169–205. Univ. Western Ontario Ser. Philos. Sci., Vol. 6, Reidel, Dordrecht, 1977. MR0498006 [83] (with Randall, C. H.): A mathematical setting for inductive reasoning. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science III (C. Hooker and W. Harper, eds.), D. Reidel Publishing Co., Dordrecht, Holland (1976), 169-205. [84] (with Randall, C. H.): The empirical logic approach to the physical sciences. Foundations of quantum mechanics and ordered linear spaces, pp. 230–249. Lecture Notes in Phys., Vol. 29,Springer, Berlin, (1974). MR0479146 [85] (with Randall, C. H.): Empirical Logic and Quantum Mechanics. Synthese 29 (1974), 81–111, also Logic and Probability in Quantum Mechanics (P. Suppes, ed.), D. Reidel Publishing Co., Dordrecht, Holland (1976). [86] (with Randall, C. H.): The stability of pure weights under conditioning, Glasg. Math. J. 15 (1974), 5–12. MR0359557 [87] (with Randall, C. H.): Operational statistics. II: Manuals of operations and their logics, J. Math. Phys. 14 (1973), 1472–1480. MR0416418 [88] (with Randall, C. H. and Janowitz, M. F.): Orthomodular generalizations of homogeneous Boolean algebras, J. Aust. Math. Soc. 15 (1973), 94–104. MR0319835 [89] (with Randall, C. H.): Operational statistics. I: Basic concepts, J. Math. Phys. 13 (1972), 1667–1675. MR0416417 [90] Free Baer ∗ -semigroups, Caribbean J. Sci. Math. 2 (1972), 25–30. MR0376911 1017

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[91] (with Randall, C. H.): States and the free orthogonality monoid, Math. Syst. Theory 6 (1972), 268–276. MR0332601 [92] (with Randall, C. H.): Conditioning maps on orthomodular lattices, Glasg. Math. J. 12 (1971), 35–42. MR0299540 [93] (with Randall, C. H.): Lexicographic orthogonality, J. Comb. Theory Ser. A 11 (1971), 157–162. MR0280420 [94] (with Randall, C. H.): An approach to empirical logic, Am. Math. Mon. 77 (1970), 363–374. MR0258688 [95] Multiplicative elements in Baer ∗ -semigroups, Math. Ann. 175 (1968), 297–302. MR0251156 [96] Semigroups co-ordinatizing orthomodular geometries, Can. J. Math. 17 (1965), 40–51. MR0204331 [97] Relative inverses in Baer ∗ -semigroups, Mich. Math. J. 10 (1963), 65–84. MR0154939 [98] A note on orthomodular lattices, Port. Math. 21 (1962), 65–72. MR0148581 [99] Conditions for the modularity of an orthomodular lattice, Pac. J. Math. 11 (1961), 889–895. MR0133270 [100] Baer ∗ -semigroups, Proc. Am. Math. Soc. 11 (1960), 648–654. MR0125808 [101] Involution Semigroups. Thesis (Ph.D.), Tulane University. (1958), pp. 126, MR2612723 [102] (with Halmos, P. R.): Advanced Problems and Solutions: Solutions: 4658, Amer. Math. Monthly 64 (1957), 50. MR1529514 Department of Mathematics and Statistics P.O. Box #10348 Louisiana Tech University Ruston, LA 71272 U.S.A E-mail : [email protected]

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