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Closure categories

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Abstract

In this work I present a condensed, but self-contained review of the categorical formulation of order structures, and the induced equivalence of the categories of closure spaces and complete atomistic lattices.

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References

  • Adámek, J., Herrlich, H., and Strecker, G. E. (1990).Abstract and Concrete Categories, Wiley, New York.

    MATH  Google Scholar 

  • Aerts, D. (1982). Description of many separated physical entities without the paradoxes encountered in quantum mechanics,Foundations of Physics,12, 1131–1170.

    Article  MathSciNet  ADS  Google Scholar 

  • Alderton, I. W. (1985). Cartesian closedness and the MacNeille completion of an initially structured category.Quaestiones Mathematicae,8, 63–78.

    Article  MATH  MathSciNet  Google Scholar 

  • Alderton, I. W. (1986). Initial completions of monotopological categories, and cartesian closedness,Quaestiones Mathematicae,8, 361–379.

    Article  MATH  MathSciNet  Google Scholar 

  • Baer, R. M. (1959). On closure operators.Archives of Mathematics,10, 261–266.

    Article  MATH  MathSciNet  Google Scholar 

  • Banaschewski, B., and Bruns, G. (1967). Categorical characterization of the MacNeille completion,Archives of Mathematics,18, 369–377.

    Article  MATH  MathSciNet  Google Scholar 

  • Barr, M., and Wells, C. (1985).Toposes, Triples and Theories, Springer-Verlag, New York.

    MATH  Google Scholar 

  • Borceux, F. (1994).Handbook of Categorical Algebra, Cambridge University Press, Cambridge.

    Google Scholar 

  • Cagliari, F., and Cicchese, M. (1983). Epireflective subcategories and semiclosure operators,Quaestiones Mathematicae,6, 295–301.

    MATH  MathSciNet  Google Scholar 

  • Čech, E. (1937). On bicompactspaces,Annals of Mathematics,38, 823–844.

    Article  MathSciNet  Google Scholar 

  • Dikranjan, D., and Giuli, E. (1987). Closure operators,Topology and Its Applications,27, 129–143.

    Article  MATH  MathSciNet  Google Scholar 

  • Dikranjan, D., and Tholen, W. (1995).Categorical Structure of Closure Operators, Kluwer, Dordrecht.

    MATH  Google Scholar 

  • Eilenberg, S., and Mac Lane, S. (1945). General theory of natural equivalences.Transactions AMS,58, 231–294.

    Article  MATH  Google Scholar 

  • Eilenberg, S., and Moore, J. C. (1965). Adjoint functors and triples,Illinois Journal of Mathematics,9, 381–398.

    MATH  MathSciNet  Google Scholar 

  • Everett, C. J. (1944). Closure operators and Galois theory in lattices,Transactions AMS,55, 514–525.

    Article  MATH  MathSciNet  Google Scholar 

  • Faure, Cl.-A. (1994). Categories of closure spaces and corresponding lattices,Cahiers de Topologie et Geometrie Differentialle Categoriques,35, 309–319.

    MATH  MathSciNet  Google Scholar 

  • Faure, Cl.-A., and Frölicher, A. (1993). Morphisms of projective geometries and of corresponding lattices,Geometriae Dedicata,47, 25–40.

    Article  MATH  MathSciNet  Google Scholar 

  • Faure, Cl.-A., and Frölicher, A. (1994). Morphisms of projective geometries and semilinear maps,Geometriae Dedicata,53, 937–969.

    Article  Google Scholar 

  • Faure, Cl.-A., and Frölicher, A. (1995). Dualities for infinite-dimensional projective geometries,Geometriae Dedicata,56, 225–236.

    Article  MATH  MathSciNet  Google Scholar 

  • Faure, Cl.-A., and Frölicher, A. (1996). The dimension theorem in axiomatic geometry.Geometriae Dedicata,66, 207–218.

    Google Scholar 

  • Faure, Cl.-A., Moore, D. J., and Piron, C. (1995). Deterministic evolutions and Schrödinger flows,Helvetica Physica Acta,68, 150–157.

    MATH  MathSciNet  Google Scholar 

  • Foulis, D. J. (1960). Baer*-semigroups,Proceedings AMS,11, 648–654.

    Article  MATH  MathSciNet  Google Scholar 

  • Godement, R. (1957).Théorie des faisceaux, Hermann, Paris.

    Google Scholar 

  • Gudder, S. P., and Michel, J. R. (1981). Representations of Baer*-semigroups,Proceedings AMS,81, 157–163.

    Article  MATH  MathSciNet  Google Scholar 

  • Herrlich, H. (1976). Initial completions,Mathematische Zeitschrif,150, 101–110.

    Article  MATH  MathSciNet  Google Scholar 

  • Hertz, P. (1922). Über Axiomensysteme für beliebige Satzsysteme. I,Mathematische Annalen,87, 246–269.

    Article  MATH  MathSciNet  Google Scholar 

  • Huber, P. J. (1961). Homotopy theory in general categories,Mathematische Annalen,144, 361–385.

    Article  MATH  MathSciNet  Google Scholar 

  • Kleisli, H. (1965). Every standard construction is induced by a pair of adjoint functors.Proceedings AMS,16, 544–546.

    Article  MATH  MathSciNet  Google Scholar 

  • Kuratowski, C. (1922). Sur l'opération\(\bar A\) de l'Analyse Situs,Fundamenta Mathematicae,3, 182–199.

    MATH  Google Scholar 

  • Mac Lane, S. (1971).Categories for the Working Mathematician, Springer-Verlag, New York.

    MATH  Google Scholar 

  • MacNeille, H. M. (1937). Partially ordered sets,Transactions AMS,42, 416–460.

    Article  MATH  MathSciNet  Google Scholar 

  • Monteiro, A., and Ribeiro, H. (1942). L'opération de fermeture et ses invariants dans les systèmes partiellement ordonnés,Portugaliae Mathematica,3, 171–183.

    MATH  MathSciNet  Google Scholar 

  • Moore, D. J. (1995). Categories of representations of physical systems.Helvetica Physica Acta,68, 658–678.

    MATH  MathSciNet  Google Scholar 

  • Moore, E. H. (1909). On a form of general analysis, with applications to linear differential and integral equations, inAtti del IV Congress. Internationale dei Matematici (Roma, 6–11 Aprile 1908) [reprinted Kraus, Nendeln (1967)].

  • Ore, O. (1943a). Some studies on closure relations,Duke Mathematical Journal,10, 761–785.

    Article  MATH  MathSciNet  Google Scholar 

  • Ore, O. (1943b). Combinations of closure relations,Annals of Mathematics,44, 514–533.

    Article  MATH  MathSciNet  Google Scholar 

  • Piron, C. (1976).Foundations of Quantum Physics, Benjamin, Reading, Massachusetts.

    MATH  Google Scholar 

  • Piron, C. (1990).Mécanique quantique bases et applications, Presses polytechniques et universitaires Romandes, Lausanne, Switzerland.

    MATH  Google Scholar 

  • Piron, C. (1995). Morphisms, hemimorphisms and Baer*-semigroups.International Journal of Theoretical Physics,34, 1681–1687.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Pool, J. C. T. (1968a). Baer*-semigroups and the logic of quantum mechanics,Communications in Mathematical Physics,9, 118–141.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Pool, J. C. T. (1968b). Semimodularity and the logic of quantum mechanics,Communications in Mathematical Physics,9, 212–228.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Riesz, F. (1909). Stetigkeitsbegriff une abstrakte Mengenlehre, inAtti del IV Congress. Internazional dei Matematici (Roma, 6–11 Aprile 1908) [reprint Kraus, Nedeln (1967)].

  • Riguet, J. (1948). Relations binaires, fermetures, correspondence de Galois,Bulletin de la Société Mathématique de France,76, 114–155.

    MATH  MathSciNet  Google Scholar 

  • Rütimann, G. T. (1975). Decomposition of projections on orthomodular lattices,Canadian Mathematical Bulletin,18, 263–267.

    MathSciNet  Google Scholar 

  • Tarski, A. (1929). Remarques sur les notions fondamentales de la méthodologie des mathématique,Annales de la Société Polonaise de Mathématique,7, 270–272.

    MATH  Google Scholar 

  • Ward, M. (1942). The closure operators of a lattice,Annals of Mathematics,43, 191–196.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Département de Physique Théorique, CH-1211, Geneva 4, Switzerland

    D. J. Moore

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Moore, D.J. Closure categories. Int J Theor Phys 36, 2707–2723 (1997). https://doi.org/10.1007/BF02435707

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