Orthomodular lattices, Foulis Semigroups and Dagger Kernel Categories

This paper is a sequel to arXiv:0902.2355 and continues the study of quantum logic via dagger kernel categories. It develops the relation between these categories and both orthomodular lattices and Foulis semigroups. The relation between the latter two notions has been uncovered in the 1960s. The current categorical perspective gives a broader context and reconstructs this relationship between orthomodular lattices and Foulis semigroups as special instance.

[1]  J. D. M. Wright.  GLEASON'S THEOREM AND ITS APPLICATIONS , 1995 .

[2]  Zahava Shmuely.  The tensor product of distributive lattices , 1979 .

[3]  P. Freyd.  Abelian categories : an introduction to the theory of functors , 1965 .

[4]  G. Rota.  Introduction to higher order categorical logic , 1988 .

[5]  Bas Spitters,et al.  Mathematical Physics A Topos for Algebraic Quantum Theory , 2009 .

[6]  C. J. Isham,et al.  Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations , 1998, quant-ph/9803055.

[7]  This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivs 3.0 Licence. To view a copy of the licence please see: http://creativecommons.0rg/licenses/by-nc-nd/3.0/ INEQIMTES IN THE DELIVERY OF SERVICES TO A FEMALE FARM CLIENTELE: SOME~~ , 2010 .

[8]  Susumu Hayashi,et al.  Adjunction of Semifunctors: Categorical Structures in Nonextensional Lambda Calculus , 1985, Theor. Comput. Sci..

[9]  M. Karoubi.  K-Theory: An Introduction , 1978 .

[10]  P. D. Finch,et al.  Quantum logic as an implication algebra , 1970, Bulletin of the Australian Mathematical Society.

[11]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[12]  Peter Selinger,et al.  Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract) , 2007, QPL.

[13]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[14]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[15]  Kôdi Husimi.  Studies on the Foundation of Quantum Mechanics. I , 1937 .

[16]  G. Kalmbach.  On Orthomodular Lattices , 1990 .

[17]  Raymond Hoofman.  Non-Stable Models of Linear Logic , 1992, LFCS.

[18]  S. A. Selesnick,et al.  An extension of the Stone representation for orthomodular lattices , 1973 .

[19]  Alexandru Baltag,et al.  LQP: the dynamic logic of quantum information , 2006, Mathematical Structures in Computer Science.

[20]  C. J. Isham,et al.  A topos foundation for theories of physics: I. Formal languages for physics , 2007 .

[21]  Bart Jacobs.  Semantics of the Second Order Lambda Calculus , 1991, Math. Struct. Comput. Sci..

[22]  David J. Foulis,et al.  Relative inverses in Baer *-semigroups. , 1963 .

[23]  Ieke Moerdijk,et al.  A Remark on the Theory of Semi-Functors , 1995, Math. Struct. Comput. Sci..

[24]  David J. Foulis.  Baer $\sp{\ast} $-semigroups , 1960 .

[25]  Grant A. Fraser.  The tensor product of distributive lattices , 1976 .

[26]  Bart Jacobs,et al.  Quantum Logic in Dagger Kernel Categories , 2009, QPL@MFPS.

[27]  J. Neumann,et al.  The Logic of Quantum Mechanics , 1936 .

[28]  C. J. Isham,et al.  A Topos Foundation for Theories of Physics: IV. Categories of Systems , 2008 .

[29]  George A. Elliott,et al.  K-theory , 1999 .

[30]  Peter Selinger,et al.  Idempotents in Dagger Categories: (Extended Abstract) , 2008, QPL.

[31]  Bob Coecke,et al.  The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes , 2001 .

[32]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[33]  C. J. Isham,et al.  A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues , 1998 .

[34]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[35]  Sylvia Pulmannová,et al.  New trends in quantum structures , 2000 .