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Ordered Sets and Complete Lattices

A Primer for Computer Science

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Book cover Algebraic and Coalgebraic Methods in the Mathematics of Program Construction

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2297))

Abstract

These notes deal with an interconnecting web of mathematical techniques all of which deserve a place in the armoury of the welleducated computer scientist. The objective is to present the ideas as a self-contained body of material, worthy of study in its own right, and at the same time to assist the learning of algebraic and coalgebraic methods, by giving prior familiarization with some of the mathematical background that arises there. Examples drawn from computer science are only hinted at: the presentation seeks to complement and not to preempt other contributions to these ACMMPC Proceedings.

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References

  1. C. J. Aarts. Galois connections presented calculationally. Master’s thesis, Eindhoven University of Technology, 1992. 52

    Google Scholar 

  2. Samson Abramsky and Achim Jung. Domain theory. In Handbook of Logic in Computer Science, volume 3, pages 1–168. Oxford University Press, 1994. 21, 72

    MathSciNet  Google Scholar 

  3. R. Backhouse, M. Bijsterveld, R. van Geldrop, and J. van der Woude. Categorical fixed point rules. http://www.win.tue.nl/pm/papers/abstract.htmlcatfixpnt, 1995. 77

  4. G. Birkhoff. Lattice Theory. American Mathematical Society, third edition, 1967. 52

    Google Scholar 

  5. C. Brink and I. M. Rewitzky. Power structures and program semantics. Monograph preprint, 1997. 43

    Google Scholar 

  6. B. A. Davey and H. A. Priestley. Distributive lattices and duality. In G. Grätzer, editor, General Lattice Theory, pages 499–517. Birkhäuser Verlag, second edition, 1998. 43, 77

    Google Scholar 

  7. A. Edelat. Domains for computation, physics and exact real arithmetic. Bulletin of Symbolic Logic, 3:401–452, 1997. 30

    Article  MathSciNet  Google Scholar 

  8. B. Ganter and R. Wille. Formal Concept Analysis. Springer-Verlag, 1999. 26, 52, 66, 67

    Google Scholar 

  9. G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, and D. S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, 1980. 21, 52, 72

    Google Scholar 

  10. B. Jónsson. A survey of Boolean algebras with operators. In I. G. Rosenberg and G. Sabidussi, editors, Algebras and Orders, volume 389 of ASI Series C, pages 239–286. NATO, 1993. 53

    Google Scholar 

  11. J.-L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixedpoint theorems and semantics: A folk tale. Information Processing Letters, 14:112–116, 1982. 68

    Article  MATH  MathSciNet  Google Scholar 

  12. Saunders Mac Lane. Categories for the Working Mathematician. Springer-Verlag, 1971. 76

    Google Scholar 

  13. G. Markowsky. Chain-complete posets and directed sets with applications. Algebra Universalis, 6:53–68, 1976. 74

    Article  MATH  MathSciNet  Google Scholar 

  14. R. N. McKenzie, G. F. McNulty, and W. E. Taylor.Algebras, Lattices, Varieties, volume I. Wadsworth & Brooks, 1987. 54

    Google Scholar 

  15. A. W. Roscoe. The Theory and Practice of Concurrency. Prentice-Hall International, 1997. 73

    Google Scholar 

  16. J. J. M. M. Rutten. A calculus of transition systems: Towards universal coalgebra. In A. Ponse, M. de Rijke, and Y. Venema, editors, Modal Logic and Process Algebra, volume 53 of CSLI Lecture Notes, pages 231–256. CSLI Publications, Stanford, 1995. 27

    Google Scholar 

  17. M. Schader, editor. Analysing and Modelling Data and Knowledge. Springer-Verlag, 1992. 67

    Google Scholar 

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Priestley, H.A. (2002). Ordered Sets and Complete Lattices. In: Backhouse, R., Crole, R., Gibbons, J. (eds) Algebraic and Coalgebraic Methods in the Mathematics of Program Construction. Lecture Notes in Computer Science, vol 2297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47797-7_2

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  • DOI: https://doi.org/10.1007/3-540-47797-7_2

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  • Print ISBN: 978-3-540-43613-3

  • Online ISBN: 978-3-540-47797-6

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