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Embedding mappings and splittings with applications

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Abstract

We present relation-algebraic specifications of injective embedding mappings and splittings of partial equivalence relations and show in each case that the axioms characterize these constructions up to isomorphism, i.e., in an essentially unique way. Based on the specifications, we develop a relational program for computing splitting and demonstrate some applications. The examples originate from a relation-algebraic treatment of processes, graph theory, and the decomposition of specific relations.

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References

  1. Behnke R., Berghammer R., Meyer E., Schneider P.: RelView–a system for calculation with relations and relational programming. In: Astesiano, E. (eds) Fundamental Approaches to Software Engineering, Lecture Notes in Computer Science, vol. 1382, pp. 318–321. Springer, Berlin (1998)

    Chapter  Google Scholar 

  2. Berghammer R.: Computation of cut completions and concept lattices using relation algebra and RelView. J. Relat. Methods Comput. Sci. 1, 50–72 (2004)

    Google Scholar 

  3. Berghammer R., Neumann F.: RelView—an OBDD-based computer algebra system for relations. In: Gansha, V.G., Mayr, E.W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, vol. 3718, pp. 40–51. Springer, Berlin (2005)

    Google Scholar 

  4. Berghammer R., Schmidt G.: Algebraic visualization of relations using RelView. In: Gansha, V.G., Mayr, E.W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, vol. 4770, pp. 58–72. Springer, Berlin (2007)

    Google Scholar 

  5. Bona M.: A Walk Through Combinatorics: An Introduction to Combinatorics and Graph Theory. World Scientific Publishing, New Jersey (2002)

    MATH  Google Scholar 

  6. Brink, C., Kahl, W., Schmidt, G. (eds): Relational Methods in Computer Science, Advances in Computing Science. Springer, Wien (1997)

    Google Scholar 

  7. Desharnais J.: Monomorphic characterization of the n-ary direct product. Inf. Sci. 119, 275–288 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. de Swart, H., Orlowska, E., Schmidt, G., Roubens, M. (eds): Theory and Applications of Relational Structures as Knowledge Instruments. Lecture Notes in Computer Science, vol. 2929. Springer, Berlin (2003)

    Google Scholar 

  9. de Swart, H., Orlowska, E., Schmidt, G., Roubens, M. (eds): Theory and Applications of Relational Structures as Knowledge Instruments II: Lecture Notes in Artifical Intelligence, vol. 4342. Springer, Berlin (2006)

    Google Scholar 

  10. Doornbos H., Backhouse R., van der Woude J.: A calculational approach to mathematical induction. Theor. Comput. Sci. 179(1–2), 103–135 (1997)

    Article  MATH  Google Scholar 

  11. Freyd P., Scedrov A.: Categories, Allegories. North-Holland, Amsterdam (1990)

    MATH  Google Scholar 

  12. Fishburn P.: On the construction of weak orders from fragmentary information. Psychometrika 38, 459–472 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fishburn P.: Interval Orders and Interval Graphs. Wiley, New York (1985)

    MATH  Google Scholar 

  14. Ganter B., Wille R.: Formal Concept Analysis. Springer, Berlin (1999)

    MATH  Google Scholar 

  15. Golumbic M.: Algorithmic Graph-Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  16. Gries D.: The Science of Computer Programming. Springer, New York (1981)

    Google Scholar 

  17. Guttmann L.: A basis for scaling qualitative data. Am. Sociol. Rev. 9, 139–150 (1944)

    Article  Google Scholar 

  18. Haeberer A., Frias M., Baum G., Veloso P.: Fork algebras. In: Brink, C., Kahl, W., Schmidt, G. (eds) Relational Methods in Computer Science, Advances in Computing Science, pp. 54–69. Springer, Wien (1997)

    Google Scholar 

  19. Maddux R.D.: Some sufficient conditions for the representability of relation algebras. Algebra Universalis 8, 162–172 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  20. Milner R.: A Calculus of Communicating Systems. Lecture Notes in Computer Science, vol. 92. Springer, Berlin (1980)

    Google Scholar 

  21. Mitas J.: Tackling the jump number of interval orders. Order 8, 115–132 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Tsoukias M., Tsoukias A., Vincke P.: Preference modelling. In: Ehrgott, M., Greco, S., Figueira, J. (eds) Multiple Criteria Decision Analysis: State of the Art Surveys, International Series in Operatios Research and Management Science, vol. 78, pp. 27–71. Springer, New York (2005)

    Google Scholar 

  23. Schmidt G.: Rectangles, fringes, and inverses. In: Berghammer, R., Möller, B., Struth, G. (eds) Relations and Kleene Algebra in Computer Science. Lecture Notes in Computer Science, vol. 4988, pp. 352–366. Springer, Berlin (2008)

    Chapter  Google Scholar 

  24. Schmidt G., Ströhlein T.: Relationen und Graphen. Springer (1989); English version: Relations and graphs. Discrete mathematics for computer scientists, EATCS Monographs on Theoretical Computer Science, Springer, Berlin (1993)

  25. Tarski A.: On the calculus of relations. J. Symb. Log. 6, 73–89 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tarski A., Givant S.: A Formalization of Set Theory Without Variables. Colloquium Publications 41. American Mathematical Society, Providence (1987)

    Google Scholar 

  27. Winter, M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertation, Fakultät für Informatik, Universität der Bundeswehr München, Dissertationsverlag NG Kopierladen GmbH (1998)

  28. Winter M.: Decomposing relations into orderings. In: Berghammer, R., Möller, B., Struth, G. (eds) Relational and Kleene-Algebraic Methods in Computer Science. Lecture Notes in Computer Science, vol. 3051, pp. 265–277. Springer, Berlin (2004)

    Google Scholar 

  29. Winter M.: A relation-algebraic theory of bisimulations. Fundam. Inform. 83(1), 1–21 (2008)

    MathSciNet  Google Scholar 

  30. Winter M.: An ordered category of processes. In: Berghammer, R., Möller, B., Struth, G. (eds) Relations and Kleene Algebra in Computer Science. Lecture Notes in Computer Science, vol. 4988, pp. 367–381. Springer, Berlin (2008)

    Chapter  Google Scholar 

  31. Zierer H.: Relation-algebraic domain constructions. Theor. Comput. Sci. 87, 163–188 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität zu Kiel, 24098, Kiel, Germany

    Rudolf Berghammer

  2. Department of Computer Science, Brock University, St. Catharines, ON, L2S 3A1, Canada

    Michael Winter

Authors
  1. Rudolf Berghammer
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  2. Michael Winter
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Correspondence to Rudolf Berghammer.

Additional information

The second author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada.

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Berghammer, R., Winter, M. Embedding mappings and splittings with applications. Acta Informatica 47, 77–110 (2010). https://doi.org/10.1007/s00236-009-0109-4

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  • DOI: https://doi.org/10.1007/s00236-009-0109-4

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