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Hypomorphic Sperner Systems and Non-Reconstructible Functions

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Abstract

A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification minors. Sperner systems being representations of certain monotone functions, infinite families of non-reconstructible functions are thus obtained. The clones of Boolean functions are completely classified in regard to reconstructibility.

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References

  1. Berge, C., Rado, R.: Note on isomorphic hypergraphs and some extensions of Whitney’s theorem to families of sets. J. Comb. Theory Ser. B 13, 226–241 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Couceiro, M., Lehtonen, E.: The arity gap of polynomial functions over bounded distributive lattices. In: 40th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2010), pp. 113–116. IEEE Computer Society, Los Alamitos (2010)

    Book  Google Scholar 

  3. Couceiro, M., Lehtonen, E., Waldhauser, T.: The arity gap of order-preserving functions and extensions of pseudo-Boolean functions. Discrete Appl. Math. 160, 383–390 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Couceiro, M., Marichal, J.-L.: Polynomial functions over bounded distributive lattices. J. Mult.-Valued Logic Soft Comput. 18, 247–256 (2012)

    MATH  MathSciNet  Google Scholar 

  5. Denecke, K., Wismath, S.L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman & Hall/CRC, Boca Raton (2002)

    MATH  Google Scholar 

  6. Ellingham, M.N.: Recent progress in edge-reconstruction. Seventeenth Manitoba Conference on Numerical Mathematics and Computing. Congr. Numer. 62, 3–20 (1988)

    MathSciNet  Google Scholar 

  7. Foldes, S., Pogosyan, G.R.: Post classes characterized by functional terms. Discrete Appl. Math. 142, 35–51 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Jablonski, S.W., Gawrilow, G.P., Kudrjawzew, W.B.: Boolesche Funktionen und Postsche Klassen. Vieweg, Braunschweig (1970)

    MATH  Google Scholar 

  9. Goodstein, R.L.: The solution of equations in a lattice. Proc. R. Soc. Edinb. Sect. A 67, 231–242 (1965/1967)

    MATH  MathSciNet  Google Scholar 

  10. Harary, F.: On the reconstruction of a graph from a collection of subgraphs. In: Theory of Graphs and Its Applications (Proc. Sympos. Smolenice, 1963), pp. 47–52,. Publ. House Czechoslovak Acad. Sci., Prague (1964)

  11. Kelly, P.J.: On Isometric Transformations. Ph.D. thesis, University of Wisconsin (1942)

  12. Kocay, W.L.: A family of nonreconstructible hypergraphs. J. Combin. Theory Ser. B 42, 46–63 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kocay, W.L., Lui, Z.M.: More non-reconstructible hypergraphs. Discrete Math. 72, 213–224 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lehtonen, E.: Totally symmetric functions are reconstructible from identification minors. Electron. J. Combin. 21(2), P2.6 (2014)

  15. Lehtonen, E.: Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings. Internat. J. Algebra Comput. 24, 11–31 (2014)

  16. Post, E.L.: The Two-Valued Iterative Systems of Mathematical Logic. Annals of Mathematical Studies, vol. 5. Princeton University Press, Princeton (1941)

    Google Scholar 

  17. Stockmeyer, P.K.: A census of nonreconstructible digraphs. I. Six related families. J. Combin. Theory Ser. B 31, 232–239 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)

    MATH  Google Scholar 

  19. Willard, R.: Essential arities of term operations in finite algebras. Discrete Math. 149, 239–259 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. LAMSADE–CNRS, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Miguel Couceiro

  2. Computer Science and Communications Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg

    Erkko Lehtonen

  3. Centro de Álgebra da Universidade de Lisboa, Avenida Professor Gama Pinto 2, 1649-003, Lisbon, Portugal

    Erkko Lehtonen

  4. Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Erkko Lehtonen

  5. Mathematics Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg

    Karsten Schölzel

Authors
  1. Miguel Couceiro
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  2. Erkko Lehtonen
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  3. Karsten Schölzel
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Correspondence to Erkko Lehtonen.

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Couceiro, M., Lehtonen, E. & Schölzel, K. Hypomorphic Sperner Systems and Non-Reconstructible Functions. Order 32, 255–292 (2015). https://doi.org/10.1007/s11083-014-9330-z

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  • DOI: https://doi.org/10.1007/s11083-014-9330-z

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