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International Conference on Relational and Algebraic Methods in Computer Science

RAMiCS 2021: Relational and Algebraic Methods in Computer Science pp 483–498Cite as

Domain Range Semigroups and Finite Representations

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 13027)

Abstract

Relational semigroups with domain and range are a useful tool for modelling nondeterministic programs. We prove that the representation class of domain-range semigroups with demonic composition is not finitely axiomatisable. We extend the result for ordered domain algebras and show that any relation algebra reduct signature containing domain, range, converse, and composition, but no negation, meet, nor join has the finite representation property. That is any finite representable structure of such a signature is representable over a finite base. We survey the results in the area of the finite representation property.

Keywords

  • Domain-range semigroups
  • Demonic composition
  • Finite representation property

J. Šemrl—The author thanks Professor Robin Hirsch for supervision and insightful conversations about the work presented.

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References

  1. Andréka, H.: On the representation problem of distributive semilattice-ordered semigroups. In: Mathematical Institute of the Hungarian Academy of Sciences, p. 174 (1988, preprint)

    Google Scholar 

  2. Bredihin, D.A., Schein, B.M.: Representations of ordered semigroups and lattices by binary relations. In: Colloquium Mathematicum, vol. 39, pp. 1–12. Institute of Mathematics Polish Academy of Sciences (1978)

    Google Scholar 

  3. Desharnais, J., Jipsen, P., Struth, G.: Domain and antidomain semigroups. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) RelMiCS 2009. LNCS, vol. 5827, pp. 73–87. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04639-1_6

    CrossRef  MATH  Google Scholar 

  4. Dijkstra, E.W., Scholten, C.S.: Predicate Calculus and Program Semantics. Springer, New York (2012). https://doi.org/10.1007/978-1-4612-3228-5

  5. Hirsch, R., Egrot, R.: Meet-completions and representations of ordered domain algebras. J. Symb. Log. (2013)

    Google Scholar 

  6. Hirsch, R., Hodkinson, I.: Relation Algebras by Games. Elsevier, Amsterdam (2002)

    MATH  Google Scholar 

  7. Hirsch, R., Mikulás, S.: Axiomatizability of representable domain algebras. J. Logic Algebraic Programm. 80(2), 75–91 (2011)

    CrossRef  MathSciNet  Google Scholar 

  8. Hirsch, R., Mikulás, S., Stokes, T.: The algebra of non-deterministic programs: demonic operators, orders and axioms. arXiv preprint arXiv:2009.12081 (2020)

  9. Hirsch, R., Šemrl, J.: Demonic lattices and semilattices in relational semigroups with ordinary composition. In: Proceedings of the 36th Annual Symposium on Logic in Computer Science, LICS, Rome, Italy. IEEE (2021)

    Google Scholar 

  10. Hirsch, R., Šemrl, J.: Finite representability of semigroups with demonic refinement. Algebra Univers. 82(2), 1–14 (2021). https://doi.org/10.1007/s00012-021-00718-5

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Hirsch, R., Stokes, T.: Axioms for signatures with domain and demonic composition. Algebra Univers. 82(2), 1–19 (2021). https://doi.org/10.1007/s00012-021-00719-4

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Jackson, M., Mikulás, S.: Domain and range for angelic and demonic compositions. J. Log. Algebraic Meth. Program. 103, 62–78 (2019)

    CrossRef  MathSciNet  Google Scholar 

  13. Mikulás, S.: Axiomatizability of algebras of binary relations. In: Classical and New Paradigms of Computation and Their Complexity Hierarchies, pp. 187–205. Springer, Cham (2004). https://doi.org/10.1007/978-1-4020-2776-5_11

  14. Neuzerling, M.: Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups. Algebra Univers. 76(4), 431–443 (2016). https://doi.org/10.1007/s00012-016-0409-9

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Rogozin, D.: The finite representation property for representable residuated semigroups. arXiv preprint arXiv:2007.13079 (2020)

  16. Zareckiĭ, K.A.: The representation of ordered semigroups by binary relations. Izvestiya Vysšhikh. Uchebnykh. Zavedeniı. Matematika 6(13), 48–50 (1959)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University College London, Gower Street, London, WC1E 6BT, UK

    Jaš Šemrl

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  1. Jaš Šemrl
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Correspondence to Jaš Šemrl .

Editor information

Editors and Affiliations

  1. École polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Université Côte d’Azur, Nice, France

    Mai Gehrke

  3. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  4. Brock University, St Catharines, ON, Canada

    Prof. Dr. Michael Winter

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Šemrl, J. (2021). Domain Range Semigroups and Finite Representations. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_29

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  • DOI: https://doi.org/10.1007/978-3-030-88701-8_29

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