Skip to main content
SpringerLink
Log in

Dependences Between Domain Constructions in Heterogeneous Relation Algebras

  • Conference paper
  • First Online:
Relational and Algebraic Methods in Computer Science (RAMiCS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13896))

  • 222 Accesses

Abstract

We show the following dependences between relational domain constructions in the framework of heterogeneous relation algebras. If all power sets and subsets exist and objects are comparable, then all sums exist. If all sums exist and atoms are rectangular, then all products exist. If all atoms are rectangular, then all subsets exist if and only if all quotients exist. We give models with rectangular atoms which rule out further dependences between these constructions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats, 2nd edn. Dover Publications (2009)

    Google Scholar 

  2. Berghammer, R., Schmidt, G., Zierer, H.: Symmetric quotients and domain constructions. Inf. Process. Lett. 33(3), 163–168 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berghammer, R., Winter, M.: Embedding mappings and splittings with applications. Acta Inf. 47(2), 77–110 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bird, R., de Moor, O.: Algebra of Programming. Prentice Hall, Hoboken (1997)

    MATH  Google Scholar 

  5. Desharnais, J.: Abstract relational semantics. Ph.D. thesis, McGill University (1989)

    Google Scholar 

  6. Freyd, P.J., Ščedrov, A.: Categories, Allegories, North-Holland Mathematical Library, vol. 39. Elsevier Science Publishers (1990)

    Google Scholar 

  7. Givant, S., Halmos, P.: Introduction to Boolean Algebras. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  8. Guttmann, W.: Multirelations with infinite computations. J. Logical Algebraic Methods Program. 83(2), 194–211 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Maddux, R.D.: Pair-dense relation algebras. Trans. Am. Math. Soc. 328(1), 83–131 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Olivier, J.P., Serrato, D.: Catégories de Dedekind: Morphismes transitifs dans les catégories de Schröder. Comptes rendus hebdomadaires des séances de l’Académie des Sciences, Série A 290, 939–941 (1980)

    Google Scholar 

  11. Schmidt, G.: Relational Mathematics. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  12. Schmidt, G., Hattensperger, C., Winter, M.: Heterogeneous relation algebra. In: Brink, C., Kahl, W., Schmidt, G. (eds.) Relational Methods in Computer Science, pp. 39–53. Springer, Wien (1997). https://doi.org/10.1007/978-3-7091-6510-2_3

    Chapter  Google Scholar 

  13. Schmidt, G., Ströhlein, T.: Relationen und Graphen. Springer, Heidelberg (1989)

    Book  MATH  Google Scholar 

  14. Schmidt, G., Winter, M.: Relational Topology. LNM, vol. 2208. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-74451-3

    Book  MATH  Google Scholar 

  15. Tarski, A.: On the calculus of relations. J. Symb. Logic 6(3), 73–89 (1941)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  17. Winter, M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdeterminismus in Programmiersprachen. Ph.D. thesis, Universität der Bundeswehr München (1998)

    Google Scholar 

  18. Winter, M.: Weak relational products. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 417–431. Springer, Heidelberg (2006). https://doi.org/10.1007/11828563_28

    Chapter  Google Scholar 

  19. Winter, M.: Products in categories of relations. J. Logic Algebraic Program. 76(1), 145–159 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

I thank the anonymous referees for pointing out related works and other helpful comments.

Author information

Authors and Affiliations

  1. Department of Computer Science and Software Engineering, University of Canterbury, Christchurch, New Zealand

    Walter Guttmann

Authors
  1. Walter Guttmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Walter Guttmann .

Editor information

Editors and Affiliations

  1. Deutsches Zentrum für Luft- und Raumfahrt, Augsburg, Germany

    Roland Glück

  2. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  3. Brock University, St Catharines, ON, Canada

    Michael Winter

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Guttmann, W. (2023). Dependences Between Domain Constructions in Heterogeneous Relation Algebras. In: Glück, R., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2023. Lecture Notes in Computer Science, vol 13896. Springer, Cham. https://doi.org/10.1007/978-3-031-28083-2_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-28083-2_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-28082-5

  • Online ISBN: 978-3-031-28083-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation