Skip to main content
SpringerLink
Log in

Relations among Matrices over a Semiring

  • Conference paper
  • First Online:
Relational and Algebraic Methods in Computer Science (RAMICS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9348))

Abstract

If I ×(S) denotes the set of (multiplicative) idempotent elements of a commutative semiring S, then a matrix over S is idempotent with respect to the Hadamard product iff all its coefficients are in I ×(S). Since the collection of idempotent matrices can be seen as an embedded structure of binary relations inside the category of matrices over S, we are interested in the relationship between the two structures. In particular, we are interested under which properties the idempotent matrices form a (distributive) allegory.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, M., Feil, T.: Turning Lights Out with Linear Algebra. Mathematics Magazine 71(4), 300–303 (1998)

    Article  MathSciNet  Google Scholar 

  2. Berghammer, R., Neumann, F.: RelView – an OBDD-based computer algebra system for relations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 40–51. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  3. Birkhoff, G.: Lattice Theory, 3rd edn., vol. XXV. American Mathematical Society Colloquium Publications (1940)

    Google Scholar 

  4. Cayley, A.: A Memoir on the Theory of Matrices. Phil. Trans. of the Royal Soc. of London 148(1), 17–37 (1858)

    Article  Google Scholar 

  5. Desharnais, J., Grinenko, A., Möller, B.: Relational style laws and constructs of linear algebra. JLAMP 83(2), 154–168 (2014)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  7. Golan, J.S.: Semirings and their Application. Kluwer (1999)

    Google Scholar 

  8. Grätzer, G.: Universal Algebra, 2nd edn. Springer (2008)

    Google Scholar 

  9. Hebish, U., Weinert, H.J.: Semirings - Algebraic Theory and Application in Computer Science. Series in Algebra, vol. 5. World Scientific (1993)

    Google Scholar 

  10. Jackson, M., Stokes, T.: Semigroups with if-then-else and halting programs. IJAC 19(7), 937–961 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Macedo, H.D., Oliveira, J.N.: Matrices As Arrows! A Biproduct Approach to Typed Linear Algebra. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) MPC 2010. LNCS, vol. 6120, pp. 271–287. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  12. Macedo, H.D., Oliveira, J.N.: Typing Linear Algebra: a Biproduct-oriented Approach. Science of Comp. Programming 78, 2160–2191 (2012)

    Article  Google Scholar 

  13. Macedo, H.D., Oliveira, J.N.: A linear algebra approach to OLAP. Formal Asp. of Comput. 27(2), 283–307 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Graduate Texts in Mathematics, vol. 5. Springer (1998)

    Google Scholar 

  15. Oliveira, J.N.: Towards a Linear Algebra of Programming. Formal Asp. of Comput. 24(4-6), 433–458 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Oliveira, J.N.: Typed linear algebra for weighted (probabilistic) automata. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 52–65. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  17. Schmidt, G., Ströhlein, T.: On kernels of graphs and solutions of games: a synopsis based on relations and fixpoints. SIAM J. Alg. Discrete Meth. (6), 54–65 (1985)

    Google Scholar 

  18. Schmidt, G., Ströhlein, T.: Relationen und Graphen. Springer (1989); English version: Relations and Graphs. Discrete Mathematics for Computer Scientists, EATCS Monographs on Theoret. Comput. Sci. Springer (1993)

    Google Scholar 

  19. Schmidt, G.: Relational Mathematics. Encyplopedia of Mathematics and Its Applications, vol. 132 (2011)

    Google Scholar 

  20. Winter, M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertationsverlag NG Kopierladen GmbH, München (1998)

    Google Scholar 

  21. Winter, M.: Relation Algebras are Matrix Algebras over a suitable Basis. University of the Federal Armed Forces Munich, Report Nr. 1998-05 (1998)

    Google Scholar 

  22. Winter, M.: A Pseudo Representation Theorem for various Categories of Relations. TAC Theory and Applications of Categories 7(2), 23–37 (2000)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Brock University, St.Catharines, Ontario, Canada, L2S 3A1

    Dylan Killingbeck & Michael Winter

  2. Department of Computer Science, Federal University of Santa Maria, Santa Maria, Brazil

    Milene Santos Teixeira

Authors
  1. Dylan Killingbeck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milene Santos Teixeira
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Winter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dylan Killingbeck .

Editor information

Editors and Affiliations

  1. McMaster University, Hamilton, Ontario, Canada

    Wolfram Kahl

  2. Brock University, St. Catharines, Ontario, Canada

    Michael Winter

  3. Universidade do Minho, Braga, Portugal

    José Oliveira

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Killingbeck, D., Teixeira, M.S., Winter, M. (2015). Relations among Matrices over a Semiring. In: Kahl, W., Winter, M., Oliveira, J. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2015. Lecture Notes in Computer Science(), vol 9348. Springer, Cham. https://doi.org/10.1007/978-3-319-24704-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-24704-5_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-24703-8

  • Online ISBN: 978-3-319-24704-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation