[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On Exponentiable Morphisms in Classical Algebra

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphisms in ideal determined varieties and extend this to ideal determined categories. Finally, we give a complete characterization of exponentiable homomorphisms of semimodules over semirings.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borceux, F., Janelidze, G., Kelly, G.M.: On the representability of actions in a semi-abelian category. Theory Appl. Categ. 14(11), 244–286 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Bourn, D., Janelidze, Z.: Approximate Mal’tsev operations. Theory Appl. Categ. 21, 152–171 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Bourn, D., Janelidze, Z.: Subtractive categories and extended subtractions. Appl. Categ. Struct. 17, 317–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clementino, M.M., Hofmann, D., Janelidze, G.: On exponentiability of étale algebraic homomorphisms. J. Pure Appl. Algebra 217, 1195–1207 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clementino, M.M., Hofmann, D., Janelidze, G.: The monads of classical algebra are seldom weakly cartesian. J. Homotopy Relat. Struct. 9(1), 175–197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Csákány, B.: Primitive classes of algebras which are equivalent to classes of semi-modules and modules (in Russian). Acta Sci. Math. (Szeged) 24, 157–164 (1963)

    MathSciNet  Google Scholar 

  7. Fichtner, K.: Eine Bemerkungüber Mannigfaltigkeiten universeller Algebren mit Idealen. Monatsb. Deutsch. Akad. Wiss. Berlin 12, 21–25 (1970)

    MathSciNet  MATH  Google Scholar 

  8. Gran, M., Janelidze, Z., Rodelo, D., Ursini, A.: Symmetry of regular diamonds, the Goursat property, and subtractivity. Theory Appl. Categ. 27(6), 80–96 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Gray, J.R.A.: Algebraic exponentiation in general categories. Appl. Categ. Struct. 20(6), 543–567 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gumm, H.P., Ursini, A.: Ideals in universal algebras. Algebra Univ. 19, 45–54 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Higgins, P.J.: Groups with multiple operators. Proc. London Math. Soc. (3) 6, 366–416 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  12. Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168, 367–386 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Janelidze, G., Márki, L., Tholen, W., Ursini, A.: Ideal-determined categories. Cah. Topol. Géom. Différ. Catég. 51(2), 115–125 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Janelidze, G., Márki, L., Ursini, A.: Ideals and clots in universal algebra and in semi-abelian categories. J. Algebra 307(1), 191–208 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Janelidze, Z.: Subtractive categories. Appl. Categ. Struct. 13, 343–350 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Janelidze, Z.: The pointed subobject functor, 3×3 lemmas, and subtractivity of spans. Theory Appl. Categ. 23(11), 221–242 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Johnson, J.S., Manes, E.G.: On modules over a semiring. J. Algebra 15, 57–67 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  18. Johnstone, P.T.: Collapsed toposes and cartesian closed varieties. J. Algebra 129, 446–480 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lane, S.M.: Duality for groups. Bull. Amer. Math. Soc. 56, 485–516 (1950)

    Article  MathSciNet  Google Scholar 

  20. Mac Lane, S.: Categories for the working mathematician, graduate texts in mathematics. Springer-Verlag, New York-Berlin (1971)

    Book  MATH  Google Scholar 

  21. Manes, E., Monads, T.: T0-spaces. Theoret. Comput. Sci. 275, 79–109 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ursini, A.: Sulle varietà di algebre con una buona teoria degli ideali. Boll. Un. Mat. Ital. (4) 6, 90–95 (1972)

    MathSciNet  MATH  Google Scholar 

  23. Ursini, A.: Osservazioni sulle varietà. B I T Boll. Un. Mat. Ital. (4) 7, 205–211 (1973)

    MathSciNet  MATH  Google Scholar 

  24. Ursini, A.: On subtractive varieties I. Algebra Univ. 31, 204–222 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Maria Manuel Clementino

  2. CIDMA, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Dirk Hofmann

  3. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

    George Janelidze

Authors
  1. Maria Manuel Clementino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dirk Hofmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. George Janelidze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Manuel Clementino.

Additional information

Dedicated to the memory of Horst Herrlich.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clementino, M.M., Hofmann, D. & Janelidze, G. On Exponentiable Morphisms in Classical Algebra. Appl Categor Struct 24, 733–742 (2016). https://doi.org/10.1007/s10485-016-9458-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-016-9458-7

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation