Applied Categorical Structures

, Volume 24, Issue 5, pp 733–742 | Cite as

On Exponentiable Morphisms in Classical Algebra

  • Maria Manuel ClementinoEmail author
  • Dirk Hofmann
  • George Janelidze


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.


Exponentiable morphism Taut monad Subtractive variety Ideal determined variety Semimodule 

Mathematics Subject Classifications (2010)

18C15 18C20 08A62 16Y60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bourn, D., Janelidze, Z.: Approximate Mal’tsev operations. Theory Appl. Categ. 21, 152–171 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bourn, D., Janelidze, Z.: Subtractive categories and extended subtractions. Appl. Categ. Struct. 17, 317–343 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Clementino, M.M., Hofmann, D., Janelidze, G.: On exponentiability of étale algebraic homomorphisms. J. Pure Appl. Algebra 217, 1195–1207 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetGoogle Scholar
  7. 7.
    Fichtner, K.: Eine Bemerkungüber Mannigfaltigkeiten universeller Algebren mit Idealen. Monatsb. Deutsch. Akad. Wiss. Berlin 12, 21–25 (1970)MathSciNetzbMATHGoogle Scholar
  8. 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)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gray, J.R.A.: Algebraic exponentiation in general categories. Appl. Categ. Struct. 20(6), 543–567 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gumm, H.P., Ursini, A.: Ideals in universal algebras. Algebra Univ. 19, 45–54 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Higgins, P.J.: Groups with multiple operators. Proc. London Math. Soc. (3) 6, 366–416 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168, 367–386 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Janelidze, G., Márki, L., Tholen, W., Ursini, A.: Ideal-determined categories. Cah. Topol. Géom. Différ. Catég. 51(2), 115–125 (2010)MathSciNetzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Janelidze, Z.: Subtractive categories. Appl. Categ. Struct. 13, 343–350 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Janelidze, Z.: The pointed subobject functor, 3×3 lemmas, and subtractivity of spans. Theory Appl. Categ. 23(11), 221–242 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Johnson, J.S., Manes, E.G.: On modules over a semiring. J. Algebra 15, 57–67 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Johnstone, P.T.: Collapsed toposes and cartesian closed varieties. J. Algebra 129, 446–480 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lane, S.M.: Duality for groups. Bull. Amer. Math. Soc. 56, 485–516 (1950)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mac Lane, S.: Categories for the working mathematician, graduate texts in mathematics. Springer-Verlag, New York-Berlin (1971)CrossRefzbMATHGoogle Scholar
  21. 21.
    Manes, E., Monads, T.: T0-spaces. Theoret. Comput. Sci. 275, 79–109 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ursini, A.: Sulle varietà di algebre con una buona teoria degli ideali. Boll. Un. Mat. Ital. (4) 6, 90–95 (1972)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ursini, A.: Osservazioni sulle varietà. B I T Boll. Un. Mat. Ital. (4) 7, 205–211 (1973)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ursini, A.: On subtractive varieties I. Algebra Univ. 31, 204–222 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Maria Manuel Clementino
    • 1
    Email author
  • Dirk Hofmann
    • 2
  • George Janelidze
    • 3
  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.CIDMA, Department of MathematicsUniversity of AveiroAveiroPortugal
  3. 3.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa

Personalised recommendations