Advertisement

Siberian Mathematical Journal

, Volume 56, Issue 3, pp 516–525 | Cite as

Axiomatizability and completeness of the class of injective acts over a commutative monoid or a group

  • A. A. StepanovaEmail author
Article

Abstract

We study monoids S over which the class of injective S-acts is axiomatizable, complete, and model complete. We prove that, for a countable commutative monoid or a countable group S, the class of injective S-acts is axiomatizable if and only if S is a finitely generated monoid. We show that there is no nontrivial monoid nor a group the class of injective acts over which is complete, model complete, or categorical.

Keywords

axiomatizable class complete class model complete class categorical class act injective act 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gould V., Mikhalev A. V., Palyutin E. A., and Stepanova A. A., “Model-theoretic properties of free, projective, and flat S-acts,” J. Math. Sci. (New York), 164, No. 2, 195–227 (2010).zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Stepanova A. A., “Axiomatizability and completeness of some classes of S-polygons,” Algebra and Logic, 30, No. 5, 379–388 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Gould V., “Axiomatisability problems for S-systems,” J. London Math. Soc., 35, 193–201 (1987).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mikhalev A. V., Ovchinnikova E. V., Palyutin E. A., and Stepanova A. A., “Model-theoretic properties of regular polygons,” J. Math. Sci. (New York), 140, No. 2, 250–285 (2007).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Stepanova A. A., “Axiomatizability and model completeness of the class of regular polygons,” Siberian Math. J., 35, No. 1, 166–177 (1994).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ershov Yu. L. and Palyutin E. A., Mathematical Logic [in Russian], Nauka, Moscow (1987).Google Scholar
  7. 7.
    Mal’cev A. I., Algebraic Systems, Springer-Verlag and Akademie-Verlag, Berlin, Heidelberg, and New York (1973).CrossRefGoogle Scholar
  8. 8.
    Clifford A. H. and Preston G. B., The Algebraic Theory of Semigroups. Vol. 1 and 2, Amer. Math. Soc., Providence (1961, 1967).CrossRefGoogle Scholar
  9. 9.
    Kilp M., Knauer U., and Mikhalev A. V., Monoids, Acts and Categories, Walter De Gruyter, Berlin and New York (2000).zbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Institute of Applied MathematicsFar-Eastern Federal UniversityVladivostokRussia

Personalised recommendations