Skip to main content
SpringerLink
Log in

Simple and pseudosimple algebras with operators

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this work, algebras with operators, i.e., algebras with an additional system of unary operations acting as endomorphisms with respect to the basic operations, are considered. We find necessary conditions for the simplicity and pseudosimplicity of an arbitrary universal algebra with a fixed operator and explore how several properties of congruences of this algebra depend on the structure of the unary reduct of a given algebra. For algebras with one ternary operation, defined in the standard way and satisfying the Mal’cev identities, and one operator (i.e., unars with Mal’cev operation), necessary and sufficient conditions for their simplicity and pseudosimplicity are obtained.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. P. Egorova, “Structure of congruences of unary algebra,” in: Ordered Sets and Lattices, Issue 5 [in Russian], Izd. Saratov. Univ., Saratov (1978), pp. 11–44.

    Google Scholar 

  2. V. K. Kartashov, “On lattices of quasivarieties of unars,” Sib. Mat. Zh., 26, No. 3, 49–62 (1985).

    MathSciNet  MATH  Google Scholar 

  3. V. K. Kartashov, “On unars with Mal’cev operation,” in: Int. Seminar “Universal Algebra and Its Applications”: Abstracts [in Russian], Volgograd (1999), p. 32.

  4. Ya. V. Khion, “Ω-ringoids, Ω-rings, and their representations,” Tr. Mosk. Mat. Obshch., 14, 3–47 (1965).

    Google Scholar 

  5. M. Kilp, U. Knauer, and A. V. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin (2000).

    MATH  Google Scholar 

  6. I. B. Kozhukhov, “Semigroups over which all acts are residually final,” Fundam. Prikl. Mat., 4, No. 4, 1335–1344 (1998).

    MathSciNet  MATH  Google Scholar 

  7. A. G. Kurosh, General Algebra. Lectures for the Academic Year 1969/70 [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  8. S. A. Liber, “Ordered n-ringoids over multi-operator groups,” Mat. Issled., 7, No. 1, 83–97 (1972).

    MathSciNet  MATH  Google Scholar 

  9. N. V. Loi and B. Wiegandt, “Subdirect irreducibility of algebras and acts with an additional unary operation,” Miskolc Math. Notes, 6, No. 2, 217–224 (2005).

    MathSciNet  MATH  Google Scholar 

  10. L. A. Skornyakov, “Generalizations of acts,” in: The Acts. III [in Russian], Preprint, Novosibirsk (1973), pp. 22–27.

  11. L. A. Skornyakov, “Unars,” Collect. Math. Soc. J. Bolyai, 29, 735–743 (1982).

    Google Scholar 

  12. L. A. Skornyakov, ed., General Algebra, Vol. 2 [in Russian], Nauka, Moscow (1991).

    MATH  Google Scholar 

  13. V. L. Usoltsev, “Minimal unary algebras with two permutable operations,” Deposited at VINITI, No. 3857-B96 (1996).

  14. G. H. Wenzel, “Subdirect irreducibility and equational compactness in unary algebras 〈A; f〉,” Arch. Math. (Basel), 21, 256–264 (1970).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Volgograd State Pedagogical University, Volgograd, Russia

    V. L. Usoltsev

Authors
  1. V. L. Usoltsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. L. Usoltsev.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 7, pp. 189–207, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Usoltsev, V.L. Simple and pseudosimple algebras with operators. J Math Sci 164, 281–293 (2010). https://doi.org/10.1007/s10958-009-9730-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-009-9730-6

Keywords

Search

Navigation