Algebra universalis

, Volume 68, Issue 3–4, pp 221–236 | Cite as

Varieties generated by modes of submodes

Open Access


In a natural way, we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (\({A, \Omega}\)) its power algebra of subsets. G. Grätzer and H. Lakser proved that for a variety \({\mathcal{V}}\), the variety \({\mathcal{V}\Sigma}\) generated by power algebras of algebras in \({\mathcal{V}}\) satisfies precisely the consequences of the linear identities true in \({\mathcal{V}}\). For certain types of algebras, the sets of their subalgebras form subalgebras of their power algebras. They are called the algebras of subalgebras. In this paper, we partially solve a long-standing problem concerning identities satisfied by the variety \({\mathcal{VS}}\) generated by algebras of subalgebras of algebras in a given variety \({\mathcal{V}}\). We prove that if a variety \({\mathcal{V}}\) is idempotent and entropic and the variety \({\mathcal{V}\Sigma}\) is locally finite, then the variety \({\mathcal{VS}}\) is defined by the idempotent and linear identities true in \({\mathcal{V}}\).

2010 Mathematics Subject Classification

Primary: 03C05 Secondary: 03G25 08A30 08A62 06E25 06A12 08B99 

Key words and phrases

idempotent entropic modes power algebras congruence relations identities varieties of finitary algebras locally finite varieties 


  1. 1.
    Adaricheva, K., Pilitowska, A., Stanovský, D.: Complex algebras of subalgebras. Algebra Logika 47(6), 655–686 (2008) (Russian). English translation: Algebra Logic 47(6), 367–383 (2008)Google Scholar
  2. 2.
    Bošnjak I., Madarász R.: On power structures. Algebra Discrete Math. 2, 14–35 (2003)Google Scholar
  3. 3.
    Brink C.: Power structures. Algebra Universalis 30, 177–216 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Goldblatt R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Gorbunov, V.A.: Algebraic Theory of Quasivarieties. Nauchnaya Kniga, Novosibirsk (1999) (Russian). English translation: Plenum, New York (1998)Google Scholar
  6. 6.
    Grätzer G., Lakser H.: Identities for globals (complex algebras) of algebras. Colloq. Math. 56, 19–29 (1988)MathSciNetMATHGoogle Scholar
  7. 7.
    Grätzer G., Whitney S.: Infinitary varieties of structures closed under the formation of complex structures. Colloq. Math. 48, 485–488 (1984)Google Scholar
  8. 8.
    Ježek, J.: A note on complex groupoids. In: Universal Algebra (Esztergom, 1977). Colloq. Math. Soc. János Bolyai, vol. 29, pp. 419–420. North-Holland, Amsterdam (1982)Google Scholar
  9. 9.
    Jónsson B., Tarski A.: Boolean algebras with operators I. Amer. J. Math. 73, 891–939 (1951)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Jónsson B., Tarski A.: Boolean algebras with operators II. Amer. J. Math. 74, 127–162 (1952)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    McKenzie R., McNulty G., Taylor W.: Algebras, Lattices, Varieties, vol. 1. Wadsworth & Brooks/Cole, Monterey (1987)Google Scholar
  12. 12.
    Pilitowska, A.: Modes of submodes. PhD thesis, Warsaw University of Technology (1996)Google Scholar
  13. 13.
    Pilitowska A.: Identities for classes of algebras closed under the complex structures. Discuss. Math. Algebra Stochastic Methods 18, 85–109 (1998)MathSciNetMATHGoogle Scholar
  14. 14.
    Pilitowska A., Zamojska-Dzienio A.: Representation of modals. Demonstratio Math. 44, 535–556 (2011)MathSciNetMATHGoogle Scholar
  15. 15.
    Romanowska A.B.: Semi-affine modes and modals. Sci. Math. Jpn. 61, 159–194 (2005)MathSciNetMATHGoogle Scholar
  16. 16.
    Romanowska A.B., Roszkowska B.: On some groupoid modes. Demonstratio Math. 20, 277–290 (1987)MathSciNetMATHGoogle Scholar
  17. 17.
    Romanowska A.B., Smith J.D.H.: Bisemilattices of subsemilattices. J. Algebra 70, 78–88 (1981)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Romanowska A.B., Smith J.D.H.: Modal Theory. Heldermann, Berlin (1985)MATHGoogle Scholar
  19. 19.
    Romanowska A.B., Smith J.D.H.: Subalgebra systems of idempotent entropic algebras. J. Algebra 120, 247–262 (1989)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Romanowska A.B., Smith J.D.H.: On the structure of subalgebra systems of idempotent entropic algebras. J. Algebra 120, 263–283 (1989)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Romanowska A.B., Smith J.D.H.: Modes. World Scientific, Singapore (2002)MATHGoogle Scholar
  22. 22.
    Shafaat A.: On varieties closed under the construction of power algebras. Bull. Austral. Math. Soc. 11, 213–218 (1974)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Szendrei A.: The operation ISKP on classes of algebras. Algebra Universalis 6, 349–353 (1976)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Trnková V.: On a representation of commutative semigroups. Semigroup Forum 10, 203–214 (1975)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

Personalised recommendations