Algebra universalis

, Volume 73, Issue 3–4, pp 369–390 | Cite as

The connection of skew Boolean algebras and discriminator varieties to Church algebras



We establish a connection between skew Boolean algebras and Church algebras. We prove that the set of all semicentral elements in a right Church algebra forms a right-handed skew Boolean algebra for the properly defined operations. The main result of this paper states that the variety of all semicentral right Church algebras of type \({\tau}\) is term equivalent to the variety of right-handed skew Boolean algebras with additional regular operations. As a corollary to this result, we show that a pointed variety is a discriminator variety if and only if it is a 0-regular variety of right-handed skew Boolean algebras.

2010 Mathematics Subject Classification

Primary: 03G10 Secondary: 08B26 

Keywords and phrases

skew Boolean algebra Church algebra discriminator variety factor congruence decomposition operator 


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  1. 1.
    Bauer A., Cvetko-Vah K.: Stone duality for skew Boolean algebras with intersections. Houston J. Math. 39, 73–109 (2013)MATHMathSciNetGoogle Scholar
  2. 2.
    Bigelow D., Burris S.N.: Boolean algebras of factor congruences. Acta Sci. Math. (Szeged) 54, 11–20 (1990)MATHMathSciNetGoogle Scholar
  3. 3.
    Bignall R.J., Leech J.: Skew Boolean algebras and discriminator varieties. Algebra Universalis 33, 387–398 (1995)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Burris S.N., Sankappanavar H.P.: A Course in Universal Algebra. Springer, Berlin (1981)CrossRefMATHGoogle Scholar
  5. 5.
    Cvetko-Vah K.: Internal decompositions of skew lattices. Comm. Algebra 35, 243–247 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cvetko-Vah K., Leech J.: Rings whose idempotents form a multiplicative set. Comm. Algebra, 40, 3288–3307 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cvetko-Vah K., Leech J., Spinks M.: Skew lattices and binary operations on functions. J. Appl. Log. 11, 253–265 (2013)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fichtner K.: Fine Bermerkung über Mannigfaltigkeiten universeller Algebren mit Idealen. Monatsb. Deutsch. Akad. Wiss. Berlin 12, 21–25 (1970)MATHMathSciNetGoogle Scholar
  9. 9.
    Kudryavtseva G: A refinement of Stone duality to skew Boolean algebras. Algebra Universalis 67, 397–416 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Ledda A., Paoli F., Salibra A.: On Semi-Boolean-like algebras. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 52, 101–120 (2013)MATHMathSciNetGoogle Scholar
  11. 11.
    Leech J.: Skew lattices in rings. Algebra Universalis 26, 48–72 (1989)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Leech J.: Skew Boolean algebras. Algebra Universalis 27, 497–506 (1990)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Leech J.: Recent developments in the theory of skew lattices. Semigroup Forum 52, 7–24 (1996)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Manzonetto, G., Salibra, A.: From \({\lambda}\)-calculus to universal algebra and back. In: Proceedings of the 33nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2008). Lecture Notes in Comput. Sci., vol. 5162, pp. 479–490. Springer, Berlin (2008)Google Scholar
  15. 15.
    McKenzie R.N., McNulty G.F., Taylor W.F.: Algebras, Lattices, Varieties, Volume I. Wadsworth Brooks, Monterey, California (1987)MATHGoogle Scholar
  16. 16.
    Salibra A., Ledda A., Paoli F., Kowalski T.: Boolean-like algebras. Algebra Universalis 69, 113–138 (2013)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Spinks, M.: On the Theory of Pre-BCK Algebras. PhD Thesis, Monash University (2003)Google Scholar
  18. 18.
    Vaggione D.: Varieties in which the Pierce stalks are directly indecomposable. J. Algebra 184, 424–434 (1996)CrossRefMATHMathSciNetGoogle Scholar

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Department of Environmental Sciences, Informatics andStatisticsUniversità Ca’Foscari VeneziaVeneziaItaly

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