Algebra universalis

, 79:88 | Cite as

Congruence computations in principal arithmetical varieties

  • Kalle KaarliEmail author
  • Alden Pixley


This paper is a continuation of the earlier paper by the same authors in which a primary result was that every arithmetical affine complete variety of finite type is a principal arithmetical variety with respect to an appropriately chosen Pixley term. The paper begins by presenting an extension of this result to all finitely generated congruences and, as an example, constructs a closed form solution formula for any finitely presented system of pairwise compatible congruences (the Chinese remainder theorem). It is also shown that in all such varieties the meet of principal congruences is also principal, and finally, if a minimal generating algebra of the variety is regular, it is shown that the variety is also regular and the join of principal congruences is again principal.


Principal arithmetical varieties Affine complete varieties Congruence computations Chinese remainder theorem Discriminator 

Mathematics Subject Classification

08A30 08A40 08B05 08B10 


  1. 1.
    Baker, K.A.: Primitive satisfaction and equational problems for lattices and other algebras. Trans. Am. Math. Soc. 190, 125–150 (1974)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bulman-Fleming, S., Werner, H.: Equational compactness in quasi-primal varieties. Algebra Univers. 7, 33–46 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chajda, I., Eigenthaler, G., Länger, H.: Congruence Classes in Universal Algebra, Research and Exposition in Mathematics, 26. Heldermann, Lemgo (2003)zbMATHGoogle Scholar
  4. 4.
    Czelakowski, J.: Protoalgebraic logics, Trends in Logic-Studia Logica Library, 10. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  5. 5.
    Day, A.: A note on the congruence extension property. Algebra Univers. 1, 234–235 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Grätzer, G.: On Boolean functions (notes on lattice theory II). Rev. Math. Pures Appl. (Bucarest) 7, 693–697 (1962)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Grätzer, G.: Two Mal’cev type theorems in universal algebra. J. Comb. Theory 8, 334–342 (1970)CrossRefGoogle Scholar
  8. 8.
    Huhn, A.P.: Schwach distributive Verbände, II. Acta Sci. Math. (Szeged) 46, 85–98 (1983)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kaarli, K.: A new characterization of arithmetical equivalence lattices. Algebra Univers. 45, 345–347 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kaarli, K., Pixley, A.: Affine complete varieties. Algebra Univers. 24, 74–90 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaarli, K., Pixley, A.: Polynomial Completeness in Algebraic Systems. CRC Press, Boca Raton (2000)CrossRefGoogle Scholar
  12. 12.
    Kaarli, K., Pixley, A.: Weakly diagonal algebras and definable principal congruences. Algebra Univers. 55, 203–212 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    McKenzie, R.M.: On spectra, and the negative solution of the decision problem for identities having a finite non-trivial model. J. Symb. Logic 40, 186–196 (1975)CrossRefGoogle Scholar
  14. 14.
    Quackenbush, R., Wolk, B.: Strong representation of congruence lattices, Algebra Univers. 1, 165–166 (1971/72)Google Scholar
  15. 15.
    Thurston, H.A.: Derived operations and congruences. Proc. London Math. Soc. (3) 8, 127–134 (1958)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of TartuTartuEstonia
  2. 2.Harvey Mudd CollegeClaremontUSA

Personalised recommendations