Algebra universalis

, 61:151 | Cite as

Algebraically expandable classes



An algebraically expandable class is a class of similar algebras axiomatizable by sentences of the form \({(\forall\exists ! \bigwedge eq)}\) . The problem investigated in this work is that of finding all algebraically expandable classes within a given variety. A complete solution is presented for a number of varieties, including the classes of Boolean algebras, Stone algebras, semilattices, distributive lattices and generalized Kleene algebras. We also study the problem for the case of discriminator varieties, where we prove that there is a lattice isomorphism between the lattice of all algebraically expandable classes of the variety and a certain lattice of subclasses of the simple members of the variety. Finally this connection is applied to calculating the algebraically expandable subclasses of the varieties of monadic algebras and P-algebras.

2000 Mathematics Subject Classification

08C10 08B10 03C40 06D30 

Keywords and phrases

Kleene algebra discriminator variety function definition sentence 


  1. 1.
    Adams M.E., Dziobiak W.: Lattices of quasivarieties of 3-element algebras. J. Algebra 166, 181–210 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, Columbia, Missouri (1974)MATHGoogle Scholar
  3. 3.
    Burris S., Sankapanavar H.: A Course in Universal Algebra. Springer Verlag, New York (1981)MATHGoogle Scholar
  4. 4.
    Cornish W.H., Fowler P.R.: Coproducts of Kleene algebras. J. Austral. Math. Soc. Ser. A 27, 209–220 (1979)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Epstein G., Horn A.: P-algebras, an abstraction from Post algebras. Algebra Universalis 7, 195–206 (1974)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Foster A.L.: An existence theorem for functionally complete universal algebras. Math. Z. 71, 69–82 (1959)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gramaglia H., Vaggione D.J.: Birkhoff-like sheaf representation for varieties of lattice expansions. Studia Logica 56, 111–131 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jónsson B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)MATHMathSciNetGoogle Scholar
  9. 9.
    John F. Kennison: Structure and costructure for strongly regular rings. J. Pure Appl. Algebra 5, 321–332 (1974)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kennison J.F., Dion Gildenhuys: Equational completion, model induced triples and pro-objects. J. Pure Appl. Algebra 1, 317–346 (1971)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. H. Krauss and D. M. Clark, Global Subdirect Products, Amer. Math. Soc. Mem. 210 (1979).Google Scholar
  12. 12.
    R. N. McKenzie, G. McNulty, and W. Taylor, Algebras, Lattices, Varieties, vol. I, Wadsworth & Brooks/Cole, Monterey, California, 1987.Google Scholar
  13. 13.
    Raphael R.: Some remarks on regular and strongly regular rings. Canad. Math. Bull. 17, 709–712 (1975)MATHMathSciNetGoogle Scholar
  14. 14.
    Vaggione D.J.: Sheaf representation and chinese remainder theorems. Algebra Universalis 29, 232–272 (1992)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Vaggione D.J.: Central elements in varieties with the Fraser-Horn property. Advances in Math. 148, 193–202 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Volger H.: Preservation theorems for limits of structures and global sections of sheaves of structures. Math. Z. 166, 27–53 (1970)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Werner H.: Discriminator algebras, algebraic representation and model theoretic properties. Akademie Verlag, Berlin (1978)MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Facultad de Matemáteca, Astronomía y FíisicaUniversidad Nacional de CórdobaCórdobaArgentina

Personalised recommendations