A Note on Complex Algebras of Semigroups

  • Peter Jipsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3051)


The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex algebras of partial (commutative) semigroups. An example is given of an 8-element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite different: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator.


Boolean Algebra Binary Operator Equational Theory Relation Algebra Algebraic Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreka, H.: Representations of distributive lattice ordered semigroups with binary relations. Algebra Universalis 28, 12–25 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Henkin, L., Monk, J.D., Tarski, A.: Cylindric algebras, Part II. North-Holland, Amsterdam (1985)zbMATHGoogle Scholar
  3. 3.
    Hirsch, R., Hodkinson, I.: Step by step — building representations in algebraic logic. J. Symbolic Logic 62, 816–847 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hirsch, R., Hodkinson, I.: Relation algebras by games. Studies in Logic and the Foundations of Mathematics, vol. 147. Elsevier Science, North-Holland (2002)zbMATHGoogle Scholar
  5. 5.
    Jipsen, P.: Computer aided investigations of relation algebras, dissertation, Vanderbilt University (1992),
  6. 6.
    Jipsen, P., Maddux, R.D.: Nonrepresentable sequential algebras. Log. J. IGPL 5(4), 565–574 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Maddux, R.D.: Nonfinite axiomatizability results for cylindric and relation algebras. J. Symbolic Logic 54(3), 951–974 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Monk, D.: On representable relation algebras. Michigan Math. J. 11, 207–210 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Reich, P.: Complex algebras of semigroups, dissertation, Iowa State University (1996)Google Scholar
  10. 10.
    Tarski, A.: On the calculus of relations. J. Symbolic Logic 6, 73–89 (1941)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Peter Jipsen
    • 1
  1. 1.Chapman UniversityOrangeUSA

Personalised recommendations