algebra universalis

, Volume 45, Issue 2–3, pp 211–219 | Cite as

The variety generated by equivalence algebras

  • J. Ježek
  • R. McKenzie


Every equivalence relation can be made into a groupoid with the same underlying set if we define the multiplication as follows: xy = x if x,y are related; otherwise, xy = y. The groupoids, obtained in this way, are called equivalence algebras. We find a finite base for the equations of equivalence algebras. The base consists of equations in four variables, and we prove that there is no base consisting of equations in three variables only. We also prove that all subdirectly irreducibles in the variety generated by equivalence algebras are embeddable into the three-element equivalence algebra, corresponding to the equivalence with two blocks on three elements.


Equivalence Relation Finite Base Equivalence Algebra Subdirectly Irreducibles 
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Copyright information

© Birkhäuser Verlag Basel, 2001

Authors and Affiliations

  • J. Ježek
    • 1
  • R. McKenzie
    • 2
  1. 1.Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic, e-mail: jezek@csmath.kaslin.mpp.cuni.czCZ
  2. 2.Vanderbilt University, Dept. of Mathematics, Nashville, Tennessee 37240, USAUS

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