Compactness of systems of equations in semigroups

  • T. Harju
  • J. KarhumÄki
  • W. Plandowski
Automata and Formal Languages III
Part of the Lecture Notes in Computer Science book series (LNCS, volume 944)


We considei systems u i = v i (i ∃ I) of equations in semigroups over finite sets of variables. A semigroup S is said to satisfy the compactness property (or CP, for short), if each system of equations has an equivalent finite subsystem. It is shown that all monoids in a variety V satisfy CP, if and only if the finitely generated monoids in V satisfy the maximal condition on congruences. We also show that if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. Finally, we give three simple examples (the bicyclic monoid, the free monogenic inverse semigroup and the Baumslag-Solitar group) which do not satisfy CP, and show that the above necessary conditions are not sufficient.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • T. Harju
    • 1
  • J. KarhumÄki
    • 1
  • W. Plandowski
    • 2
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland
  2. 2.Instytut InformatykiUniwersytet WarszawskiWarszawaPoland

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