Algebra universalis

, Volume 75, Issue 1, pp 85–106 | Cite as

Stable varieties of semigroups and groupoids



The paper deals with \({\sum}\) -composition and \({\sum}\) -essential composition of terms which lead to stable and s-stable varieties of algebras. A full description of all stable varieties of semigroups, commutative and idempotent groupoids is obtained. We use an abstract reduction system which simplifies the presentations of terms of type \({\tau = (2)}\) to study the variety of idempotent groupoids and s-stable varieties of groupoids. S-stable varieties are a variation of stable varieties, used to highlight replacement of subterms of a term in a deductive system instead of the usual replacement of variables by terms.

Key words and phrases

composition of terms essential position in terms stable variety 

2010 Mathematics Subject Classification

Primary: 08B05 Secondary: 08A02 03C05 


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  1. 1.
    Burris, S., Sankappanavar, H.: A Course in universal algebra. The millennium edition.
  2. 2.
    Evans T.: The lattice of semigroup varieties. Semigroup Forum 2, 1–43 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Howie, J.M.: Fundamentals in semigroup theory. Oxford University Press, London (2003)Google Scholar
  4. 4.
    Klop, J.W., de Vrijer, R.: Term rewriting systems. Cambridge University Press (2003)Google Scholar
  5. 5.
    Polak L.: On hyperassociativity. Algebra Universalis 36, 363–378 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Polak L.: All solid varieties of semigroups. J. Algebra 219, 421–436 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Shtrakov S., Denecke K.: Essential variables and separable sets in universal algebra. Multiple Valued Logic: An International Journal 8, 165–181 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Shtrakov S.: Essential variables and positions in terms. Algebra Universalis 61, 381–397 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceSouth-West UniversityBlagoevgradBulgaria
  2. 2.Institute of MathematicsUniversity of PotsdamPotsdamGermany

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