Semigroup Forum

, Volume 95, Issue 1, pp 245–250 | Cite as

The identities of the free product of two trivial semigroups



We exhibit an example of a finitely presented semigroup S with a minimum number of relations such that the identities of S have a finite basis while the monoid obtained by adjoining 1 to S admits no finite basis for its identities. Our example is the free product of two trivial semigroups.


Semigroup identity Finitely based semigroup Nonfinitely based semigroup Zimin word 



The work of the first author was partially supported by the CUNY Collaborative Incentive Research Grant Program (Project #94592-0001). The second author acknowledges support from the Russian Foundation for Basic Research, Project No. 14-01-00524, the Ministry of Education and Science of the Russian Federation, Project No. 1.1999.2014/K, and the Competitiveness Program of Ural Federal University. The paper was written during the second author’s stay at Hunter College of the City University of New York as Ada Peluso Visiting Professor of Mathematics and Statistics with a generous support from the Ada Peluso Endowment. The authors would like to thank the anonymous reviewer for his/her comments and suggestions.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Hunter CollegeCity University of New YorkNew YorkUSA
  2. 2.Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia

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