A group-theoretical interpretation of the word problem for free idempotent generated semigroups

Victoria Ann Rosalind Gould, Dandan Yang, Igor Dolinka

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The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup IG(E) – the ‘free-est’ semigroup with a given biordered set E of idempotents. We show that when E is finite, the word problem for IG(E) is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of IG(E). As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite E, IG(E) is always a weakly abundant semigroup satisfying the congruence condition.

Original languageEnglish
Pages (from-to)998-1041
Number of pages44
JournalAdvances in Mathematics
Early online date25 Jan 2019
Publication statusPublished - 17 Mar 2019

Bibliographical note

© 2019 Elsevier Inc. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.


  • Biordered set
  • Free idempotent generated semigroup
  • Rational subset
  • Word problem

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