On regularity and the word problem for free idempotent generated semigroups

Igor Dolinka, Robert D. Gray, Nik Ruškuc

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)
18 Downloads (Pure)


The category of all idempotent generated semigroups with a prescribed structure E of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over E, defined by a presentation over alphabet E, and denoted by IG(E). Recently, much effort has been put into investigating the structure of semigroups of the form IG(E), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(E). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗represents a regular element; if in addition one assumes that all maximal subgroups of IG(E) have decidable word problems, then the word problem in IG(E) restricted to regular words is decidable. On the other hand, we exhibit a biorder E arising from a finite idempotent semigroup S, such that the word problem for IG(E) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(E) to the subgroup membership problem in finitely presented groups.
Original languageEnglish
Pages (from-to)401–432
Number of pages32
JournalProceedings of the London Mathematical Society
Issue number3
Early online date23 Jan 2017
Publication statusPublished - Mar 2017


  • 20M05 (primary)
  • 20F05
  • 20F10 (secondary)

Cite this