On regularity and the word problem for free idempotent generated semigroups

Dolinka, Igor, Gray, Robert D. and Ruškuc, Nik (2017) On regularity and the word problem for free idempotent generated semigroups. Proceedings of the London Mathematical Society, 114 (3). 401–432. ISSN 0024-6115

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      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.

      Item Type: Article
      Uncontrolled Keywords: 20m05 (primary),20f05,20f10 (secondary)
      Faculty \ School: Faculty of Science > School of Mathematics
      Depositing User: Pure Connector
      Date Deposited: 24 Sep 2016 01:56
      Last Modified: 09 Apr 2019 11:32
      URI: https://ueaeprints.uea.ac.uk/id/eprint/60316
      DOI: 10.1112/plms.12011

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