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Abstract
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 language  English 

Pages (fromto)  401–432 
Number of pages  32 
Journal  Proceedings of the London Mathematical Society 
Volume  114 
Issue number  3 
Early online date  23 Jan 2017 
DOIs  
Publication status  Published  Mar 2017 
Keywords
 20M05 (primary)
 20F05
 20F10 (secondary)
Profiles

Robert Gray
 School of Mathematics  Reader in Pure Mathematics
 Algebra and Combinatorics  Member
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished