On regularity and the word problem for free idempotent generated semigroups

The research of the first author was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia through the grant No. 174019, and by the grant No. 0851/2015 of the Secretariat of Science and Technological Development of the Autonomous Province of Vojvodina. The research of the second author was partially supported by the EPSRC-funded project EP/N033353/1 ‘Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem’. The research of the third author was supported by the EPSRC-funded project EP/H011978/1 ‘Automata, Languages, Decidability in Algebra’.The category of all idempotent generated semigroups with a prescribed structure Ɛ of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over Ɛ, defined by a presentation over alphabet E, and denoted by IG(Ɛ). Recently, much effort has been put into investigating the structure of semigroups of the form IG(Ɛ), especially regarding their maximal subgroups. In this paper, we take these investigations in a new direction by considering the word problem for IG(Ɛ). 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(Ɛ) have decidable word problems, then the word problem in IG(Ɛ) restricted to regular words is decidable. On the other hand, we exhibit a biorder Ɛ arising from a finite idempotent semigroup S, such that the word problem for IG(Ɛ) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(Ɛ) to the subgroup membership problem in infinitely presented groups.PostprintPeer reviewe