Maximal subgroups of free idempotent generated semigroups over the full linear monoid

We show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear monoid of n x n matrices over a division ring Q has maximal subgroup isomorphic to the general linear group GL_r(Q), where n and r are positive integers with r < n/3.Comment: 37 pages; Transactions of the American Mathematical Society (to appear). arXiv admin note: text overlap with arXiv:1009.5683 by other author

Subgroups of free idempotent generated semigroups: full linear monoid

We develop some new topological tools to study maximal subgroups of free idempotent generated semigroups. As an application, we show that the rank 1 component of the free idempotent generated semigroup of the biordered set of a full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal subgroup isomorphic to the multiplicative subgroup of Q.Comment: We hope to use similar methods to study the higher rank component

On regularity and the word problem for free idempotent generated semigroups

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

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

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

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

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