[Paper Review] Subgroups of free idempotent generated semigroups need not be free

Mark Brittenham, Stuart Margolis+1 more|ArXiv.org|Aug 13, 2008

Advanced Topology and Set Theory13 references3 citations

TL;DR

This paper constructs the first known example of a non-free maximal subgroup in a free idempotent generated semigroup by using a topological approach to Nambooripad's theory of regular biordered sets. The authors associate a 2-complex to a regular biordered set derived from a 3-dimensional vector space over F₂, proving that the maximal subgroup of the resulting free regular idempotent generated semigroup is isomorphic to the free abelian group of rank 2, thus disproving a long-standing conjecture that such subgroups must be free.

ABSTRACT

We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup.

Motivation & Objective

• To disprove the conjecture that maximal subgroups of free idempotent generated semigroups are always free.
• To construct a concrete example of a regular biordered set whose associated free idempotent generated semigroup has a non-free maximal subgroup.
• To apply topological tools—specifically 2-complexes and fundamental groupoids—to study the structure of maximal subgroups in free idempotent generated semigroups.
• To extend Nambooripad’s theory of inductive groupoids by associating a 2-complex to a regular biordered set, enabling homotopical analysis of maximal subgroups.

Proposed method

• Construct a regular biordered set E from the structure of a 3-dimensional vector space over the field F₂, using idempotent elements corresponding to certain subspaces and relations.
• Associate a 2-complex K(E) to the regular biordered set E, whose fundamental groupoid is isomorphic to Nambooripad’s groupoid N(E).
• Use Bass-Serre theory to analyze the fundamental group of K(E), which corresponds to the maximal subgroup of the free regular idempotent generated semigroup RIG(E).
• Show that the fundamental group of K(E) is isomorphic to Z × Z, the free abelian group of rank 2, by analyzing the cell structure and attaching maps of the 2-complex.
• Verify that the idempotents in the semigroup act as affine functions on F₂³, and that the inverse image structure preserves the combinatorial and topological properties needed for the groupoid construction.
• Leverage known results on matrix monoids over finite fields to confirm that the same group arises as a maximal subgroup in other constructions, reinforcing the generality of the result.

Experimental results

Research questions

• RQ1 Can maximal subgroups of free idempotent generated semigroups be non-free, contradicting the long-standing conjecture that they are always free?
• RQ2 What topological or algebraic structure underlies the maximal subgroups of free idempotent generated semigroups over regular biordered sets?
• RQ3 Is there a finite regular biordered set for which the associated free regular idempotent generated semigroup has a maximal subgroup isomorphic to Z × Z?
• RQ4 How can Nambooripad’s inductive groupoid framework be combined with topological methods to analyze the group structure of maximal subgroups?
• RQ5 Can combinatorial configurations in finite geometries over F₂ yield new examples of non-free maximal subgroups in free idempotent generated semigroups?

Key findings

• The paper constructs a finite regular biordered set E derived from a 3-dimensional vector space over F₂, which gives rise to a free regular idempotent generated semigroup RIG(E).
• The maximal subgroup of RIG(E) at a specific idempotent is isomorphic to the free abelian group of rank 2, i.e., Z × Z.
• This provides the first known example of a non-free maximal subgroup in any free idempotent generated semigroup, thereby refuting the conjecture that such subgroups are always free.
• The topological construction of a 2-complex K(E) with fundamental group isomorphic to Z × Z confirms the group structure via homotopical methods.
• The idempotents in the semigroup act as affine functions on F₂³, and the inverse image structure of blocks corresponds to the action of the groupoid, preserving the topological and algebraic consistency.
• The result is generalized by showing that for any field F, the biordered set E₃(F) of 3×3 matrices over F yields a maximal subgroup isomorphic to the multiplicative group of F, which includes finite cyclic groups of order pⁿ−1 for prime powers pⁿ.