Direct products of free groups and free idempotent generated semigroups over bands

TL;DR

The paper constructs specific regular bands to embed free groups as maximal subgroups in their free idempotent generated semigroups, revealing complex subgroup structures.

Contribution

It introduces a method to realize arbitrary direct products of free groups within free idempotent generated semigroups over bands, highlighting new algebraic constructions.

Findings

Existence of regular bands with prescribed subgroup structures

Construction of semigroups containing non-abelian free groups

Demonstration of complex subgroup embeddings in idempotent generated semigroups

Abstract

For each group G which decomposes into a finitary direct product of free groups of finite rank we construct a regular band B such that the free idempotent generated semigroup over B contains a maximal subgroup isomorphic to G. In particular, there exists a (regular) band B_0 with the property that any idempotent generated semigroup whose biordered set is isomorphic to that of B_0 must have all its subgroups abelian.