Free idempotent generated semigroups: The word problem and structure via gain graphs

Building on the previous extensive study of Yang, Gould and present author, we provide a more precise insight into group-theoretical ramifications word problem for free idempotent generated semigroups over finite biordered sets. We prove that such problems are in fact equivalent to computing intersections cosets certain subgroups direct products maximal semigroup question, thus providing decidability those under assumptions related Howson property coset intersection property. also basic sketch global semigroup-theoretical structure an arbitrary semigroup, including characterisation Green’s relations key parameters non-regular $${\cal D}$$ -classes. In particular, all Schützenberger groups IG(?) set ? must be among divisors IG(?).