We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.
In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related
to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979,
as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts
with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup,
and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the
fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from
regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that
an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the
*-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some
conclusions about the lattice of congruences of a *-regular semigroup.