HS-stability and complex products in involution semigroups

When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ${}^{\ast }$-semigroups and commutative involution semigroups.