The structure of Q⁎-inverse semigroups

As a generalization of quasi-inverse semigroups in the class of regular semigroups, we consider the Q⁎-inverse semigroups which are idempotent-connected abundant semigroups with regular bands. In this paper, a construction theorem of Q⁎-inverse semigroups is given by using the wreath product of some semigroups. It is proved that a semigroup S is a Q⁎-inverse semigroup if and only if S is a spined product of an L⁎-inverse semigroup and an R⁎-inverse semigroup. Thus the structure of Q⁎-inverse semigroups is fully described and the results on quasi-inverse semigroups obtained by M. Yamada in 1973 are extended and amplified.