A wreath product decomposition of E -solid locally inverse semigroups

In this paper, a kind of restricted wreath product of an arbitrary band of groups by an arbitrary generalized inverse semigroup is introduced first. This wreath product turns out to be an E-solid semigroup, and if one comes out at the beginning with a normal band of groups, then it shows itself to be also a locally inverse semigroup. Then it is proved that every E-solid locally inverse semigroup S can be embedded in such a restricted wreath product of a normal band of groups Q by a generalized inverse semigroup K. Here K can be taken to be the greatest orthodox semigroup homomorphic image of S and Q can be taken to be the preimage of the band of idempotents of K under the canonical homomorphism of S onto its quotient K. In this way, the given E-solid locally inverse semigroup S decomposes into a restricted wreath product of its components Q and K whose structure derives directly from that of S.