On a generalized partial wreath product of semigroups

The construction of wreath product is well–known and widely used both for groups and semigroups. It gives a possibility to get new group (semigroups) from old. The semigroup of partial automorphisms of a rooted tree of a special kind is isomorphic to partial wreath product of inverse symmetric semigroups. However, sometimes it is necessary to consider a construction similar to wreath product, but which allows to get wreath product of a semigroup (group) with a set of semigroups (groups). It is shown In the present paper a generalization of partial wreath product of semigroups is introduced. It is proved that the generalized partial wreath product of semigroups is a semigroup. Besides, it is shown that the generalized wreath product of inverse semigroups is an inverse semigroup. It is also shown that under certain conditions the generalized wreath product of semigroup is isomorphic to the direct product of wreath products of semigroups. The description of idempotents of a generalized partial wreath product is provided.