The Automaticity of union of disjoint Semigroups

An essential question has been arisen whether the classes of automatic semigroups are closed under certain semigroup constructions. We give here a necessary and sufficient condition for automaticity of some special cases of the disjoint union of two automatic semigroups. In particular, for two non-monoid semigroups S and T we consider the normal ideal extention of S by T. Also, we here study the behaviour of decomposable semigroups as a disjoint union of their sub-semigroups