Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10901134) and the Science Foundation of the Department of Education of Yunnan Province (Grant No.2011Y478).
In this paper we study the isomorphisms of two $*$-bisimple type $A~ \omega^{2}$-semigroups such that $\mathcal{D^{*}}=\widetilde{D}$ and obtain a criterion for isomorphisms of two such semigroups.