Endomorphisms of Clifford semigroups with injective structure homomorphisms

In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups $G_{\alpha}\cup G_{\beta}$ ($\alpha >\beta $) with an injective structure homomorphism, where $G_{\alpha}$ has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.