Extensions of endomorphisms of

For a compact space we consider extending endomorphisms of the algebra to be endomorphisms of Arens-Hoffman and Cole extensions of . Given a non-linear, monic polynomial , with semi-simple, we show that if an endomorphism of extends to the Arens-Hoffman extension with respect to , then it also extends to the simple Cole extension with respect to . We show that the converse to this is false. For a locally connected, metric we characterize the algebraically closed in terms of the extendability of endomorphisms to Arens-Hoffman and to simple Cole extensions.