As a well-known semigroup, Hamilton semigroup acts on continuous functions in an algebra system. According to the characteristics of Hamilton semigroup, some basic properties of the semigroup and graph theoretical representation are used to investigate the structure of endomorphism to Hamilton semigroup. First of all, a new multiplication, which can solve the problem with computing endomorphism for Hamilton semigroup, is defined and then the main result that the endomorphism of Hamilton semigroup is also a Hamilton semigroup is given. Moreover, by adopting the partial order to the endomorphism of Hamilton semigroup, an expression of graph theory is established and that the Hasse graph for endomorphism semigroup is an oriented forest is proved.