Hamilton半群是一种重要的代数结构。针对Hamilton半群的特点，利用其半群性质和图论结果对其自同态的结构进行了研究。首先定义了其自同态的一种乘法运算，并证明了Hamilton半群的自同态也构成一个Hamilton半群。其次，在引入半序关系之后，给出了Hamilton半群的自同态半群的一个图论表示，即关于半序关系的覆盖图是有向森林。

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.