| In this paper, we discuss the structure of finite poset and give two new proofs of the Dilworth theorem.In part Ⅰ, we first introduce the concepts of "independent set" (greatest uncomparable set), the "degree" of a poset and "top set". Then we obtain the "layer theorem of poset". By means of the properties of top set and by applying the method of induction to the degree of poset, we derive a new proof of the Dilworth theorem (theorem 2).In part Ⅱ, according to various different cases of the indepedent set of poset, we classify posets into two types named A-type and B-type, and prove that any poset can be expressed in unions of finite B-type posets, and from this result we give another new proof of the Dilworth theorem. In part Ⅲ. we use properties of top set to give a simplified proof of the original proof of the Dilworth theorem. |
