序半群的阿基米德半群的半格分解

On Semilattice Decompositions into Archimedean Subsemigroups of Ordered Semigroups

本文通过一个序半群S上的一些二元关系以及它的理想的根集的性质该序半群是阿基米德半群的半格,特别地是阿基米德半群的链的刻划,证明了S是阿基米德链当且仅当S是准素的.通过序半群的m-系的概念,证明了S的任意半素理想是含它的所有素理想的交,并通过该结论,证明了S是阿基米德半群的链当且仅当S是阿基米德半群的半格且S的所有素理想关于集包含关系构成链.作为应用,该结论在一般的半群（没有序）[1]中也成立.

In this paper, some characterizations for an ordered semigroup S to be a semilattice of archimedean semigroups, in particular, a chain of archimedeam subsemi-groups are given by some binary relations on S and the properties of radical subset of the ideals of S. The result that every semiprime ideal of S is the intersection of prime ideals containing it is proved by the concept of m-system of S. Furthermore, using the prime radical theorem, we prove that an ordered semigroup S is a chain of archimedean semigroups if and only if S is a semillatice of archimedean semigroups and all prime ieals of S is a chain. As an application the corresponding results [1] on semigroups without orders can be obtained by moderate modificaitons.