偏序集上的相对理想及其分解

Relative Ideal on Poset and Its Decomposition

作为Domain理论中理想这一经典概念自然的推广,文章借助一致集和相对定向集引入相对理想和一致理想的概念,分析理想、一致理想和相对理想三者之间的内在联系.利用极大理想的定义思想引入极大相对理想和局部极大相对理想的概念,对极大相对理想和局部极大相对理想二者的存在性进行证明.最后给出并证明偏序集上相对理想的一个分解定理:设P是偏序集,则其每一个真相对理想都可以表示成若干个相对局部极大相对理想的交.

As a natural extension of the ideal in domain theory,we firstly will introduce relative ideal and uni⁃form ideal with the help of uniform set and relative directional set,and analyze the internal relations among ideal,uniform ideal and relative ideal.Secondly,we introduce maximal relative ideal and local maximum rela⁃tive ideal by using the idea of maximal ideal,and prove the existence of the maximal relative ideal and the lo⁃cal maximal relative ideal.Finally,we show the decomposition theorem of the relative ideal on a poset:let P be a poset,then every proper relative ideal can be expressed as the intersection of some relative local maxi⁃mal relative ideals.