偏序集的主理想连续性和闭区间连续性

CLOSED INTERVAL CONTINUITY AND PRINCIPAL IDEAL CONTINUITY ON POSETS

在偏序集上引入并考察了主理想连续性（ 相应地：代数性） 和闭区间连续性（ 相应地：代数性） ． 证明了主理想连续性（ 代数性） 和通常的偏序集连续性（ 代数性） 是等价的． 构造了反例说明闭区间连续性与通常的偏序集连续性互不蕴涵． 证明了连续偏序集（ 代数偏序集） 如果非空有限集有多值并，则必定是闭区间连续集（ 代数集） ；而闭区间连续性（ 代数性） 附加下方控制条件，则蕴涵通常连续性（ 代数性） ． 得到了Ｓｃｏｔｔ Ｄｏｍａｉｎ 的两个新的等价刻画．

In this paper, principal ideal continuity (respectively, algebraicity) and closed interval continuity (respectively, algebraicity) on posets are introduced and examined. It is proved that principal ideal continuity (algebraicity) is equivalent to the common continuity (algebraicity) on posets. Some counterexamples are constructed to show that closed ideal continuity and the common continuity on posets can not imply each other. It is also proved that continuous (algebraic) posets with multiple sups for non empty finite subsets are closed interval continuous (algebraic) and that closed interval continuous (algebraic) posets fulfilling the controlling condition from below are continuous (algebraic) posets as well. Two new characterizations for Scott Domain are also given.