完全格

Multi tool use
在数学中,完全格是在其中所有子集都有上确界(并)和下确界(交)的偏序集。完全格出现于数学和计算机科学的很多应用中。作为格的特殊实例,在序理论和泛代数中都有所研究。
完全格一定不能混淆于完全偏序(cpo),它构成严格的更加一般的一个偏序集合类别。更特殊的完全格是完全布尔代数和完全Heyting代数(locale)。
形式定义
偏序集合(L, ≤)是完全格,如果L的所有子集A在(L, ≤)中都有最大下界(下确界,交)和最小上界(上确界,并)二者。它们被表示为:
⋀{displaystyle bigwedge }
A(交)和⋁{displaystyle bigvee }
A(并)。
注意在A是空集的特殊情况下,L的任何元素都是空集的上界和下界,A的交将是L的最大元素。类似的,空集的并生成最小元素。因为定义还确保了二元交和并的存在,完全格因为形成了特殊种类的有界格。
上述定义的更多蕴涵在关于序理论中完备性性质的文章中讨论。
例子
- 给定集合的幂集,按包含排序。上确界给出自这些子集的并集而下确界给出自这些子集的交集。
单位区间[0,1]和扩展的实数轴,通过平常的全序和普通的上确界和下确界。实际上,全序集合(带有它的序拓扑)作为拓扑空间是紧致的,如果它作为一个格是完全的。
- 非负整数按整除排序。这个格最小元是1,因为它可以被任何其他数整除。可能令人惊奇的是,最大元是0,因为它可以被任何数整除。有限集合的上确界给出自最小公倍數而下确界给出自最大公约数。对于无限集合,上确界将总是0而下确界可以大于1。例如,所有偶数的集合有2作为最大公约数。如果从这个结构中去掉0它仍是格但不再是完全的。
- 任何给定群的子群在包含关系下。(尽管这里的下确界是平常的集合论交集,但子群的集合的上确界是子群的集合论并集所生成的子群,而不是集合论并集自身)。如果e是G的单位元,则平凡的群{e}是G的极小子群。而极大子群是群G自身。
模的子模按包含排序。上确界给出自子模的和而下确界给出自交集。
- 环的理想子环按包含排序。上确界给出自理想子环的和而下确界给出自交集。
拓扑空间的开集按包含排序。上确界给出自开集的并而下确界给出自交集的内部。
实数或复数的向量空间的凸集按包含排序。下确界给出自凸集的交集而上确界给出自并集的凸包。
- 在集合上拓扑按包含排序。下确界给出自拓扑的交集,而上确界给出自拓扑的并集所生成的拓扑。
- 在集合上的所有传递关系的格。
多重集的子多重集的格。
- 在集合上的所有等价关系的格;等价关系~被认为比≈更小(或"更细"),如果x~y总是蕴涵x≈y。
引用
参见
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