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预序关系

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预序关系(简称预序,又称先序preorder)、在数学中,是一类接近于偏序关系的二元关系,但仅满足自反性和传递性而不满足反对称性。偏序的大多数理论均可扩展到预序。

定义

考虑集合 P 及其上的二元关系 。若 具有自反性和传递性,则称 预序。具体来说,对任意 P 的元素 abc,下列性质成立:

a a (自反性)
a bb c,则 a c (传递性)

带预序的集合称为预序集合。同时满足反对称性(若 a bb a,则 a = b)的预序为偏序

说明

作为特例,空集上的空关系为一预序。空集加上空关系构成一预序集。

导出偏序

将预序集的等价元素等同起来,可得到由该预序集所导出的偏序集。具体过程如下:定义预序集 X 上的等价关系 ,使得 a b 当且仅当 a bb a。定义所得商集 (所有 等价类构成的集合)上的序关系 ,使得[x] [y] 当且仅当 x y。由 的构造可知, 的定义与所选等价类的代表元素无关,故上述定义明确。易证该关系为一偏序。

举例

参见

参考文献

  • Schröder, Bernd S. W., Ordered Sets: An Introduction, Boston: Birkhäuser, 2002, ISBN 0-8176-4128-9 
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预序关系
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