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A subset of a partially ordered set is said to be cofinal if for every there exists some such that . Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements.
A subset of a partially ordered set is said to be a lower set of if it is downward closed: if and then . Every lower set of a finite ordered set is equal to the smallest lower set containing all maximal elements of .
Read more about this topic: Maximal Element
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