Infimum and supremum
Minimal upper bounds

Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a totally ordered set, like the real numbers, the concepts are the same.

As an example, let S {\displaystyle S} be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from S {\displaystyle S} together with the set of integers Z {\displaystyle \mathbb {Z} } and the set of positive real numbers R +, {\displaystyle \mathbb {R} ^{+},} ordered by subset inclusion as above. Then clearly both Z {\displaystyle \mathbb {Z} } and R + {\displaystyle \mathbb {R} ^{+}} are greater than all finite sets of natural numbers. Yet, neither is R + {\displaystyle \mathbb {R} ^{+}} smaller than Z {\displaystyle \mathbb {Z} } nor is the converse true: both sets are minimal upper bounds but none is a supremum.
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