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## Maximal and minimal elements.

In this subsection, we consider a poset .

Definition 4.4.5
1. An element in is called a maximal element in if there exist no such that .
2. An element in is called a minimal element in if there exist no such that .

The Hasse diagram of a (finite) poset is a useful tool for finding maximal and minimal elements: they are respectively top and bottom elements of the diagram. For example, in , is a minimal element and is a maximal element. Note, however, that this example is quite special: there is a unique maximal element and a unique minimal element. This situation is not general. Consider the set ordered by division. Its Hasse diagram is displayed in Figure 9.

It appears clearly that has three minimal elements (namely , and ) and three maximal elements (namely , and ).

Definition 4.4.6
1. An element is the greatest element of if, for any element , we have .
2. An element is the least element of if, for any element , we have .

Example 4.4.7
1. The set has a least element (namely ) and a greatest element (namely ).
2. The poset , ordered by division, has neither a least nor a greatest element.
3. The poset , ordered by division, has as its least element and has no greatest element.

Definition 4.4.8   Let be a poset. The set is well-ordered is every non empty subset of has a least element.

For example, the set of all natural numbers, ordered by is well-ordered.

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root 2002-06-10