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Concept
Greatest element and least element
Summary
In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually, that is, it is an element of that is smaller than every other element of
Let be a preordered set and let
An element is said to be if and if it also satisfies:
for all
By switching the side of the relation that is on in the above definition, the definition of a least element of is obtained. Explicitly, an element is said to be if and if it also satisfies:
for all
If is also a partially ordered set then can have at most one greatest element and it can have at most one least element. Whenever a greatest element of exists and is unique then this element is called greatest element of . The terminology least element of is defined similarly.
If has a greatest element (resp. a least element) then this element is also called (resp. ) of
Greatest elements are closely related to upper bounds.
Let be a preordered set and let
An is an element such that and for all Importantly, an upper bound of in is required to be an element of
If then is a greatest element of if and only if is an upper bound of in In particular, any greatest element of is also an upper bound of (in ) but an upper bound of in is a greatest element of if and only if it to
In the particular case where the definition of " is an upper bound of " becomes: is an element such that and for all which is to the definition of a greatest element given before.
Thus is a greatest element of if and only if is an upper bound of .
If is an upper bound of that is not an upper bound of (which can happen if and only if ) then can be a greatest element of (however, it may be possible that some other element a greatest element of ).
In particular, it is possible for to simultaneously have a greatest element for there to exist some upper bound of .
Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers.
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