Greatest element and least element
In mathematics, especially in order theory, the greatest element of a subset
of a partially ordered set (poset) is an element of
and if it also satisfies: By switching the side of the relation that
Greatest elements are closely related to upper bounds.
Importantly, an upper bound of
which is completely identical to the definition of a greatest element given before.
to simultaneously not have a greatest element and 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.
This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either.
is a partially ordered set then
is defined to mean a maximal element of the subset
Like upper bounds and maximal elements, greatest elements may fail to exist.
In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum.
Similar conclusions hold for least elements.
One of the most important differences between a greatest element
Because preorders are reflexive (which means that
In general, however, preordered sets (and even directed partially ordered sets) may have elements that are incomparable.
This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important if statement.
is a set containing at least two (distinct) elements and define a partial order
holds, which shows that all pairs of distinct (i.e. non-equal) elements in
[note 1] In contrast, if a preordered set
is also partially ordered then it is possible to conclude that
However, the uniqueness conclusion is no longer guaranteed if the preordered set
is a non-empty set and define a preorder
The directed preordered set
has multiple distinct greatest elements.
The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively.
The notation of 0 and 1 is used preferably when the poset is a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top.
The existence of least and greatest elements is a special completeness property of a partial order.
Further introductory information is found in the article on order theory.
