Supremum

In mathematics, the supremum and infimum terms or least upper bound and greatest lower bound occur in the study partially ordered sets. Clearly, the supremum, an upper bound that is less than all other possible upper bounds. If one exists, it is uniquely determined. According to the infimum is a lower bound that is greater than all other lower bounds. The concept is used in different variations in almost all mathematical sciences.

• 2.1 Uniqueness and existence
• 2.2 properties with respect to a ambient epsilon

• 4.1 Real Numbers
• 4.2 Other semi ordered sets

Definitions

Suprema ( and infima ) of amounts

View

The supremum (in German " Supreme" ) an amount is related to the maximum of a set and is - clearly spoken - an element (above ) of all other elements is " on" all or "beyond". The term " the other " is to indicate that the supremum does not have to be the largest element "among others", but certainly can be outside ( "beyond" ) the crowd. And because there can be several elements that correspond to this view, is chosen for uniqueness of the smallest element that has this property; so to speak, the element that is "closest " or "directly" above all others - the supremum so called a " directly In Lying ". Elements which, although they are above all elements of a set, but not necessarily in an immediate way, hot upper bounds. This then gives the definition of the supremum as least upper bound of a set.

The infimum ( German " lower limit " ) of a set is defined analogously, as " immediately Among Lying " or greatest lower bound.

In the real case

This view is easy to sets of real numbers transferred (as subsets of the real numbers ): Let

The set of real numbers is less than 2 2 Then the supremum of X ( in ). Because 2 is an upper bound of X since they ( actually even strictly greater ) is greater than or equal to each element of X - ie " about lies ." However, in contrast to the number 4, which is also an upper limit, there is not a number less than 2, which is also an upper bound of X. Therefore, two least upper bound of X, therefore supremum.

With a small modification, the relationship of supremum and maximum is then clear. The maximum is in fact the largest element " among all the elements " of a quantity:

X is apparently no maximum, since it reflects to any real number is a real number that is greater than, for example, with the choice. The number 2 is the supremum indeed greater than all elements of X, but is not in X, since it is not strictly smaller than themselves. Now consider the quantity

SO 2 X maximum, as it is less than or equal to itself and there is no number greater than 2 which is less than or equal to 2. Likewise, 2 but also the supremum of X as already of X, since the same conditions are met such as there.

In fact, each maximum is always supremum. Therefore, it is also common, not elemental to define the term maximum, but to appoint him as a special case of the supremum, if this is even element of the set whose supremum is it. - The same is valid for the minimum.

Generally

However, upper and lower bounds and suprema and infima can be considered not only on the real numbers, but more generally on partially ordered sets. The formal definitions are as follows:

If M is a partially ordered set with partial order and T is a subset of M then:

If M is the set of real numbers, then:

• If T is bounded from above, not empty, then T has a least upper bound ( proof idea below) and they are called upper bound or supremum of T - in characters.
• If T is bounded from below and not empty, then T has a greatest lower bound ( proof analog) and they are called lower bound or infimum of T - in characters.
• If T is bounded above and contain the supremum of T in T, is defined as the supremum as a maximum of T, in characters.
• If T is bounded from below and include the infimum of T in T, is defined as the infimum as a minimum of T, in characters.
• Is the T upwards unlimited, you would write (see infinity ). However, the symbol ∞ is not a real number and not the supremum of T in the sense defined here: ∞ as Supremumswert is just the formal notation that no upper bound is available, see also in advanced real numbers. It is sometimes referred to in this context ∞ as " inauthentic supremum ".
• If T is down indefinitely, to write analogously.

Suprema ( and infima ) of pictures

Pictures generally

The concept of supremum on quantities shall apply mutatis mutandis also to illustrations (functions). Because the image of a figure is always a lot. Namely, for a picture

The amount

The so-called element images, that is, the images of the individual elements of X under the mapping f

Now if Y is a partially ordered set, then we define

As the supremum of f on X - if it exists in Y.

Similarly is defined the infimum of f on X.

The defining property of the supremum can be formulated as a Galois connection between and: for all and is

This is equipped with the pointwise order and.

Follow as pictures

Summing up a sequence a1, a2, a3, ... of elements of Y as picture

On - so according to

- It is clear from the definition of the supremum ( infimum ) of pictures instantly the definition of the supremum ( infimum ) of a sequence (an ) - if it exists in Y.

Properties

Uniqueness and existence

If b is an upper bound of T and c > b, then c is also an upper bound of T. Conversely, if c is no upper bound of T and b < c, then b is also no upper bound of T. The same is true for lower bounds.

The supremum of T is (in the case of its existence ) is uniquely determined. The same is true for the infimum of T.

It is possible that a subset T of a partially ordered set M has several minimal upper bounds, ie upper bounds, so that each small element is no upper bound. However, when T is greater than a minimum upper limit, there is no smallest upper bound, ie no supremum of T. An example is the set with the partial order. Here, the two minimal upper bounds c and d

Properties with respect to an epsilon - ambient

Let be a nonempty subset of the real numbers, then also applies to the

• Supremum of:

• Infimum of:

Existence of the supremum for bounded subsets of the real numbers

The existence of the supremum of a bounded subset of the real numbers can be shown in several ways:

First, one can determine the existence of supremum and infimum for bounded subsets of the real numbers just as an axiom. This requirement is often called Supremumsaxiom or completeness axiom.

Assuming the axiom that each of intervals exactly defines a real number may be as follows: One constructs a nest of intervals that includes the supremum. To this end, we construct two sequences, one of which, is first increasing and does not consist of upper bounds of the second, is monotonically decreasing and consists of upper bounds of, so that is still considered that the distances corresponding sequence elements go to 0 ( by each considered the middle of the interval and decide whether it is an upper bound or not). This yields the common limit of the two sequences as the smallest upper bound of, because: Every element of each element is less than or equal to the upper sequence, ie less than or equal, therefore an upper limit of the. And any real number that is less than, is less than at least one element ( for a certain amount ) of the lower sequence, so no upper limit.

An equivalent formulation for the existence of the supremum is the intersection axiom after each Dedekind cut is generated by a real number.

Examples

Real Numbers

The following examples are based on subsets of the real numbers.

Other semi- ordered sets

In any non-empty bounded above or below subset has a supremum and infimum. Looking at other quantities on which order relations are defined, so this is not mandatory:

• The set of rational numbers is ordered with respect to the natural order total. The quantity is limited, for example by the number up, but has no supremum in.
• In the set of natural numbers with their natural order every number is both lower and upper bound of the empty set. Therefore, no infimum, but probably true.
• In respect of the inclusion partially ordered set, the amount is limited both by the member and by upward. A supremum, ie a least upper bound of, does not exist in however.