# Monotonous illustration

In mathematics, a **monotonous mapping** is a mapping between two semi-ordered sets , in which the order of the respective picture elements of the target set can be inferred from the order of two elements of the definition set . If the order of the elements is retained, one speaks of an **isotonic** or **order-preserving mapping** or an **order homomorphism** . If the order is **reversed** , one speaks of an **antitonic** or **order-reversing mapping** .

Well-known examples of monotonic mappings are (not necessarily strictly) monotonic real functions . The concept of monotony is also applied more generally to vector-valued functions , operators , sequences of numbers , sequences of sets and sequences of functions .

## definition

If and are two semi-ordered sets , then a mapping is called *isotonic* , *order-* preserving or an *order homomorphism* , if for all elements

holds true, and *antitone* or *reversal* , if for all

applies. A mapping is called *monotonic* if it is isotonic or antitonic. If the corresponding strict orders and are defined, a mapping is called *strictly isotonic* if for all elements

applies, and *strictly antiton* if for all

applies. A mapping is called *strictly monotonic* if it is strictly isotonic or strictly antitone.

## Examples

### Monotonous consequences

- A mapping from to defined by is monotonic if and only if the sequence is a monotonic sequence .
- If there is an arbitrary set and its power set , an order relation can be defined on the power set by the subset relation . A mapping from to defined by is monotonic if and only if the set sequence is a monotonic set sequence .
- An order can be defined on a set of real-valued functions with a domain

- .

- A mapping from to defined by is monotonic if and only if the function sequence is a monotonic function sequence .

### Monotonous functions

- The monotonic mappings from to are exactly the monotonic real functions .
- If one considers orders on the which are defined by generalized inequality , then monotonic mappings from to are exactly the K-monotonic functions .
- Monotonic mappings which are provided with the Loewner partial order after mapping the space of the symmetrical real matrices are called matrix-monotonic functions .
- Measures on an algebra over a basic set are monotonous mappings from to .
- External dimensions on the basic set are monotonous mappings from to .

## properties

An isotonic mapping represents an order homomorphism , whereas an antitonic mapping an order antihomomorphism . A bijective isotonic mapping, the inverse of which is also isotonic, is an order isomorphism , a bijective antitonic mapping with an antitonic inverse is an order anti-isomorphism .

The inverse of a bijective isotonic mapping does not necessarily have to be isotonic itself. For example, if with and with and are the (identical) mapping , then is isotonic, but not, because it does not imply . The same applies to the antitonicity of the inverse of a bijective antitonic mapping. Therefore, the isotonic or the antitonic of the inverse must be explicitly required for iso- and anti-isomorphisms.

The execution of two isotonic mappings one after the other and is isotonic again. After the identity map is isotonic, represents the amount of isotonic self-images with the sequential execution as linking a monoid (the Endomorphismenmonoid ). In general form partially ordered sets with isotonic pictures a (Cartesian completed) category . The bijective isotonic self-mappings with isotonic inverses form a group (the automorphism group ) with the sequential execution as a link . The successive execution of two antitonic images is not antitonic again, but isotonic. The sequential execution of an isotonic with an antitonic mapping is always antitonic, regardless of the sequence.

## Related terms

A mapping between two semi-ordered sets and , for which the converse

applies to all , means orderly reflective . An order-reflecting image is always injective . An image that both preserves order and reflects order, so for them

applies to all , is called order embedding . A surjective order embedding is an order isomorphism and one then writes . Only applies to an order embedding .

## literature

- Rudolf Berghammer: Regulations and Associations . Basics, procedures and applications. Springer Vieweg, Wiesbaden 2013, ISBN 978-3-658-02710-0 , doi : 10.1007 / 978-3-658-02711-7 .
- Steven Roman: Lattices and Ordered Sets . Springer, 2008, ISBN 978-0-387-78900-2 , doi : 10.1007 / 978-0-387-78901-9 .
- Bernhard Ganter:
*Discrete Mathematics: Ordered Quantities*, Springer, 2013, ISBN 978-3-642-37500-2