Monotonous illustration

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A monotonically increasing real function (red) is isotonic and a monotonically decreasing real function (blue) is antitone with respect to the ≤-order on the real numbers

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 .


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.


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 .


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 .