# Monotonous illustration

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 .

## definition

If and are two semi-ordered sets , then a mapping is called isotonic , order- preserving or an order homomorphism , if for all elements${\ displaystyle (G, \ leq)}$${\ displaystyle (H, \ preceq)}$${\ displaystyle \ phi \ colon G \ rightarrow H}$ ${\ displaystyle a, b \ in G}$

${\ displaystyle a \ leq b \ Rightarrow \ phi (a) \ preceq \ phi (b)}$

holds true, and antitone or reversal , if for all${\ displaystyle a, b \ in G}$

${\ displaystyle a \ leq b \ Rightarrow \ phi (b) \ preceq \ phi (a)}$

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${\ displaystyle <}$${\ displaystyle \ prec}$${\ displaystyle \ phi}$ ${\ displaystyle a, b \ in G}$

${\ displaystyle a

applies, and strictly antiton if for all${\ displaystyle a, b \ in G}$

${\ displaystyle a

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 .${\ displaystyle (\ mathbb {N}, \ leq)}$${\ displaystyle (\ mathbb {R}, \ leq)}$${\ displaystyle \ psi (i) = a_ {i}}$${\ displaystyle (a_ {i}) _ {i \ in \ mathbb {N}}}$
• If there is an arbitrary set and its power set , an order relation can be defined on the power set by the subset relation . A mapping from to defined by is monotonic if and only if the set sequence is a monotonic set sequence .${\ displaystyle M}$${\ displaystyle {\ mathcal {P}} (M)}$${\ displaystyle \ subset}$${\ displaystyle (\ mathbb {N}, \ leq)}$${\ displaystyle ({\ mathcal {P}} (M), \ subset)}$${\ displaystyle \ psi (i) = A_ {i}}$ ${\ displaystyle (A_ {i}) _ {i \ in \ mathbb {N}}}$
• An order can be defined on a set of real-valued functions with a domain${\ displaystyle F}$${\ displaystyle D}$
${\ displaystyle f_ {1} \ leq _ {f} f_ {2} \ iff f_ {1} (x) \ leq f_ {2} (x) {\ text {for all}} x \ in D}$.
A mapping from to defined by is monotonic if and only if the function sequence is a monotonic function sequence .${\ displaystyle (\ mathbb {N}, \ leq)}$${\ displaystyle (F, \ leq _ {f})}$${\ displaystyle \ psi (i) = f_ {i}}$ ${\ displaystyle (f_ {i}) _ {i \ in \ mathbb {N}}}$

### Monotonous functions

• The monotonic mappings from to are exactly the monotonic real functions .${\ displaystyle (\ mathbb {R}, \ leq)}$${\ displaystyle (\ mathbb {R}, \ leq)}$
• If one considers orders on the which are defined by generalized inequality , then monotonic mappings from to are exactly the K-monotonic functions .${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ preccurlyeq _ {K}}$${\ displaystyle (\ mathbb {R} ^ {n}, \ preccurlyeq _ {K})}$${\ displaystyle (\ mathbb {R}, \ leq)}$
• Monotonic mappings which are provided with the Loewner partial order after mapping the space of the symmetrical real matrices are called matrix-monotonic functions .${\ displaystyle S ^ {n}}$${\ displaystyle (\ mathbb {R}, \ leq)}$
• Measures on an algebra over a basic set are monotonous mappings from to .${\ displaystyle \ sigma}$ ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle \ Omega}$${\ displaystyle ({\ mathcal {A}}, \ subset)}$${\ displaystyle (\ mathbb {R} _ {+} \ cup \ {+ \ infty \}, \ leq)}$
• External dimensions on the basic set are monotonous mappings from to .${\ displaystyle \ Omega}$${\ displaystyle ({\ mathcal {P}} (\ Omega), \ subset)}$${\ displaystyle (\ mathbb {R} _ {+} \ cup \ {+ \ infty \}, \ leq)}$

## 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. ${\ displaystyle \ phi ^ {- 1}}$${\ displaystyle \ phi}$${\ displaystyle G = \ {a, b, c \}}$${\ displaystyle a \ leq b, a \ leq c}$${\ displaystyle H = \ {a, b, c \}}$${\ displaystyle a \ preceq b \ preceq c}$${\ displaystyle \ phi \ colon G \ to H}$${\ displaystyle \ phi (a) = a, \ phi (b) = b, \ phi (c) = c}$${\ displaystyle \ phi}$${\ displaystyle \ phi ^ {- 1}}$${\ displaystyle b \ preceq c}$${\ displaystyle b \ leq c}$

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. ${\ displaystyle \ psi \ circ \ phi}$${\ displaystyle \ phi \ colon F \ to G}$${\ displaystyle \ psi \ colon G \ to H}$ ${\ displaystyle \ operatorname {id} \ colon G \ to G}$${\ displaystyle \ phi \ colon G \ to G}$

## Related terms

A mapping between two semi-ordered sets and , for which the converse ${\ displaystyle \ phi \ colon G \ rightarrow H}$${\ displaystyle (G, \ leq)}$${\ displaystyle (H, \ preceq)}$

${\ displaystyle a \ leq b \ Leftarrow \ phi (a) \ preceq \ phi (b)}$

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 ${\ displaystyle a, b \ in G}$

${\ displaystyle a \ leq b \ Leftrightarrow \ phi (a) \ preceq \ phi (b)}$

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 . ${\ displaystyle a, b \ in G}$${\ displaystyle (G, \ leq) \ cong (H, \ preceq)}$${\ displaystyle (G, \ leq) \ cong (\ phi (G), \ preceq)}$