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

${\displaystyle a<b\Rightarrow \phi (a)\prec \phi (b)}$

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

${\displaystyle a<b\Rightarrow \phi (b)\prec \phi (a)}$

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 ${\displaystyle F}$ ${\displaystyle D}$

${\displaystyle f_{1}\leq _{f}f_{2}\iff f_{1}(x)\leq f_{2}(x){\text{ für alle }}x\in D}$.

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

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