Order isomorphism

An order isomorphism is a term from order theory , a sub-area of ​​mathematics. It enables the unambiguous transfer of less than or equal relations between sets.

definition

If two partial orders and are given, a mapping is called ${\displaystyle (G,\leq _{G})}$ ${\displaystyle (H,\leq _{H})}$

${\displaystyle \psi \colon G\rightarrow H}$

an order isomorphism if there is a bijective isotonic map whose inverse map is also an isotonic map. ${\displaystyle \psi }$ ${\displaystyle \psi ^{-1}}$

If an order isomorphism exists between and , the existence can also be expressed with and and are referred to as order isomorphism . If an order isomorphism maps a set to itself, it is an automorphism and is also called order automorphism . ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle G\cong H}$ ${\displaystyle G}$ ${\displaystyle H}$

Examples

• The identical mapping of every partial / total order is at the same time an order automorphism. ${\displaystyle \operatorname {id} _{G}\colon G\rightarrow G,a\mapsto a}$
• No order isomorphism can be explained between restricted open and restricted semi-open or closed intervals , because the latter have smallest and / or largest elements , the former do not.
• Let be a function that maps from into the set of all square numbers: The function reads new: There is also an inverse function of this new function : Thus is bijective. Because is bijective and isotonic and because the orders and are total, there is also an order isomorphism. ${\displaystyle g}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle Q}$
  ${\displaystyle Q=\left\{n^{2}\mid n\in \mathbb {N} \right\}\subset \mathbb {N} }$
  ${\displaystyle g\colon \mathbb {N} \rightarrow Q,x\mapsto x^{2}}$
  ${\displaystyle g}$ ${\displaystyle g^{-1}\colon Q\rightarrow \mathbb {N} ,x\mapsto {\sqrt {x}}}$
  ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle (\mathbb {N} ,\leq )}$ ${\displaystyle (Q,\leq )}$ ${\displaystyle g}$
• The identical figure is a bijective antitone figure between and . ${\displaystyle \operatorname {id} _{\mathbb {R} }\colon \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto x}$ ${\displaystyle (\mathbb {R} ,\leq )}$ ${\displaystyle (\mathbb {R} ,\geq )}$
• The function of the additively inverse element is an involution and thus also a bijection. is an antitonic map of in itself and also an isotonic map of to . Furthermore, there is even an isomorphism of order, since the order relations are total orders and there is bijective. This applies to the whole numbers , the rational numbers and the real numbers , among others . ${\displaystyle f\colon M\rightarrow M,x\mapsto -x}$ ${\displaystyle f}$ ${\displaystyle (M,\leq )}$ ${\displaystyle (M,\leq )}$ ${\displaystyle (M,\geq )}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle M=\mathbb {Z} }$ ${\displaystyle M=\mathbb {Q} }$ ${\displaystyle M=\mathbb {R} }$
• The component-wise-less-or-equal-relation on any n-tuples forms a real partial order that does not meet the totality criterion. The function is obviously bijective, the inverse function is . On is also both and isotonic, which distinguishes and as order isomorphisms - more precisely as an order automorphism, because both the definition and the target sets are . ${\displaystyle (a_{1},\cdots ,a_{n})\leq ^{n}(b_{1},\cdots ,b_{n}):\Longleftrightarrow \forall i\in [1,n]\cap \mathbb {N} :a_{i}\leq b_{i}}$ ${\displaystyle n\geq 2}$ ${\displaystyle \Psi \colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2},(x,y)\mapsto (2x,2y)}$ ${\displaystyle \Psi ^{-1}\colon \mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2},(x,y)\mapsto \left({\frac {x}{2}},{\frac {y}{2}}\right)}$ ${\displaystyle (\mathbb {R} ^{2},\leq ^{2})}$ ${\displaystyle \Psi }$ ${\displaystyle \Psi ^{-1}}$ ${\displaystyle \Psi }$ ${\displaystyle \Psi ^{-1}}$ ${\displaystyle \mathbb {R} ^{2}}$

composition

If an order isomorphism between and and if an order isomorphism between and , then there is also an order isomorphism between and . The property - that it is order isomorphisms - guarantees that the images are bijective, which means that the function created by the composition must also be bijective. The bijectivity also guarantees that the image of is equal to the target set of . ${\displaystyle f\colon U\rightarrow V}$ ${\displaystyle (U,\leq _{U})}$ ${\displaystyle (V,\leq _{V})}$ ${\displaystyle g\colon V\rightarrow W}$ ${\displaystyle (V,\leq _{V})}$ ${\displaystyle (W,\leq _{W})}$ ${\displaystyle f\circ g\colon U\rightarrow W}$ ${\displaystyle (U,\leq _{U})}$ ${\displaystyle (W,\leq _{W})}$ ${\displaystyle f}$ ${\displaystyle f}$

properties

• Because of the bijectivity, it is true that

${\displaystyle \forall a\in G:a=\psi ^{-1}\left(\psi (a)\right)}$

applies and also:

${\displaystyle \forall a\in H:a=\psi \left(\psi ^{-1}(a)\right)}$

• If and are total orders and there is an isotonic bijection , then this is automatically also an isomorphism of order, or is also isotonic. ${\displaystyle (G,\leq _{G})}$ ${\displaystyle (H,\leq _{G})}$ ${\displaystyle \gamma \colon G\rightarrow H}$ ${\displaystyle \gamma ^{-1}}$
• It can be shown that every finite set is order isomorphic to the set of natural numbers up to the cardinality of the set. Formally:

${\displaystyle \left(M,\leq _{M}\right)\cong \left(\left\{1,\dots ,\left|M\right|\right\},\leq \right)}$.

literature

• Rudolf Berghammer: Orders, associations and relations with applications . Springer + Vieweg, 2nd edition 2012. ISBN 978-3658006181