Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Examples:: [[dictionary order relation]] ---- > [!definition] Definition. ([[order type]]) > Suppose that $A$ and $B$ are sets with order relations lt;_{A}$ and lt;_B$ respectively. We say that $A$ and $B$ have the same **order type** if there is a [[bijection|bijective correspondence]] between them that respects order; that is, if there exists a [[bijection]] $f:A \to B$ such that $a_{1} <_{A} a_{2} \implies f(a_{1})<_{B} f(a_{2}).$ > [!basicexample] > The [[interval]] $(-1,1) \subset \rr$ has the same **order type** as $\rr$ itself, for the function $f:(-1,1)\to \rr$ given by $f(x):=\frac{x}{1-x^{2}}$ is an order-respecting [[bijection]]— taking $x,y \in (-1,1)$ with $x<y$ we may verify $\frac{x}{1-x^{2}} < \frac{y}{1-y^{2}}$ easily by checking that $f$ is strictly [[increasing]] on $(-1,1)$. > [!note] Problem. > *Both $\{ 1,2 \} \times \mathbb{N}$ and $\mathbb{N} \times \{ 1,2 \}$ are [[well-ordered set]]s in the [[dictionary order relation|dictionary order]]. Do they have the same [[order type]]?* > **Solution**. Yes— we claim the 'transpose' $f:\{ 1,2 \} \times \nn \to \nn \times \{ 1,2 \}$ given by $f\big( (x,y) \big):=(y,x)$ is the order-respecting [[bijection]] desired. Denote by lt;_{1}$ the [[dictionary order relation]] on $\dom f$ (i.e., $(a_{1},b_{1})<_1(a_{2},b_{2})$ when $a_{1}<a_{2}$ or $a_{1}=a_{2}$ with $b_{1}<b_{2}$) and by lt;_{2}$ the [[dictionary order relation]] on $\im f$ (i.e., $(b_{1},a_{1}) <_{2} (b_{2},a_{2})$ when $a_{1}<a_{2}$ or $a_{1}=a_{2}$ with $b_{1}<b_{2}$) and let $(a_{1},b_{1}), (a_{2},b_{2}) \in \{ 1,2 \} \times \nn$ with $(a_{1},b_{1})<_{1}(a_{2},b_{2})$; then $(b_{1},a_{1})<_{2}(b_{2},a_{2})$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```