User:Vahedi.vahid/sandbox

We recall construction of a Krasner hyperfield, as a quotient structure of a classical field by a subgroup. Let $(F,+,\cdot)$ be a field and $G$ be a subgroup of $(F^{*}, \cdot),$ where $F^{*}=F\setminus\{0\}.$ Take $\frac{F}{G}=\{aG\mid a\in F\}$ with the hyperoperation and the multiplication defined by: i) $aG\oplus bG=\{cG| c\in aG+bG \},$ ii) $aG\odot bG=abG,$ for all $aG,bG\in \frac{F}{G}$. Then $(\frac{F}{G}, \oplus, \odot)$ is a hyperfield. From now on, we denote $\bar{a}=aG,$ for all $aG\in \frac{F}{G}$ and the constructed hyperfield $(\frac{F}{G}, \oplus, \odot)$ by $\bar{F},$ and call it the Krasner hyperfield. Moreover, we denote the inverse of $\bar{a}$ relative to $\oplus$ by $\circleddash\bar{a}$ and, for $\bar{a} \neq \bar{0},$ the multiplicative inverse $\bar{a}^{-1}$ by $\dfrac{1}{\bar{a}}$. Besides, we will use the notation $\bar{S}=\{\bar s | s\in S\}$ and $\bar{T}=\{\bar t | t\in T\}$ for all $S\subseteq F,T\subseteq F^2$. Let $ \bar F $ be the Krasner hyperfield associated with the field $F$ and $ (\bar A,\bar B)\in \bar F^{2}$. Define the generalized homography transformation on $F$ as $\bar 1\in (\bar x\circleddash\bar A)\odot(\bar y\circleddash\bar B)$ on $\bar{F}$, and call it the hyperhomography relation. We call the set $H_{\bar A,\bar B} (\bar F)=\{(\bar x, \bar y)\in \bar F^{2}\mid \bar 1\in (\bar x\circleddash\bar A)\odot(\bar y\circleddash\bar B)\}$ hyperhomography, while $H_{a,b}(F)=\{(x, y)\in F^{2}\mid y=f_{a,b}(x)=b+\frac{1}{x-a} \}$ is a homography, for all $a\in \bar A$ and $b\in\bar B$. Notice that the hyperhomography $H_{\bar A,\bar B} (\bar F)$ is a generalization of a homography $H_{a,b} (F)$, because if so $G=\{1\}$ then $(x,y)\in H_{A,B} (F)$ is equivalent with $y=f_{A,B}(x)=B+\frac{1}{x-A}$, i.e., $(x-A)(y-B)=1$ and we have $H_{\bar A,\bar B} (\bar F)=H_{A,B} (F)$. Besides, by applying special condition on hyperhomographies these can equipped to hypergroup structure. That if $G=\{1\}$ this hypergroup structure on hyperhomographies to group structure on homographies is reduced. See for more information.