Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ and three distinct constants $0,1,\infty\in L$ satisfying the following axioms:
$\forall x,z\in L\;\forall y\in L\setminus\{\infty\}\;\;\big(x+(y+z)=(x+y)+z\big)$;
$\forall x,y\in L\;\;(x+y=y+x)$;
$\forall x\in L\;(x+0=x=0+x)$;
$\forall x\in L\;\exists y\in L\;(x+y=0=y+x)$;
$\forall x,z\in L\;\forall y\in L\setminus\{0,\infty\}\;\;(x\cdot(y\cdot z)=(x\cdot y)\cdot z)$;
$\forall x,y\in L \;\;(x\cdot y=y\cdot x)$;
$\forall x\in L\; (x\cdot 1=x=1\cdot x)$;
$\forall x\in L\;\exists y\in L\;(x\cdot y=1=y\cdot x)$;
$\forall x\in L\setminus\{0,\infty\}\;\forall y,z\in L\;\;\big(x\cdot(y+z)=x\cdot y+x\cdot z\big)$.
$0\cdot 0=0$ and $\infty\cdot\infty=\infty$.
Added in Edit: One of my collegues told me that there exists a related algebraic structure, called a wheel, whose purpose is to express the algebraic structure of fields extended by infinity.
There was a discussion in comments why I force the equalities $0\cdot\infty=1$ and $\infty+\infty=0$ and do not leave that product and sum undefined. For $0\cdot\infty=1$ the reason is simple: when we calculate the cross ratio of the quadruple $[a,b,c,d]$, we use the formula $$\frac{(c-a)\cdot(d-b)}{(d-a)\cdot(c-b)}.$$ If one of the numbers $a,b,c,d$ is infinity, then we use exactly those laws: $x+\infty=\infty$ for $x\ne\infty$, and $\frac{\infty}{\infty}=1$. The latter equality is equivalent to $0\cdot\infty=\frac1{\infty}\cdot\infty=\frac{\infty}{\infty}=1$. So, in order to make such calculations legal, in is better to define the addition and multiplication everywhere but do not require the associativity and distributivity in degenerated cases because they crack at those degenerate cases anyway.
Now I have a proof (a bit long though) that for the algebraic structure $L$ satisfying the above ten axioms, we have the equalities: $$0\cdot\infty=1,\quad \infty+\infty=0,\quad\mbox{and}\quad x\cdot\infty=\infty$$ for any non-zero $x\in L$. Moreover, if $L$ contains more than four elements, then we also have $x+\infty=\infty$ for all $x\in L\setminus\{\infty\}$ and $L\setminus\{\infty\}$ is subfield of $L$.
For the cardinalities $n\in\{3,4\}$ there exist two counterexamples: just take the ring $\mathbb Z_n$ and proclaim the element $2$ to be the infinity. Then modify the multiplication in order to guarantee that $0\cdot \infty=1$ and $x\cdot \infty=\infty$ for every non-zero $x$. Then the obtained algebraic structure satisfies my ten axioms but $\mathbb Z_n\setminus\{\infty\}$ is not a subfield of $\mathbb Z_n$. But those seem to be the only counterexamples.
