Algebraic structure of the extended real line $\overline{\Bbb R}$. The extended real line $\overline{\Bbb R}$ is defined to be the set $\overline{\Bbb R}=\Bbb R\cup\{\infty,-\infty\}$, where the adjoined symbols $\{\infty,-\infty\}$ represents the "points at infinity" in both positive and negative direction.
$\overline{\Bbb R}$ can be given a topology by declaring that apart from the usual open basis, we let $(a,\infty]$ and $[-\infty,a)$ be open for any $a\in\Bbb R$. This is the two-point compactification, making $\overline{\Bbb R}$ a compact topological space.
However, the algebraic structure of $\overline{\Bbb R}$ seems rather unique. We declare that for any $a\in \Bbb R$,
$$\begin{align}
\infty+a &= \infty \\
-\infty+a &= -\infty \\
\infty+\infty &= \infty \\
-\infty-\infty &= -\infty \\
\frac a{\infty} &= 0 =\frac a{-\infty}
\end{align}$$
 and for $\infty\ge a>0> b\ge -\infty$,
$$\begin{align}
a\cdot\infty &= \infty \\
a\cdot(-\infty) &= -\infty \\
b\cdot\infty &= -\infty \\
b\cdot(-\infty) &= \infty \\
0\cdot\infty &=0 = 0\cdot(-\infty).\\
\end{align}$$
All other combinations, like $\infty-\infty$ or $\frac{\infty}{\infty}$, are left undefined.
Yes, these all make sense but I just want to know if it fits into any bigger framework? This clearly is not in accordance with "basics" algebraic structures that we studied in our undergraduate years.
$-\infty$ is not the additive inverse of $\infty$, neither is $\frac a{-\infty}=a\cdot{-\infty}^{-1}$  since ${-\infty}$ does not have a multiplicative inverse.

Is there a general theory to this kind of algebraic structure? 

I am thinking about boolean algebra since $1$ in a boolean algebra also exhibits this kind of `absorbing' behaviour. Since I lack any deep knowledge in the field of algebra, I hope that someone here might be able to give an insight into this.
PS. I tagged "logic" since I think it looks similar to boolean algebra. Please tell me if this is somehow not appropriate.
 A: While algebraic structures with partially defined operations are studied, it is much nicer to stick to structures where the binary operation is possible between any pair of elements.
That said, one natural version of what you're describing is the semiring $[0, \infty]$ defined by the operations you've described on just the nonnegative extended real line.
Another natural version is the projective real line where you just adjoin a symbol $\infty$ which acts like a point that glues together the ends of the real line. (There is no $-\infty$ added here, and the ordering of the reals becomes less important as we have "bent it into a circle.")
In this second picture, the problem of dealing with $-\infty$ has disappeared. Secondly, the operations using $\infty$ have a nice explanation with linear-fractional transformations using real numbers. A general linear-fractional transformation looks like this: 
$x\mapsto\frac{ax+b}{cx+d}$ where $a,b,c,d$ are fixed real numbers such that $ad-bc\neq 0$.
If $c=0$, then the transformation maps $\mathbb R$ onto $\mathbb R$, and we can safely extend the map to send $\infty\mapsto \infty$ to get a map of the entire projective line onto itself.
When $c\neq 0$, this expression maps $\mathbb R\setminus\{\frac{-d}c\}$ onto $\mathbb R\setminus\{\frac{a}c\}$. To complete the map we would have to figure out where to send $\infty$ and $\frac{-d}{c}$. The nice choice is to send $\frac{-d}{c}\to \infty$ and $\infty\to \frac{a}{c}$.
With these conventions you can work out the consistent definitions of operations including $\infty$ so that most operations take place on the entire projective real line.
