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.