0

I wonder whether are there known algebraic systems where there are non-infinity (not satisfying $x+a=x$) elements such that their power or product is infinity (an algebraic object that satisfies $x+a=x$, like in extended reals)?

I can think about $\overline{\mathbb R}^2$, which seemingly has divisors of infinity like $(1,\infty)$, but what about those with elements that squared give infinity?

Shaun
  • 44,997
Anixx
  • 9,119
  • Your question isn't very clear: (1) don't the extended reals already provide an example of what you are looking for? (2) what does $\overline{\Bbb{R}}^2$ mean? – Rob Arthan Jun 22 '22 at 22:08
  • @RobArthan oh, (1) is a typo, fixed. – Anixx Jun 22 '22 at 22:13
  • @RobArthan (2)-extended reals, direct product with itself. – Anixx Jun 22 '22 at 22:14
  • 2
    I am still not sure exactly what you are looking for (because I don't know what axioms you are expecting your algebras to satisfy). But how about $R = {0, 1, 2, 3}$ with addition defined by $ \min(x + y, 3)$ and multiplication defined by $\min(xy, 3)$? Then the additive annihilator $\infty = 3$ and $2 \neq \infty$, but $2^2 = \infty$. – Rob Arthan Jun 22 '22 at 22:26
  • 1
    @Anixx -- to be clear, you're looking for a system with $+$ and $\times$, where $a + \infty = \infty$ for all $a$, and where there's an element which squares to $\infty$? – HallaSurvivor Jun 22 '22 at 22:40
  • @HallaSurvivor yes. That would be interesting. And I already have a conjectured set. – Anixx Jun 22 '22 at 22:43
  • @HallaSurvivor something like this: https://math.stackexchange.com/questions/4467187/anti-dual-numbers-and-what-are-their-properties but that system is not distributive (even with non-infinity-divisors). I wonder if it is inevitable. – Anixx Jun 23 '22 at 03:49

1 Answers1

1

Such structure $(S,+,\cdot)$ exists.

Take any non trivial ring $A$ with unit having an element $e\in A$ such that $e^2=0_A$.

Set $S=A$, $+=\cdot_A$, $\cdot=\cdot_A$, $\infty=0_A$.

Then for all $a\in A$, $a+\infty=a\cdot 0_A=0_A=\infty$, and $e\cdot e=e^2=0_A=\infty$. However, not all element is $\infty$. Indeed, $1_A\neq \infty$. Otherwise, $1_A=0_A$ and $A$ would be trivial.

As a concrete example, you may take $A=\mathbb{Z}/n^2\mathbb{Z}$, and $e=\bar{n}$ (there are other examples as well in the same spirit)

However, I don't understand why you are looking for such structures. They have extremely limited interest from a computational point of view. For example $(S,+)$ cannot be a group (because of the existence of $\infty$), and I am pretty sure that you cannot have distributivity properties, even if a proof does not come into my mind right away...

GreginGre
  • 15,028
  • Thanks for the answer. I still cannot understand some notation though. What is $\overline{n}$ for instance? And yes, I also thought that distributivity is not possible there. I have two structures of this kind in my mind (for the both I already have code in Mathematica defining multiplication). They are the first two examples via this link: https://math.stackexchange.com/questions/4462705/lemniscate-numbers-and-others-what-would-be-the-properties That's the motivation. – Anixx Jun 23 '22 at 12:51
  • Another motivation is to have simple structure that would describe divergent formal power series and integrals, I thought that divisors of infinity may play a role of these objects, just like dual unity plays a role of infinitesimal. – Anixx Jun 23 '22 at 12:55
  • well, if $m$ is an integer, $\bar{m}$ is it class modulo $n^2$. – GreginGre Jun 23 '22 at 14:28