# Is there a good name for this type of relation?

I am defining the relation $\sim$ to be

$$f \sim g \Longleftrightarrow \lim_{n \to \infty} \frac f g = 1$$

In my paper, I am asserting that if $f \sim f_0$, then

$$f + g \sim f_0 + g\\ fg \sim f_0{g}\\ \frac f g \sim \frac {f_0} g$$

Basically, it works like an algebraic equation: you can add, multiply, or divide both sides by the same quantity. Is there a name for this type of relation?

I'm asking this because I'm trying to write a terse description of this in a header, but the best that comes to mind is "Both Sides of a $\sim$ Relation may be Added, Multiplied, or Divided by the Same Quantity". I need to write something like "$\sim$ is an ___ Relation".

edit: I decided to put "$\sim$ Relations can be Algebraically Manipulated" for the header. Does that sound appropriate?

• What you're looking for might be congruence. – Adayah Nov 12 '17 at 18:45
• @Adayah For those not familiar with abstract algebra, it's not immediately clear what it is when you read the description... I decided to put "$\sim$ Relations can be Algebraically Manipulated" for the header. Does that sound appropriate? – James Ko Nov 12 '17 at 21:09
• The established name for this is "asymptotic equivalence". See here for example. – EuYu Nov 12 '17 at 21:43
• Also, I just want to mention that the property $f_0 + g \sim f + g$ is not true without further assumptions. For example, $f(x) = x+\log x$ and $f_0(x)=x$ are certainly asymptotically equivalent. Now take $g(x) = -x$. Then $f + g$ and $f_0 + g$ are certainly not asymptotically equivalent. – EuYu Nov 12 '17 at 21:51
• @EuYu That is true. I left out that I constrained the functions to be positive. You misread my question though, I was not asking was $\sim$ was called, I was asking what to call a particular property of $\sim$. – James Ko Nov 12 '17 at 22:11