It occured to me that when you perform division in some algebraic system, such as $\frac a b = c$ in $\mathbb R$, the division itself represents a relation of sorts between $a$ and $b$, and once you calculate this relation, the resulting element $c$, being 'merely' the relation or some kind of representation of it, has lost the information about what either $a$ or $b$ may have been.
So division destroys or weakens information. Other operations have similar peculiarities. Multiplication such as $a b = c$ is very 'lossy' in $\mathbb R$, but not as lossy in $\mathbb N$ since the set of possible divisors of $c$ is finite.
So my question is, are there any formalizations which account for (or may be able to account for) this particular aspect of mathematical operations (or functions/relations in general)?