Information-theoretic aspects of mathematical systems? 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)?

Your particular example of information being lost in multiplication or division is really the reason one studies ideals in ring theory, normal subgroups in group theory, and kernels in general. But I think prime ideals explain it best.

The prime factorization of a number represents how much information was lost in creating it. A prime number only has one factorization, so if it is a product, we know the factors. The more prime factors an element has, the more information is lost in creating it.
Mathematicians generalized this to 'prime ideals' where a class of products (called an ideal) has information loss measured by the number of of prime ideals in its decomposition.
Fields have the greatest information loss, as every element can occur from a product with any given factors.
Ideals can also be used in another way to destroy information. Quotienting by an ideal causes you to lose information in proportion to the size of the ideal. Quotienting the integers by the ideal of multiples of 10 tells you the last digit of a number, while quotienting hy multiples of 2 only tells you if a number is odd or even.

