I was thinking about algebraic structures and what happens when we apply functions on them. (I dont know if I'm using good terms here but I haven't studied algebra yet)
I was particulary interested in loss of some information when we apply some functions.
Here is my example. we have Group denominated as $\mathbb{G}$ . and binary function $f: G \times G \rightarrow G $
also let have function $\xi : \mathbb{A} \rightarrow \mathbb{N}$ where $\mathbb{A}$ is any algebraic structure and $\mathbb{N}$ is 'amount' of different states in algebraic structurecan be (or simply $|\mathbb{A}|$)
So for example : $ \xi(\mathbb{N}) = \infty$ but also $\xi(\mathbb{N} \times \mathbb{N}) = 2\xi(\mathbb{N})$
Now lets define function $\Xi: \mathbb{F} \rightarrow \mathbb{Z}$ where first argument is function with 'information input' of $\xi(in)$ and 'informational output' $\xi(out)$ and with output $ = \frac{\xi(in)}{\xi(out)}$
So for example a binary function $ \mathbb{F}$ has always $\Xi(\mathbb{F}) = \frac{1}{2}$
Which field of algebra is looking at this informational loss etc??