Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism
An Example similar to the construction I found was this:
Lets take define $\mathcal U $ as the union of infinite pairwise disjoint sets $U_i$
$$\mathcal U=\bigcup_{i \in \mathbb N}U_i$$
and we take a one-to-one function $f:\mathcal U \rightarrow \mathcal U$ with this property:
for every element of $x\in\mathcal U$ if $x\in U_k $ then his image $f(x)\in U_{k+1}$ and thus we have
$$f[U_k]=U_{k+1}$$
Now if we build an algebraic structure $\mathbb U_0$ on $U_0$ lets say adding an unit element and defining on it a binary operation and lets say an order relation $\mathbb U_0=(U_0,k_0,*_0,<_0)$ in this way we are able to define another algebraic structure $\mathbb U_1$ on $U_1=f[U_0]$ that is homomorphic to $\mathbb U_0$.
I can perform this construction infinite times defining $\mathbb U_{n+1}$ from $\mathbb U_{n}$ in the following way
*i)*$\mathbb U_0:=(U_0,k_0,*_0.<_0)$
ii) $\mathbb U_{n+1}=(f[U_n],f(k_n),*_{n+1} ,<_{n+1})$
iii) For every $a,b \in U_n$ then $f(a*_nb)=f(a)*_{n+1}f(b)$
iv) $f(a)<_{n+1}f(b)$ only if $a<_nb$
What I obtain (I guess) is an infinite collection of algebraic structures $\{\mathbb U_i\}_{i \in \mathbb N}$ all pairwise homomorphic in this way
$\mathbb U_n$ is homomorphic to $\mathbb U_{n+1}$ via $f$
$\mathbb U_t$ is homomorphic to $\mathbb U_{s}$ via $f^{\circ s-t}$
This is most important property that holds for all this kind of constructions.
I used a collection of structures with binary operations only as example but maybe these constructions can be made on more general objects
Q1-There is a branch of mathematics, maybe a theory, that use often similar costructions? Are these interesting and usefull objects somewhere in the mathematical landscape or are useless?
Q2- If yes maybe these constructions are already well studied (maybe in Category Theory), can you give me some references?
EDIT
Someone voted for the closure of this question because is too broad.
If it is then I didn't know it. Thats why I ask this question. If this subject is too common then there should be a good formalization of these constructions and I am in fact asking for good references too, since I did not found nothing on wikipedia. (Probably I don't know the terminology and that explains the Tags)