# Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism

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)

• Can you specify a bit? What was the title of that text or who wrote that? Maybe another example or this example more detailed would help us.. – Berci Feb 28 '14 at 14:15
• Ok, i'll add more examples in the next hours. – MphLee Feb 28 '14 at 14:20
• Updated with a more concrete example @Berci I must remember that this is only an example but I'm interested in the concept behind it. By the way I don't think that knowing the autor can be usefull for this, what I know is that the book he wrote is the only one where I saw this contruction ...but that doesn't mean nothing since my knowledge of math books is very small. – MphLee Feb 28 '14 at 17:23
• Thanks for updating, it is getting to make much more sense now. Well, can you confirm that $f$ being 'one-to-one' between $U_n$ and $U_{n+1}$ wants to mean that it is a bijection $U_n\to U_{n+1}$ when restricted to $U_n$? If yes, then the word you used 'homomorphic' would rather be 'isomorphic', and in this case, this example seems not particularly interesting, since you just copy one arbitrary structure $\Bbb U_0$ for infinitely many times. (You could instead also take the set $\Bbb N\times{\Bbb U_0}$..) – Berci Mar 2 '14 at 11:46
• @MphLee: I was looking over my Q/A and I remembered this question. I'd like to offer a bounty for it, but I'd like one piece of clarification: would it be fair to say that you are looking for a general theory which describes chains $U_1\to U_2\to\cdots$ of surjective homomorphisms? – Eric Stucky Dec 10 '14 at 9:08