What is the relation between these two functions? Are they isomorphic?

Question

Suppose I have an infinite sequence $a_{i}\in A$, and two functions, $\Theta:\mathbb{N}\rightarrow \mathbb{N}$, and $\vartheta: a_{i}\mapsto a_{j}$, so that $\forall (i\geq 1):\{ \vartheta(a_{i})=a_{(\Theta(i))} \}$. Clearly, $\vartheta$ is operating on the indexes of a sequence (i.e., $\mathbb{N}$) in exactly the same way that $\Theta$ is operating on $\mathbb{N}$. Do we say that $\vartheta$ and $\Theta$ are "isomorphic"? "homomorphic"? What term(s) do we use to describe the relationship between $\vartheta$ and $\Theta$?

Answer 1

A sequence in $A$ is a map $a: \mathbb N \to A$ where $a(i)=a_i$.

The relationship between $\vartheta$ and $\Theta$ is just $\vartheta =a \circ \Theta$.

If $\Theta$ is increasing, then $a \circ \Theta$ is a subsequence of $a$.

The terms "isomorphic" and "homomorphic" are not usually applied in this context.

Answer 2

$\Theta :\mathbb{N}\rightarrow \mathbb{N}$ takes a natural number and returns a natural number.

$\vartheta :A\rightarrow A$ is defined in terms of $\Theta$, taking $a_{i}$ and returning $a_{\Theta(i)}$. If $\Theta$ is a bijection*, then $\vartheta$ becomes a permutation of the sequence $\{a_{i}\}_{i\in \mathbb{N}}$ moving elements around in the sequence.

*$\Theta$ needs to be a bijection because a sequence is a function $a_{n}:\mathbb{N} \rightarrow A$, and if $ \Theta$ isn’t, then $\{a_{\Theta(i)}\}_{i \in \mathbb{N}} = \{\vartheta(a_{i})\}_{i \in \mathbb{N}}$ is no longer a sequence.