So I am trying to learn for an exam, and I found an exercise but without solutions and I can't really get behind the topic:
Let $G = (V,E)$ be a connected graph with $v \geq 2$ Vertices. $P(G)$ is the name of all paths in $G$.
Determine a composition $+$ on $P(G)$ so that $\langle P(G),+ \rangle$ is an algebra and that a surjective homomorphism exists between the algebra and $\langle \sum^*,\circ \rangle$ on all $\sum$ with $v$ letters.
What is meant with Alphabet? Is that some theoretical computer science problem? It's quite interesting though, so I would like to know how the proof works.
So I found a "definition" (not really) of the term Alphabet: Let $\sum$ be an finite alphabet and $\sum^*$ the set of all finite words on $\sum$.
You can think of an alphabet $\Sigma$ as simply a set of "letters". Any set can be used as an alphabet. A word in the alphabet $\Sigma$ is an ordered list of symbols $w_1w_2\ldots w_n$ where each of the $w_i \in \Sigma$.
Given any two words $u$ and $v$ made from the same alphabet $\Sigma$, you can form their concatenation $u+v$, which is just the word made of the letters from $u$ followed by the letters from $v$.
$\Sigma^*$ is the set of words of any length which you can make from the alphabet $\Sigma$ (including as a special case the "empty word" $\epsilon$ which has no letters in it.) Algebraically, it's something like the free monoid on the set $\Sigma$, with concatenation as the monoid operator.
As a concrete example, you can have an alphabet $\Sigma = \{a, b\}$. Then the set $\Sigma^*$ of all possible words from $\Sigma$ contains elements such as a, b, ab, abb, abbbbabab, and so on.
I am not sure what $n$ stands for in your question. Perhaps instead of $n$, it should be $v\equiv|V|$.