So I come across this "$;$" in almost every definition. Now I've seen that I actually don't really understand what it means.
Below is a definition of an optimization problem, which is defined by $\prod = (D,f,opt) $, this is the last subdefinition. Does the symbol "$;$" mean that the second part of the definition is the same as the first part but in other notation, or what exactly does it mean?
$$Opt(\prod) = \{x^* \in D(\prod); f(x^{*})=v^{*}(\prod) \}$$
If you need to know the background of this is: $v^{*}( \prod ) = opt\{ f(x), x \in D \} $ where this is an optimal solution to the problem and $D( \prod ) = D$ is a set of allowed solutions to the problem. This are the two subdefinitions and the third is the one written first, $Opt(\prod) = ...$
The question is also general for every math problem about this notation.
Without further context, I would interpret this as "set builder notation". Usually I would denote this with a bar $\mid$, but I have seen sets with $;$. That is, for the set of natural numbers greater than 5 we might write $$ \{x \in \mathbb N \mid x > 5\}, $$ but we might also write $$ \{ x \in \mathbb N ; x > 5\}. $$ Similarly, in your case, I would interpret the set you describe as the set of those $x^*$ which are elements of $D(\Pi)$ such that the condition $f(x^*) = v(\Pi)$ holds.